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{k : Type.{u}} [inst._@.Mathlib.Algebra.Category.ModuleCat.Algebra.1160953746._hygCtx._hyg.3 : Field.{u} k] {A : Type.{w}} [inst._@.Mathlib.Algebra.Category.ModuleCat.Algebra.1160953746._hygCtx._hyg.7 : Ring.{w} A] [inst._@.Mathlib.Algebra.Category.ModuleCat.Algebra.1160953746._hygCtx._hyg.10 : Algebra.{u, w} k A (Semifield.toCommSemiring.{u} k (Field.toSemifield.{u} k inst._@.Mathlib.Algebra.Category.ModuleCat.Algebra.1160953746._hygCtx._hyg.3)) (Ring.toSemiring.{w} A inst._@.Mathlib.Algebra.Category.ModuleCat.Algebra.1160953746._hygCtx._hyg.7)], CategoryTheory.Linear.{u, v, max (succ v) w} k (DivisionSemiring.toSemiring.{u} k (Semifield.toDivisionSemiring.{u} k (Field.toSemifield.{u} k inst._@.Mathlib.Algebra.Category.ModuleCat.Algebra.1160953746._hygCtx._hyg.3))) (ModuleCat.{v, w} A inst._@.Mathlib.Algebra.Category.ModuleCat.Algebra.1160953746._hygCtx._hyg.7) (ModuleCat.moduleCategory.{v, w} A inst._@.Mathlib.Algebra.Category.ModuleCat.Algebra.1160953746._hygCtx._hyg.7) 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A)","typeReferences":[["Semifield","toCommSemiring"],["CategoryTheory","Linear"],["ModuleCat","instPreadditive"],["Field"],["Field","toSemifield"],["ModuleCat","moduleCategory"],["Semifield","toDivisionSemiring"],["DivisionSemiring","toSemiring"],["ModuleCat"],["Ring","toSemiring"],["Algebra"],["Ring"]],"valueReferences":[["ModuleCat","moduleOfAlgebraModule"],["ModuleCat","instPreadditive"],["Module"],["ModuleCat","linearOverField","_proof_3"],["ModuleCat","moduleCategory"],["DivisionSemiring","toSemiring"],["CategoryTheory","Category","toCategoryStruct"],["ModuleCat","linearOverField","_proof_2"],["ModuleCat","Hom","instModule"],["CategoryTheory","CategoryStruct","toQuiver"],["Quiver","Hom"],["CategoryTheory","Linear","mk"],["Field","toSemifield"],["AddCommGroup","toAddCommMonoid"],["inferInstance"],["CategoryTheory","Preadditive","homGroup"],["Semifield","toDivisionSemiring"],["ModuleCat","linearOverField","_proof_1"],["ModuleCat"]]},{"isProp":true,"kind":"theorem","name":["ModuleCat","linearOverField","_proof_3"],"typeFallback":"forall {k : Type.{u_3}} [inst._@.Mathlib.Algebra.Category.ModuleCat.Algebra.1160953746._hygCtx._hyg.3 : Field.{u_3} k] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Category.ModuleCat.Algebra.1160953746._hygCtx._hyg.7 : Ring.{u_2} A] [inst._@.Mathlib.Algebra.Category.ModuleCat.Algebra.1160953746._hygCtx._hyg.10 : Algebra.{u_3, u_2} k A (Semifield.toCommSemiring.{u_3} k (Field.toSemifield.{u_3} k inst._@.Mathlib.Algebra.Category.ModuleCat.Algebra.1160953746._hygCtx._hyg.3)) (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Category.ModuleCat.Algebra.1160953746._hygCtx._hyg.7)] (X : ModuleCat.{u_1, u_2} A inst._@.Mathlib.Algebra.Category.ModuleCat.Algebra.1160953746._hygCtx._hyg.7) (Y : ModuleCat.{u_1, u_2} A inst._@.Mathlib.Algebra.Category.ModuleCat.Algebra.1160953746._hygCtx._hyg.7) (Z : ModuleCat.{u_1, u_2} A inst._@.Mathlib.Algebra.Category.ModuleCat.Algebra.1160953746._hygCtx._hyg.7) (f : Quiver.Hom.{u_1, max (succ u_1) u_2} (ModuleCat.{u_1, u_2} A 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inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.9)))) n) a))) (gc_ceil : GaloisConnection.{u_4, 0} α Nat (PartialOrder.toPreorder.{u_4} α inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.12) Nat.instPreorder ceil (Nat.cast.{u_4} α (AddMonoidWithOne.toNatCast.{u_4} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u_4} α (NonAssocSemiring.toAddCommMonoidWithOne.{u_4} α (Semiring.toNonAssocSemiring.{u_4} α inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.9)))))), motive (FloorSemiring.mk.{u_4} α inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.12 floor ceil floor_of_neg gc_floor gc_ceil)) -> (motive t)","typeFull":"{α : Type u_4} →\n  [inst : Semiring α] →\n    [inst_1 : PartialOrder α] →\n      {motive : FloorSemiring α → Sort u} →\n        (t : FloorSemiring α) →\n          ((floor ceil : α → ℕ) →\n              (floor_of_neg : ∀ {a : α}, a < 0 → floor a = 0) →\n                (gc_floor : ∀ {a : α} {n : ℕ}, 0 ≤ a → (n ≤ floor a ↔ ↑n ≤ a)) →\n                  (gc_ceil : GaloisConnection ceil Nat.cast) →\n                    motive\n                      { floor := floor, ceil := ceil, floor_of_neg := floor_of_neg, gc_floor := gc_floor,\n                        gc_ceil := gc_ceil }) →\n            motive t","typeReadable":"{α : Type u_4} →\n  [inst : Semiring α] →\n    [inst_1 : PartialOrder α] →\n      {motive : FloorSemiring α → Sort u} →\n        (t : FloorSemiring α) →\n          ((floor ceil : α → ℕ) →\n              (floor_of_neg : ∀ {a : α}, a < 0 → floor a = 0) →\n                (gc_floor : ∀ {a : α} {n : ℕ}, 0 ≤ a → (n ≤ floor a ↔ ↑n ≤ a)) →\n                  (gc_ceil : GaloisConnection ceil Nat.cast) →\n                    motive\n                      { floor := floor, ceil := ceil, floor_of_neg := floor_of_neg, gc_floor := gc_floor,\n                        gc_ceil := gc_ceil }) →\n            motive 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{inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.9 : Semiring.{u_4} α} {inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.12 : PartialOrder.{u_4} α} {P : Sort.{u}} {floor : α -> Nat} {ceil : α -> Nat} {floor_of_neg : forall {a : α}, (LT.lt.{u_4} α (Preorder.toLT.{u_4} α (PartialOrder.toPreorder.{u_4} α inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.12)) a (OfNat.ofNat.{u_4} α 0 (Zero.toOfNat0.{u_4} α (MulZeroClass.toZero.{u_4} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_4} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} α (Semiring.toNonAssocSemiring.{u_4} α inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.9))))))) -> (Eq.{1} Nat (floor a) (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))} {gc_floor : forall {a : α} {n : Nat}, (LE.le.{u_4} α (Preorder.toLE.{u_4} α (PartialOrder.toPreorder.{u_4} α inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.12)) (OfNat.ofNat.{u_4} α 0 (Zero.toOfNat0.{u_4} α (MulZeroClass.toZero.{u_4} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_4} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} α (Semiring.toNonAssocSemiring.{u_4} α inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.9)))))) a) -> (Iff (LE.le.{0} Nat instLENat n (floor a)) (LE.le.{u_4} α (Preorder.toLE.{u_4} α (PartialOrder.toPreorder.{u_4} α inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.12)) (Nat.cast.{u_4} α (AddMonoidWithOne.toNatCast.{u_4} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u_4} α (NonAssocSemiring.toAddCommMonoidWithOne.{u_4} α (Semiring.toNonAssocSemiring.{u_4} α inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.9)))) n) a))} {gc_ceil : GaloisConnection.{u_4, 0} α Nat (PartialOrder.toPreorder.{u_4} α inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.12) Nat.instPreorder ceil (Nat.cast.{u_4} α (AddMonoidWithOne.toNatCast.{u_4} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u_4} α (NonAssocSemiring.toAddCommMonoidWithOne.{u_4} α (Semiring.toNonAssocSemiring.{u_4} α inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.9)))))} {floor' : α -> Nat} {ceil' : α -> Nat} {floor_of_neg' : forall {a : α}, (LT.lt.{u_4} α (Preorder.toLT.{u_4} α (PartialOrder.toPreorder.{u_4} α inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.12)) a (OfNat.ofNat.{u_4} α 0 (Zero.toOfNat0.{u_4} α (MulZeroClass.toZero.{u_4} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_4} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} α (Semiring.toNonAssocSemiring.{u_4} α inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.9))))))) -> (Eq.{1} Nat (floor' a) (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))} {gc_floor' : forall {a : α} {n : Nat}, (LE.le.{u_4} α (Preorder.toLE.{u_4} α (PartialOrder.toPreorder.{u_4} α inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.12)) (OfNat.ofNat.{u_4} α 0 (Zero.toOfNat0.{u_4} α (MulZeroClass.toZero.{u_4} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_4} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} α (Semiring.toNonAssocSemiring.{u_4} α inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.9)))))) a) -> (Iff (LE.le.{0} Nat instLENat n (floor' a)) (LE.le.{u_4} α (Preorder.toLE.{u_4} α (PartialOrder.toPreorder.{u_4} α inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.12)) (Nat.cast.{u_4} α (AddMonoidWithOne.toNatCast.{u_4} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u_4} α (NonAssocSemiring.toAddCommMonoidWithOne.{u_4} α (Semiring.toNonAssocSemiring.{u_4} α inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.9)))) n) a))} {gc_ceil' : GaloisConnection.{u_4, 0} α Nat (PartialOrder.toPreorder.{u_4} α inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.12) Nat.instPreorder ceil' (Nat.cast.{u_4} α (AddMonoidWithOne.toNatCast.{u_4} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u_4} α (NonAssocSemiring.toAddCommMonoidWithOne.{u_4} α (Semiring.toNonAssocSemiring.{u_4} α inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.9)))))}, (Eq.{succ u_4} (FloorSemiring.{u_4} α inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.12) (FloorSemiring.mk.{u_4} α inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.12 floor ceil floor_of_neg gc_floor gc_ceil) (FloorSemiring.mk.{u_4} α inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Order.Floor.Defs.2810192196._hygCtx._hyg.12 floor' ceil' floor_of_neg' gc_floor' gc_ceil')) -> ((HEq.{succ u_4} (α -> Nat) floor (α -> Nat) floor') -> (HEq.{succ u_4} (α -> Nat) ceil (α -> Nat) ceil') -> P) -> P","typeFull":"{α : Type u_4} →\n  {inst : Semiring α} →\n    {inst_1 : PartialOrder α} →\n      {P : Sort u} →\n        {floor ceil : α → ℕ} →\n          {floor_of_neg : ∀ {a : α}, a < 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{α : Type u_4} →\n    [inst : Semiring α] →\n      [inst_1 : PartialOrder α] →\n        FloorSemiring α →\n          {α' : Type u_4} → [inst' : Semiring α'] → [inst'_1 : PartialOrder α'] → FloorSemiring α' → Sort u","typeReferences":[["FloorSemiring"],["PartialOrder"],["Semiring"]],"valueReferences":[["Nat"],["HEq"],["FloorSemiring","casesOn"]]},{"isProp":true,"kind":"theorem","name":["exists_floor'"],"typeFallback":"forall {α : Type.{u_4}} [inst._@.Mathlib.Algebra.Order.Floor.Defs.1385415486._hygCtx._hyg.6 : Ring.{u_4} α] [inst._@.Mathlib.Algebra.Order.Floor.Defs.1385415486._hygCtx._hyg.9 : PartialOrder.{u_4} α] [inst._@.Mathlib.Algebra.Order.Floor.Defs.1385415486._hygCtx._hyg.12 : IsStrictOrderedRing.{u_4} α (Ring.toSemiring.{u_4} α inst._@.Mathlib.Algebra.Order.Floor.Defs.1385415486._hygCtx._hyg.6) inst._@.Mathlib.Algebra.Order.Floor.Defs.1385415486._hygCtx._hyg.9] (x : α), (Exists.{1} Int (fun (n : Int) => LE.le.{u_4} α (Preorder.toLE.{u_4} α (PartialOrder.toPreorder.{u_4} α 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{E : Type.{u}} [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2142052623._hygCtx._hyg.3 : NormedAddCommGroup.{u} E] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2142052623._hygCtx._hyg.6 : NormedSpace.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2142052623._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2142052623._hygCtx._hyg.12 : CompleteSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2142052623._hygCtx._hyg.3)))] {R : Real} {c : Complex} {w : Complex} {f : Complex -> E} {s : Set.{0} Complex}, (Set.Countable.{0} Complex s) -> (Membership.mem.{0, 0} Complex (Set.{0} Complex) (Set.instMembership.{0} Complex) (SDiff.sdiff.{0} (Set.{0} Complex) (Set.instSDiff.{0} Complex) (Metric.ball.{0} Complex 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Zero"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["instHMul"],["ContinuousOn","inv₀"],["Metric","sphere_subset_closedBall"],["AddZero","toZero"],["Set","Countable","insert"],["sub_eq_zero"],["sub_ne_zero"],["UniformSpace","toTopologicalSpace"],["SubNegZeroMonoid","toNegZeroClass"],["ContinuousOn","circleIntegrable"],["congrArg"],["Complex","instDecidableEq"],["SeminormedCommRing","toNonUnitalSeminormedCommRing"],["NormedField","toField"],["circleIntegral","integral_congr"],["Dist","dist"],["MonoidWithZero","toMonoid"],["Complex","instNatCast"],["instHSMul"],["IsTopologicalDivisionRing","toContinuousInv₀"],["Zero","toOfNat0"],["continuousOn_dslope"],["LT","lt","le"],["Not"],["Inv","inv"],["Semiring","toMonoidWithZero"],["ne_of_lt"],["NormedCommRing","toSeminormedCommRing"],["NormedSpace","toIsBoundedSMul"],["InnerProductSpace","toNormedSpace"],["Complex","instMul"],["NormedAddCommGroup","toSeminormedAddCommGroup"],["Complex","instSub"],["NormedCommRing","toNonUnitalNormedCommRing"]]},{"isProp":true,"kind":"theorem","name":["DifferentiableOn","contDiffOn"],"typeFallback":"forall {E : Type.{u}} [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.4185742909._hygCtx._hyg.3 : NormedAddCommGroup.{u} E] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.4185742909._hygCtx._hyg.6 : NormedSpace.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.4185742909._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.4185742909._hygCtx._hyg.12 : CompleteSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.4185742909._hygCtx._hyg.3)))] {s : Set.{0} Complex} {f : Complex -> E} {n : WithTop.{0} ENat}, (DifferentiableOn.{0, 0, u} Complex (DenselyNormedField.toNontriviallyNormedField.{0} Complex Complex.instDenselyNormedField) Complex Complex.addCommGroup Complex.instModuleSelf (UniformSpace.toTopologicalSpace.{0} Complex 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{𝕜 : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.FDeriv.Basic.3347205837._hygCtx._hyg.3 : NontriviallyNormedField.{u_1} 𝕜] {E : Type.{u_2}} [inst._@.Mathlib.Analysis.Calculus.FDeriv.Basic.3347205837._hygCtx._hyg.7 : AddCommGroup.{u_2} E] [inst._@.Mathlib.Analysis.Calculus.FDeriv.Basic.3347205837._hygCtx._hyg.10 : Module.{u_1, u_2} 𝕜 E (DivisionSemiring.toSemiring.{u_1} 𝕜 (Semifield.toDivisionSemiring.{u_1} 𝕜 (Field.toSemifield.{u_1} 𝕜 (NormedField.toField.{u_1} 𝕜 (NontriviallyNormedField.toNormedField.{u_1} 𝕜 inst._@.Mathlib.Analysis.Calculus.FDeriv.Basic.3347205837._hygCtx._hyg.3))))) (AddCommGroup.toAddCommMonoid.{u_2} E inst._@.Mathlib.Analysis.Calculus.FDeriv.Basic.3347205837._hygCtx._hyg.7)] [inst._@.Mathlib.Analysis.Calculus.FDeriv.Basic.3347205837._hygCtx._hyg.14 : TopologicalSpace.{u_2} E] {F : Type.{u_3}} [inst._@.Mathlib.Analysis.Calculus.FDeriv.Basic.3347205837._hygCtx._hyg.18 : AddCommGroup.{u_3} F] [inst._@.Mathlib.Analysis.Calculus.FDeriv.Basic.3347205837._hygCtx._hyg.21 : Module.{u_1, u_3} 𝕜 F (DivisionSemiring.toSemiring.{u_1} 𝕜 (Semifield.toDivisionSemiring.{u_1} 𝕜 (Field.toSemifield.{u_1} 𝕜 (NormedField.toField.{u_1} 𝕜 (NontriviallyNormedField.toNormedField.{u_1} 𝕜 inst._@.Mathlib.Analysis.Calculus.FDeriv.Basic.3347205837._hygCtx._hyg.3))))) (AddCommGroup.toAddCommMonoid.{u_3} F inst._@.Mathlib.Analysis.Calculus.FDeriv.Basic.3347205837._hygCtx._hyg.18)] [inst._@.Mathlib.Analysis.Calculus.FDeriv.Basic.3347205837._hygCtx._hyg.25 : TopologicalSpace.{u_3} F] {f : E -> F}, Eq.{1} Prop (Differentiable.{u_1, u_2, u_3} 𝕜 inst._@.Mathlib.Analysis.Calculus.FDeriv.Basic.3347205837._hygCtx._hyg.3 E inst._@.Mathlib.Analysis.Calculus.FDeriv.Basic.3347205837._hygCtx._hyg.7 inst._@.Mathlib.Analysis.Calculus.FDeriv.Basic.3347205837._hygCtx._hyg.10 inst._@.Mathlib.Analysis.Calculus.FDeriv.Basic.3347205837._hygCtx._hyg.14 F inst._@.Mathlib.Analysis.Calculus.FDeriv.Basic.3347205837._hygCtx._hyg.18 inst._@.Mathlib.Analysis.Calculus.FDeriv.Basic.3347205837._hygCtx._hyg.21 inst._@.Mathlib.Analysis.Calculus.FDeriv.Basic.3347205837._hygCtx._hyg.25 f) (DifferentiableOn.{u_1, u_2, u_3} 𝕜 inst._@.Mathlib.Analysis.Calculus.FDeriv.Basic.3347205837._hygCtx._hyg.3 E inst._@.Mathlib.Analysis.Calculus.FDeriv.Basic.3347205837._hygCtx._hyg.7 inst._@.Mathlib.Analysis.Calculus.FDeriv.Basic.3347205837._hygCtx._hyg.10 inst._@.Mathlib.Analysis.Calculus.FDeriv.Basic.3347205837._hygCtx._hyg.14 F inst._@.Mathlib.Analysis.Calculus.FDeriv.Basic.3347205837._hygCtx._hyg.18 inst._@.Mathlib.Analysis.Calculus.FDeriv.Basic.3347205837._hygCtx._hyg.21 inst._@.Mathlib.Analysis.Calculus.FDeriv.Basic.3347205837._hygCtx._hyg.25 f (Set.univ.{u_2} E))","typeFull":"∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : AddCommGroup E] [inst_2 : Module 𝕜 E]\n  [inst_3 : TopologicalSpace E] {F : Type u_3} [inst_4 : 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[inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1704782924._hygCtx._hyg.12 : CompleteSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1704782924._hygCtx._hyg.3)))] {R : Real} {c : Complex} {w : Complex} {f : Complex -> E}, (DiffContOnCl.{0, 0, u} Complex Complex E (DenselyNormedField.toNontriviallyNormedField.{0} Complex Complex.instDenselyNormedField) Complex.instNormedAddCommGroup inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1704782924._hygCtx._hyg.3 (InnerProductSpace.toNormedSpace.{0, 0} Complex Complex Complex.instRCLike (NormedAddCommGroup.toSeminormedAddCommGroup.{0} Complex Complex.instNormedAddCommGroup) (RCLike.innerProductSpace.{0} Complex Complex.instRCLike)) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1704782924._hygCtx._hyg.6 f (Metric.ball.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))) c R)) -> (Membership.mem.{0, 0} Complex (Set.{0} Complex) (Set.instMembership.{0} Complex) (Metric.ball.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))) c R) w) -> (Eq.{succ u} E (HSMul.hSMul.{0, u, u} Complex E E (instHSMul.{0, u} Complex E (SMulZeroClass.toSMul.{0, u} Complex E (AddZero.toZero.{u} E (AddZeroClass.toAddZero.{u} E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1704782924._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1704782924._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1704782924._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1704782924._hygCtx._hyg.3))))))) (DistribSMul.toSMulZeroClass.{0, u} Complex E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E 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DiffContOnCl ℂ f (Metric.ball c R) →\n    w ∈ Metric.ball c R → (2 * ↑Real.pi * Complex.I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z = f w","typeReadable":"∀ {E : Type u} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] [CompleteSpace E] {R : ℝ} {c w : ℂ} {f : ℂ → E},\n  DiffContOnCl ℂ f (Metric.ball c R) →\n    w ∈ Metric.ball c R → (2 * ↑Real.pi * Complex.I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z = f 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{E : Type.{u}} [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1671878455._hygCtx._hyg.3 : NormedAddCommGroup.{u} E] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1671878455._hygCtx._hyg.6 : NormedSpace.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1671878455._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1671878455._hygCtx._hyg.12 : CompleteSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1671878455._hygCtx._hyg.3)))] {R : NNReal} {c : Complex} {f : Complex -> E} {s : Set.{0} Complex}, (Set.Countable.{0} Complex s) -> (ContinuousOn.{0, u} Complex E (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex 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{E : Type.{u}} [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3000083063._hygCtx._hyg.3 : NormedAddCommGroup.{u} E] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3000083063._hygCtx._hyg.6 : NormedSpace.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3000083063._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3000083063._hygCtx._hyg.12 : CompleteSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3000083063._hygCtx._hyg.3)))] {R : Real} {f : Complex -> E} {c : Complex}, (LT.lt.{0} Real Real.instLT (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)) R) -> (DifferentiableOn.{0, 0, u} Complex (DenselyNormedField.toNontriviallyNormedField.{0} Complex Complex.instDenselyNormedField) Complex Complex.addCommGroup 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(x : Complex) (y : Complex), Eq.{1} Complex (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.exp x) (Complex.exp y)) (Complex.exp (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) x y))","typeFull":"∀ (x y : ℂ), Complex.exp x * Complex.exp y = Complex.exp (x + y)","typeReadable":"∀ (x y : ℂ), Complex.exp x * Complex.exp y = Complex.exp (x + y)","typeReferences":[["HAdd","hAdd"],["Complex"],["Complex","instAdd"],["instHAdd"],["instHMul"],["HMul","hMul"],["Complex","exp"],["Eq"],["Complex","instMul"]],"valueReferences":[["HAdd","hAdd"],["Complex"],["Complex","exp_add"],["Complex","instAdd"],["instHAdd"],["Eq","symm"],["instHMul"],["HMul","hMul"],["Complex","exp"],["Complex","instMul"]]},{"isProp":true,"kind":"theorem","name":["Complex","differentiable_on_off_countable_deriv_eq_smul_circleIntegral"],"typeFallback":"forall {E : Type.{u}} [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1629092713._hygCtx._hyg.3 : NormedAddCommGroup.{u} E] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1629092713._hygCtx._hyg.6 : NormedSpace.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1629092713._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1629092713._hygCtx._hyg.12 : CompleteSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1629092713._hygCtx._hyg.3)))] {R : Real} {f : Complex -> E} {c : Complex} {s : Set.{0} Complex}, (LT.lt.{0} Real Real.instLT (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)) R) -> (Set.Countable.{0} Complex s) -> (ContinuousOn.{0, u} Complex E (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex 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inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2507199099._hygCtx._hyg.3))))) (intervalIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2507199099._hygCtx._hyg.3 (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2507199099._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2507199099._hygCtx._hyg.6) (fun (x : Real) => f (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) (Complex.ofReal x) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.ofReal (Complex.im z)) Complex.I))) (Complex.re z) (Complex.re w) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace)) (intervalIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2507199099._hygCtx._hyg.3 (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2507199099._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2507199099._hygCtx._hyg.6) (fun (x : Real) => f (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) (Complex.ofReal x) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.ofReal (Complex.im w)) Complex.I))) (Complex.re z) (Complex.re w) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace))) (HSMul.hSMul.{0, u, u} Complex E E (instHSMul.{0, u} Complex E (SMulZeroClass.toSMul.{0, u} Complex E (AddZero.toZero.{u} E (AddZeroClass.toAddZero.{u} E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2507199099._hygCtx._hyg.3)))) 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E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2507199099._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2507199099._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2507199099._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2507199099._hygCtx._hyg.3))))) (DistribMulAction.toDistribSMul.{0, u} Complex E (MonoidWithZero.toMonoid.{0} Complex 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(NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2507199099._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2507199099._hygCtx._hyg.6)))))) Complex.I (intervalIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2507199099._hygCtx._hyg.3 (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2507199099._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2507199099._hygCtx._hyg.6) (fun (y : Real) => f (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) (Complex.ofReal (Complex.re w)) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.ofReal y) Complex.I))) (Complex.im z) (Complex.im w) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace)))) (HSMul.hSMul.{0, u, u} Complex E E (instHSMul.{0, u} Complex E (SMulZeroClass.toSMul.{0, u} Complex E 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{r₁ : NNReal} {r₂ : NNReal}, Eq.{1} Prop (LT.lt.{0} Real Real.instLT (NNReal.toReal r₁) (NNReal.toReal r₂)) (LT.lt.{0} NNReal (Preorder.toLT.{0} NNReal (PartialOrder.toPreorder.{0} NNReal NNReal.instPartialOrder)) r₁ r₂)","typeFull":"∀ {r₁ r₂ : NNReal}, (↑r₁ < ↑r₂) = (r₁ < r₂)","typeReadable":"∀ {r₁ r₂ : NNReal}, (↑r₁ < ↑r₂) = (r₁ < r₂)","typeReferences":[["LT","lt"],["PartialOrder","toPreorder"],["Real"],["Real","instLT"],["NNReal"],["Preorder","toLT"],["NNReal","toReal"],["NNReal","instPartialOrder"],["Eq"]],"valueReferences":[["LT","lt"],["PartialOrder","toPreorder"],["Real"],["Real","instLT"],["NNReal"],["Preorder","toLT"],["NNReal","toReal"],["NNReal","instPartialOrder"],["propext"],["NNReal","coe_lt_coe"]]},{"isProp":true,"kind":"theorem","name":["Complex","integral_boundary_rect_eq_zero_of_continuousOn_of_differentiableOn"],"typeFallback":"forall {E : Type.{u}} [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3 : NormedAddCommGroup.{u} E] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.6 : NormedSpace.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3)] (f : Complex -> E) (z : Complex) (w : Complex), (ContinuousOn.{0, u} Complex E (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3)))) f (Complex.reProdIm (Set.uIcc.{0} Real Real.lattice (Complex.re z) (Complex.re w)) (Set.uIcc.{0} Real Real.lattice (Complex.im z) (Complex.im w)))) -> (DifferentiableOn.{0, 0, u} Complex (DenselyNormedField.toNontriviallyNormedField.{0} Complex Complex.instDenselyNormedField) Complex Complex.addCommGroup Complex.instModuleSelf (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) E (NormedAddCommGroup.toAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3) (NormedSpace.toModule.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.6) (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3)))) f (Complex.reProdIm (Set.Ioo.{0} Real Real.instPreorder (Min.min.{0} Real Real.instMin (Complex.re z) (Complex.re w)) (Max.max.{0} Real Real.instMax (Complex.re z) (Complex.re w))) (Set.Ioo.{0} Real Real.instPreorder (Min.min.{0} Real Real.instMin (Complex.im z) (Complex.im w)) (Max.max.{0} Real Real.instMax (Complex.im z) (Complex.im w))))) -> (Eq.{succ u} E (HSub.hSub.{u, u, u} E E E (instHSub.{u} E (SubNegMonoid.toSub.{u} E (AddGroup.toSubNegMonoid.{u} E (NormedAddGroup.toAddGroup.{u} E (NormedAddCommGroup.toNormedAddGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3))))) (HAdd.hAdd.{u, u, u} E E E (instHAdd.{u} E (AddCommMagma.toAdd.{u} E (AddCommSemigroup.toAddCommMagma.{u} E (AddCommMonoid.toAddCommSemigroup.{u} E (ESeminormedAddCommMonoid.toAddCommMonoid.{u} E 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inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3 (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.6) (fun (x : Real) => f (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) (Complex.ofReal x) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.ofReal (Complex.im z)) Complex.I))) (Complex.re z) (Complex.re w) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace)) (intervalIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3 (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.6) (fun (x : Real) => f (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) (Complex.ofReal x) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.ofReal (Complex.im w)) Complex.I))) (Complex.re z) (Complex.re w) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace))) (HSMul.hSMul.{0, u, u} Complex E E (instHSMul.{0, u} Complex E (SMulZeroClass.toSMul.{0, u} Complex E (AddZero.toZero.{u} E (AddZeroClass.toAddZero.{u} E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3))))))) (DistribSMul.toSMulZeroClass.{0, u} Complex E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3)))) 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(PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3)))) (Module.toDistribMulAction.{0, u} Complex E Complex.instSemiring (ESeminormedAddCommMonoid.toAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3))) (NormedSpace.toModule.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.6)))))) Complex.I (intervalIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3 (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.6) (fun (y : Real) => f (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) (Complex.ofReal (Complex.re w)) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.ofReal y) Complex.I))) (Complex.im z) (Complex.im w) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace)))) (HSMul.hSMul.{0, u, u} Complex E E (instHSMul.{0, u} Complex E (SMulZeroClass.toSMul.{0, u} Complex E (AddZero.toZero.{u} E (AddZeroClass.toAddZero.{u} E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3))))))) (DistribSMul.toSMulZeroClass.{0, u} Complex E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3))))) (DistribMulAction.toDistribSMul.{0, u} Complex E (MonoidWithZero.toMonoid.{0} Complex (Semiring.toMonoidWithZero.{0} Complex Complex.instSemiring)) (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3)))) (Module.toDistribMulAction.{0, u} Complex E Complex.instSemiring (ESeminormedAddCommMonoid.toAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3))) (NormedSpace.toModule.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.6)))))) Complex.I (intervalIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3 (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.6) (fun (y : Real) => f (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) (Complex.ofReal (Complex.re z)) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.ofReal y) Complex.I))) (Complex.im z) (Complex.im w) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace)))) (OfNat.ofNat.{u} E 0 (Zero.toOfNat0.{u} E (NegZeroClass.toZero.{u} E (SubNegZeroMonoid.toNegZeroClass.{u} E (SubtractionMonoid.toSubNegZeroMonoid.{u} E (SubtractionCommMonoid.toSubtractionMonoid.{u} E (AddCommGroup.toDivisionAddCommMonoid.{u} E (NormedAddCommGroup.toAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2709478306._hygCtx._hyg.3)))))))))","typeFull":"∀ {E : Type u} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] (f : ℂ → E) (z w : ℂ),\n  ContinuousOn f (Set.uIcc z.re w.re ×ℂ Set.uIcc z.im w.im) →\n    DifferentiableOn ℂ f (Set.Ioo (min z.re w.re) (max z.re w.re) ×ℂ Set.Ioo (min z.im w.im) (max z.im w.im)) →\n      (((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * Complex.I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * Complex.I)) +\n            Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * Complex.I)) -\n          Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * Complex.I) =\n        0","typeReadable":"∀ {E : Type u} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] (f : ℂ → E) (z w : ℂ),\n  ContinuousOn f (Set.uIcc z.re w.re ×ℂ Set.uIcc z.im w.im) →\n    DifferentiableOn ℂ f (Set.Ioo (min z.re w.re) (max z.re w.re) ×ℂ Set.Ioo (min z.im w.im) (max z.im w.im)) →\n      (((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * Complex.I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * Complex.I)) +\n            Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * Complex.I)) -\n          Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * Complex.I) =\n        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{E : Type.{u}} [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2615959168._hygCtx._hyg.3 : NormedAddCommGroup.{u} E] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2615959168._hygCtx._hyg.6 : NormedSpace.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2615959168._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2615959168._hygCtx._hyg.12 : CompleteSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2615959168._hygCtx._hyg.3)))] {f : Complex -> E} {s : Set.{0} Complex}, (IsOpen.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex 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Type u_9} {β : Type u_10} {inst : Semigroup α} [self : SemigroupAction α β] (x y : α) (b : β),\n  x • y • b = (x * y) • b","typeReadable":"∀ {α : Type u_9} {β : Type u_10} {inst : Semigroup α} [self : SemigroupAction α β] (x y : α) (b : β),\n  x • y • b = (x * y) • b","typeReferences":[["HSMul","hSMul"],["SemigroupAction"],["instHMul"],["HMul","hMul"],["SemigroupAction","toSMul"],["instHSMul"],["Semigroup"],["Eq"],["Semigroup","toMul"]],"valueReferences":[["HSMul","hSMul"],["Eq","symm"],["instHMul"],["HMul","hMul"],["SemigroupAction","toSMul"],["instHSMul"],["SemigroupAction","mul_smul"],["Semigroup","toMul"]]},{"isProp":true,"kind":"theorem","name":["DifferentiableOn","circleIntegral_sub_inv_smul"],"typeFallback":"forall {E : Type.{u}} [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2456697007._hygCtx._hyg.3 : NormedAddCommGroup.{u} E] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2456697007._hygCtx._hyg.6 : NormedSpace.{0, u} Complex E Complex.instNormedField 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inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2456697007._hygCtx._hyg.3)))) (Module.toDistribMulAction.{0, u} Complex E Complex.instSemiring (ESeminormedAddCommMonoid.toAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2456697007._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2456697007._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2456697007._hygCtx._hyg.3))) (NormedSpace.toModule.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2456697007._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2456697007._hygCtx._hyg.6)))))) (Inv.inv.{0} Complex Complex.instInv (HSub.hSub.{0, 0, 0} Complex Complex Complex (instHSub.{0} Complex Complex.instSub) z w)) (f z)) c R) (HSMul.hSMul.{0, u, u} Complex E E (instHSMul.{0, u} Complex E (SMulZeroClass.toSMul.{0, u} Complex E (AddZero.toZero.{u} E (AddZeroClass.toAddZero.{u} E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2456697007._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E 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inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2456697007._hygCtx._hyg.6)))))) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (OfNat.ofNat.{0} Complex 2 (instOfNatAtLeastTwo.{0} Complex 2 Complex.instNatCast (Nat.instAtLeastTwoHAddOfNat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (Nat.instNeZeroSucc (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) (Complex.ofReal Real.pi)) Complex.I) (f w)))","typeFull":"∀ {E : Type u} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] [CompleteSpace E] {R : ℝ} {c w : ℂ} {f : ℂ → E},\n  DifferentiableOn ℂ f (Metric.closedBall c R) →\n    w ∈ Metric.ball c R → ∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z = (2 * ↑Real.pi * Complex.I) • f w","typeReadable":"∀ {E : Type u} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] [CompleteSpace E] {R : ℝ} {c w : ℂ} {f : ℂ → E},\n  DifferentiableOn ℂ f (Metric.closedBall c R) →\n    w ∈ Metric.ball c R → ∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z = (2 * ↑Real.pi * Complex.I) • f w","typeReferences":[["circleIntegral"],["PseudoMetricSpace","toUniformSpace"],["Complex","instDenselyNormedField"],["Membership","mem"],["HMul","hMul"],["SMulZeroClass","toSMul"],["Complex","instSemiring"],["Real","pi"],["NormedAddCommGroup","toAddCommGroup"],["Complex","ofReal"],["Complex"],["NormedSpace"],["DenselyNormedField","toNontriviallyNormedField"],["HSub","hSub"],["NormedAddCommGroup"],["NormedField","toNormedCommRing"],["Complex","instModuleSelf"],["DistribSMul","toSMulZeroClass"],["NormedAddCommGroup","toENormedAddCommMonoid"],["NormedSpace","toModule"],["Real"],["Complex","instInv"],["SeminormedAddCommGroup","toPseudoMetricSpace"],["DistribMulAction","toDistribSMul"],["SeminormedCommRing","toSeminormedRing"],["AddZeroClass","toAddZero"],["Complex","instNormedField"],["Set","instMembership"],["Nat"],["HSMul","hSMul"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["instHMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["ESeminormedAddCommMonoid","toESeminormedAddMonoid"],["UniformSpace","toTopologicalSpace"],["DifferentiableOn"],["Metric","ball"],["Nat","instNeZeroSucc"],["SeminormedRing","toPseudoMetricSpace"],["CompleteSpace"],["instOfNatNat"],["MonoidWithZero","toMonoid"],["Complex","instNatCast"],["instHSMul"],["Eq"],["ESeminormedAddMonoid","toAddMonoid"],["Complex","I"],["Inv","inv"],["Set"],["instOfNatAtLeastTwo"],["Semiring","toMonoidWithZero"],["OfNat","ofNat"],["Complex","addCommGroup"],["Metric","closedBall"],["Module","toDistribMulAction"],["NormedCommRing","toSeminormedCommRing"],["instHSub"],["Complex","instMul"],["Complex","instSub"],["NormedAddCommGroup","toSeminormedAddCommGroup"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["Nat","instAtLeastTwoHAddOfNat"]],"valueReferences":[["RCLike","innerProductSpace"],["Complex","instDenselyNormedField"],["PseudoMetricSpace","toUniformSpace"],["UniformSpace","toTopologicalSpace"],["Metric","ball"],["NormedAddCommGroup","toAddCommGroup"],["Complex"],["SeminormedRing","toPseudoMetricSpace"],["Complex","instNormedAddCommGroup"],["DenselyNormedField","toNontriviallyNormedField"],["DifferentiableOn","diffContOnCl"],["closure"],["NormedField","toNormedCommRing"],["Complex","instModuleSelf"],["DifferentiableOn","mono"],["NormedSpace","toModule"],["SeminormedAddCommGroup","toPseudoMetricSpace"],["DiffContOnCl","circleIntegral_sub_inv_smul"],["SeminormedCommRing","toSeminormedRing"],["Complex","instNormedField"],["Complex","addCommGroup"],["Metric","closedBall"],["Metric","closure_ball_subset_closedBall"],["NormedCommRing","toSeminormedCommRing"],["NormedAddCommGroup","toSeminormedAddCommGroup"],["InnerProductSpace","toNormedSpace"],["Complex","instRCLike"]]},{"isProp":true,"kind":"theorem","name":["Complex","circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable"],"typeFallback":"forall {E : Type.{u}} [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2947557891._hygCtx._hyg.3 : NormedAddCommGroup.{u} E] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2947557891._hygCtx._hyg.6 : NormedSpace.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2947557891._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2947557891._hygCtx._hyg.12 : CompleteSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2947557891._hygCtx._hyg.3)))] {R : Real}, (LT.lt.{0} Real Real.instLT (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)) R) -> (forall {f : Complex -> E} {c : Complex} {s : Set.{0} Complex}, (Set.Countable.{0} Complex s) -> (ContinuousOn.{0, u} Complex E (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2947557891._hygCtx._hyg.3)))) f (Metric.closedBall.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))) c R)) -> (forall (z : Complex), (Membership.mem.{0, 0} Complex (Set.{0} Complex) (Set.instMembership.{0} Complex) (SDiff.sdiff.{0} (Set.{0} Complex) (Set.instSDiff.{0} Complex) (Metric.ball.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))) c R) s) z) -> (DifferentiableAt.{0, 0, u} Complex (DenselyNormedField.toNontriviallyNormedField.{0} Complex Complex.instDenselyNormedField) Complex Complex.addCommGroup Complex.instModuleSelf (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) E (NormedAddCommGroup.toAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2947557891._hygCtx._hyg.3) (NormedSpace.toModule.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2947557891._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2947557891._hygCtx._hyg.6) (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2947557891._hygCtx._hyg.3)))) f z)) -> (Eq.{succ u} E (circleIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2947557891._hygCtx._hyg.3 inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2947557891._hygCtx._hyg.6 (fun (z : Complex) => HSMul.hSMul.{0, u, u} Complex E E (instHSMul.{0, u} Complex E (SMulZeroClass.toSMul.{0, u} Complex E (AddZero.toZero.{u} E (AddZeroClass.toAddZero.{u} E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E 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(NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2947557891._hygCtx._hyg.3))) (NormedSpace.toModule.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2947557891._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2947557891._hygCtx._hyg.6)))))) (Inv.inv.{0} Complex Complex.instInv (HSub.hSub.{0, 0, 0} Complex Complex Complex (instHSub.{0} Complex Complex.instSub) z c)) (f z)) c R) (HSMul.hSMul.{0, u, u} Complex E E (instHSMul.{0, u} Complex E (SMulZeroClass.toSMul.{0, u} Complex E (AddZero.toZero.{u} E (AddZeroClass.toAddZero.{u} E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E 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{E : Type.{u}} [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.3 : NormedAddCommGroup.{u} E] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.6 : NormedSpace.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.12 : CompleteSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.3)))] {c : Complex} {R : Real}, (LT.lt.{0} Real Real.instLT (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)) R) -> (forall {f : Complex -> E} {y : E} {s : Set.{0} Complex}, (Set.Countable.{0} Complex s) -> (ContinuousOn.{0, u} Complex E (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.3)))) f (SDiff.sdiff.{0} (Set.{0} Complex) (Set.instSDiff.{0} Complex) (Metric.closedBall.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))) c R) (Singleton.singleton.{0, 0} Complex (Set.{0} Complex) (Set.instSingletonSet.{0} Complex) c))) -> (forall (z : Complex), (Membership.mem.{0, 0} Complex (Set.{0} Complex) (Set.instMembership.{0} Complex) (SDiff.sdiff.{0} (Set.{0} Complex) (Set.instSDiff.{0} Complex) (SDiff.sdiff.{0} (Set.{0} Complex) (Set.instSDiff.{0} Complex) (Metric.ball.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))) c R) (Singleton.singleton.{0, 0} Complex (Set.{0} Complex) (Set.instSingletonSet.{0} Complex) c)) s) z) -> (DifferentiableAt.{0, 0, u} Complex (DenselyNormedField.toNontriviallyNormedField.{0} Complex Complex.instDenselyNormedField) Complex Complex.addCommGroup Complex.instModuleSelf (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) E (NormedAddCommGroup.toAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.3) (NormedSpace.toModule.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.6) (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.3)))) f z)) -> (Filter.Tendsto.{0, u} Complex E f (nhdsWithin.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) c (Compl.compl.{0} (Set.{0} Complex) (Set.instCompl.{0} Complex) (Singleton.singleton.{0, 0} Complex (Set.{0} Complex) (Set.instSingletonSet.{0} Complex) c))) (nhds.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.3)))) y)) -> (Eq.{succ u} E (circleIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.3 inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.6 (fun (z : Complex) => HSMul.hSMul.{0, u, u} Complex E E (instHSMul.{0, u} Complex E (SMulZeroClass.toSMul.{0, u} Complex E (AddZero.toZero.{u} E (AddZeroClass.toAddZero.{u} E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E 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(DistribSMul.toSMulZeroClass.{0, u} Complex E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E 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(NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.3))) (NormedSpace.toModule.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.6)))))) (Inv.inv.{0} Complex Complex.instInv (HSub.hSub.{0, 0, 0} Complex Complex Complex (instHSub.{0} Complex Complex.instSub) z c)) (f z)) c R) (HSMul.hSMul.{0, u, u} Complex E E (instHSMul.{0, u} Complex E (SMulZeroClass.toSMul.{0, u} Complex E (AddZero.toZero.{u} E (AddZeroClass.toAddZero.{u} E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E 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(AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.3))))) (DistribMulAction.toDistribSMul.{0, u} Complex E (MonoidWithZero.toMonoid.{0} Complex (Semiring.toMonoidWithZero.{0} Complex Complex.instSemiring)) (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.3)))) (Module.toDistribMulAction.{0, u} Complex E Complex.instSemiring (ESeminormedAddCommMonoid.toAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3131034186._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E 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{E : Type.{u}} [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2319345309._hygCtx._hyg.3 : NormedAddCommGroup.{u} E] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2319345309._hygCtx._hyg.6 : NormedSpace.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2319345309._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2319345309._hygCtx._hyg.12 : CompleteSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2319345309._hygCtx._hyg.3)))] {f : Complex -> E} {s : Set.{0} Complex}, (IsOpen.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex 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Prop (Eq.{1} Complex Complex.I (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZero))) False","typeFull":"(Complex.I = 0) = False","typeReadable":"(Complex.I = 0) = False","typeReferences":[["Complex"],["Complex","instZero"],["Complex","I"],["False"],["Zero","toOfNat0"],["Eq"],["OfNat","ofNat"]],"valueReferences":[["Complex"],["Complex","instZero"],["Complex","I"],["eq_false"],["Zero","toOfNat0"],["Complex","I_ne_zero"],["Eq"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["Complex","circleIntegral_sub_inv_smul_of_differentiable_on_off_countable"],"typeFallback":"forall {E : Type.{u}} [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.654568467._hygCtx._hyg.3 : NormedAddCommGroup.{u} E] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.654568467._hygCtx._hyg.6 : NormedSpace.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.654568467._hygCtx._hyg.3)] 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NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] [CompleteSpace E] {s : Set ℂ} {f : ℂ → E},\n  DifferentiableOn ℂ f s → IsOpen s → AnalyticOn ℂ f 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inst._@.Mathlib.Analysis.Complex.CauchyIntegral.19214370._hygCtx._hyg.3))) (NormedSpace.toModule.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.19214370._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.19214370._hygCtx._hyg.6)))))) Complex.I (intervalIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.19214370._hygCtx._hyg.3 (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.19214370._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.19214370._hygCtx._hyg.6) (fun (y : Real) => f (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) (Complex.ofReal (Complex.re w)) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.ofReal y) Complex.I))) (Complex.im z) (Complex.im w) (MeasureTheory.MeasureSpace.volume.{0} 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em","name":["Complex","analyticAt_iff_eventually_differentiableAt"],"typeFallback":"forall {E : Type.{u}} [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3655923508._hygCtx._hyg.3 : NormedAddCommGroup.{u} E] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3655923508._hygCtx._hyg.6 : NormedSpace.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3655923508._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3655923508._hygCtx._hyg.12 : CompleteSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3655923508._hygCtx._hyg.3)))] {f : Complex -> E} {c : Complex}, Iff (AnalyticAt.{0, 0, u} Complex Complex E (DenselyNormedField.toNontriviallyNormedField.{0} Complex Complex.instDenselyNormedField) Complex.instNormedAddCommGroup (InnerProductSpace.toNormedSpace.{0, 0} Complex Complex Complex.instRCLike 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DifferentiableAt ℂ f z","typeReadable":"∀ {E : Type u} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] [CompleteSpace E] {f : ℂ → E} {c : ℂ},\n  AnalyticAt ℂ f c ↔ ∀ᶠ (z : ℂ) in nhds c, DifferentiableAt ℂ f 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{G : Type.{u_3}} [inst._@.Mathlib.Algebra.Group.Basic.1964277017._hygCtx._hyg.6 : AddGroup.{u_3} G] {a : G} {b : G}, Eq.{1} Prop (Eq.{succ u_3} G (HSub.hSub.{u_3, u_3, u_3} G G G (instHSub.{u_3} G (SubNegMonoid.toSub.{u_3} G (AddGroup.toSubNegMonoid.{u_3} G inst._@.Mathlib.Algebra.Group.Basic.1964277017._hygCtx._hyg.6))) a b) (OfNat.ofNat.{u_3} G 0 (Zero.toOfNat0.{u_3} G (NegZeroClass.toZero.{u_3} G (SubNegZeroMonoid.toNegZeroClass.{u_3} G (SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (AddGroup.toSubtractionMonoid.{u_3} G inst._@.Mathlib.Algebra.Group.Basic.1964277017._hygCtx._hyg.6))))))) (Eq.{succ u_3} G a b)","typeFull":"∀ {G : Type u_3} [inst : AddGroup G] {a b : G}, (a - b = 0) = (a = b)","typeReadable":"∀ {G : Type u_3} [inst : AddGroup G] {a b : G}, (a - b = 0) = (a = 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(NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3412838271._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3412838271._hygCtx._hyg.12 : CompleteSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3412838271._hygCtx._hyg.3)))] {R : Real} {c : Complex} {w : Complex} {f : Complex -> E}, (DiffContOnCl.{0, 0, u} Complex Complex E (DenselyNormedField.toNontriviallyNormedField.{0} Complex Complex.instDenselyNormedField) Complex.instNormedAddCommGroup inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3412838271._hygCtx._hyg.3 (InnerProductSpace.toNormedSpace.{0, 0} Complex Complex Complex.instRCLike (NormedAddCommGroup.toSeminormedAddCommGroup.{0} Complex Complex.instNormedAddCommGroup) (RCLike.innerProductSpace.{0} Complex Complex.instRCLike)) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3412838271._hygCtx._hyg.6 f (Metric.ball.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))) c R)) -> (Membership.mem.{0, 0} Complex (Set.{0} Complex) (Set.instMembership.{0} Complex) (Metric.ball.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))) c R) w) -> (Eq.{succ u} E (circleIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3412838271._hygCtx._hyg.3 inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3412838271._hygCtx._hyg.6 (fun (z : Complex) => HSMul.hSMul.{0, u, u} Complex E E (instHSMul.{0, u} Complex E (SMulZeroClass.toSMul.{0, u} Complex E (AddZero.toZero.{u} E 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(Complex.ofReal Real.pi)) Complex.I) (f w)))","typeFull":"∀ {E : Type u} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] [CompleteSpace E] {R : ℝ} {c w : ℂ} {f : ℂ → E},\n  DiffContOnCl ℂ f (Metric.ball c R) →\n    w ∈ Metric.ball c R → ∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z = (2 * ↑Real.pi * Complex.I) • f w","typeReadable":"∀ {E : Type u} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] [CompleteSpace E] {R : ℝ} {c w : ℂ} {f : ℂ → E},\n  DiffContOnCl ℂ f (Metric.ball c R) →\n    w ∈ Metric.ball c R → ∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z = (2 * ↑Real.pi * Complex.I) • f 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{E : Type.{u}} [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3 : NormedAddCommGroup.{u} E] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.6 : NormedSpace.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)] (f : Complex -> E) (z : Complex) (w : Complex), (DifferentiableOn.{0, 0, u} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Complex Complex.addCommGroup (NormedSpace.toModule.{0, 0} Real Complex Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (InnerProductSpace.toNormedSpace.{0, 0} Real Complex 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(NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.6)) (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) f (Complex.reProdIm (Set.uIcc.{0} Real Real.lattice (Complex.re z) (Complex.re w)) (Set.uIcc.{0} Real Real.lattice (Complex.im z) (Complex.im w)))) -> (MeasureTheory.IntegrableOn.{0, u} Complex E (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Complex (measureSpaceOfInnerProductSpace.{0} Complex Complex.instNormedAddCommGroup instInnerProductSpaceRealComplex (FiniteDimensional.rclike_to_real.{0} Complex Complex.instRCLike) Complex.measurableSpace Complex.borelSpace)) (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (SeminormedAddGroup.toContinuousENorm.{u} E (SeminormedAddCommGroup.toSeminormedAddGroup.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3))) (fun (z : Complex) => HSub.hSub.{u, u, u} E E E (instHSub.{u} E (SubNegMonoid.toSub.{u} E (AddGroup.toSubNegMonoid.{u} E (NormedAddGroup.toAddGroup.{u} E (NormedAddCommGroup.toNormedAddGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3))))) (HSMul.hSMul.{0, u, u} Complex E E (instHSMul.{0, u} Complex E (SMulZeroClass.toSMul.{0, u} Complex E (AddZero.toZero.{u} E (AddZeroClass.toAddZero.{u} E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E 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(DistribSMul.toSMulZeroClass.{0, u} Complex E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3))))) (DistribMulAction.toDistribSMul.{0, u} Complex E (MonoidWithZero.toMonoid.{0} Complex (Semiring.toMonoidWithZero.{0} Complex Complex.instSemiring)) (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) 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(NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3))) (NormedSpace.toModule.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.6)))))) Complex.I (DFunLike.coe.{succ u, 1, succ u} (ContinuousLinearMap.{0, 0, 0, u} Real Real (DivisionSemiring.toSemiring.{0} Real (Semifield.toDivisionSemiring.{0} Real (Field.toSemifield.{0} Real (NormedField.toField.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField)))))) (DivisionSemiring.toSemiring.{0} Real (Semifield.toDivisionSemiring.{0} Real (Field.toSemifield.{0} Real 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(NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) instInnerProductSpaceRealComplex)) (NormedSpace.toModule.{0, u} Real E Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.6))) Complex (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Complex) => E) (ContinuousLinearMap.funLike.{0, 0, 0, u} Real Real (DivisionSemiring.toSemiring.{0} Real (Semifield.toDivisionSemiring.{0} Real (Field.toSemifield.{0} Real (NormedField.toField.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField)))))) (DivisionSemiring.toSemiring.{0} Real (Semifield.toDivisionSemiring.{0} Real (Field.toSemifield.{0} Real (NormedField.toField.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField)))))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (DivisionSemiring.toSemiring.{0} Real (Semifield.toDivisionSemiring.{0} Real (Field.toSemifield.{0} Real (NormedField.toField.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField)))))))) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (AddCommGroup.toAddCommMonoid.{u} E (NormedAddCommGroup.toAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)) (NormedSpace.toModule.{0, 0} Real Complex Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (InnerProductSpace.toNormedSpace.{0, 0} Real Complex Real.instRCLike (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) instInnerProductSpaceRealComplex)) (NormedSpace.toModule.{0, u} Real E Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.6))) (fderiv.{0, 0, u} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Complex Complex.addCommGroup (NormedSpace.toModule.{0, 0} Real Complex Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (InnerProductSpace.toNormedSpace.{0, 0} Real Complex Real.instRCLike (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) instInnerProductSpaceRealComplex)) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) E (NormedAddCommGroup.toAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) (NormedSpace.toModule.{0, u} Real E Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.6)) (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) f z) (OfNat.ofNat.{0} Complex 1 (One.toOfNat1.{0} Complex Complex.instOne)))) (DFunLike.coe.{succ u, 1, succ u} (ContinuousLinearMap.{0, 0, 0, u} Real Real (DivisionSemiring.toSemiring.{0} Real (Semifield.toDivisionSemiring.{0} Real (Field.toSemifield.{0} Real (NormedField.toField.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField)))))) (DivisionSemiring.toSemiring.{0} Real (Semifield.toDivisionSemiring.{0} Real (Field.toSemifield.{0} Real (NormedField.toField.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField)))))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (DivisionSemiring.toSemiring.{0} Real (Semifield.toDivisionSemiring.{0} Real (Field.toSemifield.{0} Real (NormedField.toField.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField)))))))) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (AddCommGroup.toAddCommMonoid.{u} E (NormedAddCommGroup.toAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)) (NormedSpace.toModule.{0, 0} Real Complex Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (InnerProductSpace.toNormedSpace.{0, 0} Real Complex Real.instRCLike (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) instInnerProductSpaceRealComplex)) (NormedSpace.toModule.{0, u} Real E Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.6))) Complex (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Complex) => E) (ContinuousLinearMap.funLike.{0, 0, 0, u} Real Real (DivisionSemiring.toSemiring.{0} Real (Semifield.toDivisionSemiring.{0} Real (Field.toSemifield.{0} Real (NormedField.toField.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField)))))) (DivisionSemiring.toSemiring.{0} Real (Semifield.toDivisionSemiring.{0} Real (Field.toSemifield.{0} Real (NormedField.toField.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField)))))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (DivisionSemiring.toSemiring.{0} Real (Semifield.toDivisionSemiring.{0} Real (Field.toSemifield.{0} Real (NormedField.toField.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField)))))))) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (AddCommGroup.toAddCommMonoid.{u} E (NormedAddCommGroup.toAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)) (NormedSpace.toModule.{0, 0} Real Complex Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (InnerProductSpace.toNormedSpace.{0, 0} Real Complex Real.instRCLike (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) instInnerProductSpaceRealComplex)) (NormedSpace.toModule.{0, u} Real E Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.6))) (fderiv.{0, 0, u} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Complex Complex.addCommGroup (NormedSpace.toModule.{0, 0} Real Complex Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (InnerProductSpace.toNormedSpace.{0, 0} Real Complex Real.instRCLike (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) instInnerProductSpaceRealComplex)) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) E (NormedAddCommGroup.toAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) (NormedSpace.toModule.{0, u} Real E Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.6)) (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) f z) Complex.I)) (Complex.reProdIm (Set.uIcc.{0} Real Real.lattice (Complex.re z) (Complex.re w)) (Set.uIcc.{0} Real Real.lattice (Complex.im z) (Complex.im w))) (MeasureTheory.MeasureSpace.volume.{0} Complex (measureSpaceOfInnerProductSpace.{0} Complex Complex.instNormedAddCommGroup instInnerProductSpaceRealComplex (FiniteDimensional.rclike_to_real.{0} Complex Complex.instRCLike) Complex.measurableSpace Complex.borelSpace))) -> (Eq.{succ u} E (HSub.hSub.{u, u, u} E E E (instHSub.{u} E (SubNegMonoid.toSub.{u} E (AddGroup.toSubNegMonoid.{u} E (NormedAddGroup.toAddGroup.{u} E (NormedAddCommGroup.toNormedAddGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3))))) (HAdd.hAdd.{u, u, u} E E E (instHAdd.{u} E (AddCommMagma.toAdd.{u} E (AddCommSemigroup.toAddCommMagma.{u} E (AddCommMonoid.toAddCommSemigroup.{u} E (ESeminormedAddCommMonoid.toAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3))))))) (HSub.hSub.{u, u, u} E E E (instHSub.{u} E (SubNegMonoid.toSub.{u} E (AddGroup.toSubNegMonoid.{u} E (NormedAddGroup.toAddGroup.{u} E (NormedAddCommGroup.toNormedAddGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3))))) (intervalIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3 (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.6) (fun (x : Real) => f (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) (Complex.ofReal x) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.ofReal (Complex.im z)) Complex.I))) (Complex.re z) (Complex.re w) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace)) (intervalIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3 (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.6) (fun (x : Real) => f (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) (Complex.ofReal x) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.ofReal (Complex.im w)) Complex.I))) (Complex.re z) (Complex.re w) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace))) (HSMul.hSMul.{0, u, u} Complex E E (instHSMul.{0, u} Complex E (SMulZeroClass.toSMul.{0, u} Complex E (AddZero.toZero.{u} E (AddZeroClass.toAddZero.{u} E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3))))))) (DistribSMul.toSMulZeroClass.{0, u} Complex E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3))))) (DistribMulAction.toDistribSMul.{0, u} Complex E (MonoidWithZero.toMonoid.{0} Complex (Semiring.toMonoidWithZero.{0} Complex Complex.instSemiring)) (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (Module.toDistribMulAction.{0, u} Complex E Complex.instSemiring (ESeminormedAddCommMonoid.toAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3))) (NormedSpace.toModule.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.6)))))) Complex.I (intervalIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3 (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.6) (fun (y : Real) => f (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) (Complex.ofReal (Complex.re w)) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.ofReal y) Complex.I))) (Complex.im z) (Complex.im w) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace)))) (HSMul.hSMul.{0, u, u} Complex E E (instHSMul.{0, u} Complex E (SMulZeroClass.toSMul.{0, u} Complex E (AddZero.toZero.{u} E (AddZeroClass.toAddZero.{u} E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3))))))) (DistribSMul.toSMulZeroClass.{0, u} Complex E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3))))) (DistribMulAction.toDistribSMul.{0, u} Complex E (MonoidWithZero.toMonoid.{0} Complex (Semiring.toMonoidWithZero.{0} Complex Complex.instSemiring)) (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (Module.toDistribMulAction.{0, u} Complex E Complex.instSemiring (ESeminormedAddCommMonoid.toAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3))) (NormedSpace.toModule.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.6)))))) Complex.I (intervalIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3 (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.6) (fun (y : Real) => f (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) (Complex.ofReal (Complex.re z)) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.ofReal y) Complex.I))) (Complex.im z) (Complex.im w) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace)))) (intervalIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3 (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.6) (fun (x : Real) => intervalIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3 (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.6) (fun (y : Real) => HSub.hSub.{u, u, u} E E E (instHSub.{u} E (SubNegMonoid.toSub.{u} E (AddGroup.toSubNegMonoid.{u} E (NormedAddGroup.toAddGroup.{u} E (NormedAddCommGroup.toNormedAddGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3))))) (HSMul.hSMul.{0, u, u} Complex E E (instHSMul.{0, u} Complex E (SMulZeroClass.toSMul.{0, u} Complex E (AddZero.toZero.{u} E (AddZeroClass.toAddZero.{u} E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3))))))) (DistribSMul.toSMulZeroClass.{0, u} Complex E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3))))) (DistribMulAction.toDistribSMul.{0, u} Complex E (MonoidWithZero.toMonoid.{0} Complex (Semiring.toMonoidWithZero.{0} Complex Complex.instSemiring)) (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (Module.toDistribMulAction.{0, u} Complex E Complex.instSemiring (ESeminormedAddCommMonoid.toAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3))) (NormedSpace.toModule.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.6)))))) Complex.I (DFunLike.coe.{succ u, 1, succ u} (ContinuousLinearMap.{0, 0, 0, u} Real Real (DivisionSemiring.toSemiring.{0} Real (Semifield.toDivisionSemiring.{0} 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(InnerProductSpace.toNormedSpace.{0, 0} Real Complex Real.instRCLike (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) instInnerProductSpaceRealComplex)) (NormedSpace.toModule.{0, u} Real E Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.6))) Complex (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Complex) => E) (ContinuousLinearMap.funLike.{0, 0, 0, u} Real Real 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Real.denselyNormedField) Complex Complex.addCommGroup (NormedSpace.toModule.{0, 0} Real Complex Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (InnerProductSpace.toNormedSpace.{0, 0} Real Complex Real.instRCLike (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) instInnerProductSpaceRealComplex)) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex 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Complex (instHAdd.{0} Complex Complex.instAdd) (Complex.ofReal x) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.ofReal y) Complex.I))) (OfNat.ofNat.{0} Complex 1 (One.toOfNat1.{0} Complex Complex.instOne)))) (DFunLike.coe.{succ u, 1, succ u} (ContinuousLinearMap.{0, 0, 0, u} Real Real (DivisionSemiring.toSemiring.{0} Real (Semifield.toDivisionSemiring.{0} Real (Field.toSemifield.{0} Real (NormedField.toField.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField)))))) (DivisionSemiring.toSemiring.{0} Real (Semifield.toDivisionSemiring.{0} Real (Field.toSemifield.{0} Real (NormedField.toField.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField)))))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (DivisionSemiring.toSemiring.{0} Real (Semifield.toDivisionSemiring.{0} 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(NormedSpace.toModule.{0, 0} Real Complex Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (InnerProductSpace.toNormedSpace.{0, 0} Real Complex Real.instRCLike (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) instInnerProductSpaceRealComplex)) (NormedSpace.toModule.{0, u} Real E Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.6))) Complex (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Complex) => E) (ContinuousLinearMap.funLike.{0, 0, 0, u} Real Real (DivisionSemiring.toSemiring.{0} Real (Semifield.toDivisionSemiring.{0} Real (Field.toSemifield.{0} Real (NormedField.toField.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField)))))) (DivisionSemiring.toSemiring.{0} Real (Semifield.toDivisionSemiring.{0} Real (Field.toSemifield.{0} Real (NormedField.toField.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField)))))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (DivisionSemiring.toSemiring.{0} Real (Semifield.toDivisionSemiring.{0} Real (Field.toSemifield.{0} Real (NormedField.toField.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField)))))))) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) (AddCommGroup.toAddCommMonoid.{u} E (NormedAddCommGroup.toAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)) (NormedSpace.toModule.{0, 0} Real Complex Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (InnerProductSpace.toNormedSpace.{0, 0} Real Complex Real.instRCLike (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) instInnerProductSpaceRealComplex)) (NormedSpace.toModule.{0, u} Real E Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.6))) (fderiv.{0, 0, u} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Complex Complex.addCommGroup (NormedSpace.toModule.{0, 0} Real Complex Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (InnerProductSpace.toNormedSpace.{0, 0} Real Complex Real.instRCLike (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) instInnerProductSpaceRealComplex)) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) E (NormedAddCommGroup.toAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) (NormedSpace.toModule.{0, u} Real E Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.6)) (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1611038366._hygCtx._hyg.3)))) f (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) (Complex.ofReal x) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.ofReal y) Complex.I))) Complex.I)) (Complex.im z) (Complex.im w) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace)) (Complex.re z) (Complex.re w) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace)))","typeFull":"∀ {E : Type u} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] (f : ℂ → E) (z w : ℂ),\n  DifferentiableOn ℝ f (Set.uIcc z.re w.re ×ℂ Set.uIcc z.im w.im) →\n    MeasureTheory.IntegrableOn (fun z => Complex.I • (fderiv ℝ f z) 1 - (fderiv ℝ f z) Complex.I)\n        (Set.uIcc z.re w.re ×ℂ Set.uIcc z.im w.im) MeasureTheory.volume →\n      (((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * Complex.I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * Complex.I)) +\n            Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * Complex.I)) -\n          Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * Complex.I) =\n        ∫ (x : ℝ) in z.re..w.re,\n          ∫ (y : ℝ) in z.im..w.im,\n            Complex.I • (fderiv ℝ f (↑x + ↑y * Complex.I)) 1 - (fderiv ℝ f (↑x + ↑y * Complex.I)) Complex.I","typeReadable":"∀ {E : Type u} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] (f : ℂ → E) (z w : ℂ),\n  DifferentiableOn ℝ f (Set.uIcc z.re w.re ×ℂ Set.uIcc z.im w.im) →\n    MeasureTheory.IntegrableOn (fun z => Complex.I • (fderiv ℝ f z) 1 - (fderiv ℝ f z) Complex.I)\n        (Set.uIcc z.re w.re ×ℂ Set.uIcc z.im w.im) MeasureTheory.volume →\n      (((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * Complex.I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * Complex.I)) +\n            Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * Complex.I)) -\n          Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * Complex.I) =\n        ∫ (x : ℝ) in z.re..w.re,\n          ∫ (y : ℝ) in z.im..w.im,\n            Complex.I • (fderiv ℝ f (↑x + ↑y * Complex.I)) 1 - (fderiv ℝ f (↑x + ↑y * Complex.I)) 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{E : Type.{u}} [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.3 : NormedAddCommGroup.{u} E] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.6 : NormedSpace.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.12 : CompleteSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.3)))] {R : Real} {c : Complex} {w : Complex} {f : Complex -> E} {s : Set.{0} Complex}, (Set.Countable.{0} Complex s) -> (Membership.mem.{0, 0} Complex (Set.{0} Complex) (Set.instMembership.{0} Complex) (Metric.ball.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))) c R) w) -> (ContinuousOn.{0, u} Complex E (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.3)))) f (Metric.closedBall.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))) c R)) -> (forall (x : Complex), (Membership.mem.{0, 0} Complex (Set.{0} Complex) (Set.instMembership.{0} Complex) (SDiff.sdiff.{0} (Set.{0} Complex) (Set.instSDiff.{0} Complex) (Metric.ball.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))) c R) s) x) -> (DifferentiableAt.{0, 0, u} Complex (DenselyNormedField.toNontriviallyNormedField.{0} Complex Complex.instDenselyNormedField) Complex Complex.addCommGroup Complex.instModuleSelf (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) E (NormedAddCommGroup.toAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.3) (NormedSpace.toModule.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.6) (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.3)))) f x)) -> (Eq.{succ u} E (HSMul.hSMul.{0, u, u} Complex E E (instHSMul.{0, u} Complex E (SMulZeroClass.toSMul.{0, u} Complex E (AddZero.toZero.{u} E (AddZeroClass.toAddZero.{u} E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.3))))))) (DistribSMul.toSMulZeroClass.{0, u} Complex E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.3))))) (DistribMulAction.toDistribSMul.{0, u} Complex E (MonoidWithZero.toMonoid.{0} Complex (Semiring.toMonoidWithZero.{0} Complex Complex.instSemiring)) (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.3)))) (Module.toDistribMulAction.{0, u} Complex E Complex.instSemiring (ESeminormedAddCommMonoid.toAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.3))) (NormedSpace.toModule.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.6)))))) (Inv.inv.{0} Complex Complex.instInv (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (OfNat.ofNat.{0} Complex 2 (instOfNatAtLeastTwo.{0} Complex 2 Complex.instNatCast (Nat.instAtLeastTwoHAddOfNat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (Nat.instNeZeroSucc (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) (Complex.ofReal Real.pi)) Complex.I)) (circleIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3567328302._hygCtx._hyg.3 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inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2257845486._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2257845486._hygCtx._hyg.6) (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2257845486._hygCtx._hyg.3)))) (deriv.{0, u} Complex (DenselyNormedField.toNontriviallyNormedField.{0} Complex Complex.instDenselyNormedField) E (NormedAddCommGroup.toAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2257845486._hygCtx._hyg.3) (NormedSpace.toModule.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2257845486._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2257845486._hygCtx._hyg.6) (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.2257845486._hygCtx._hyg.3)))) f) s)","typeFull":"∀ {E : Type u} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] [CompleteSpace E] {s : Set ℂ} {f : ℂ → E},\n  DifferentiableOn ℂ f s → IsOpen s → DifferentiableOn ℂ (deriv f) s","typeReadable":"∀ {E : Type u} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] [CompleteSpace E] {s : Set ℂ} {f : ℂ → E},\n  DifferentiableOn ℂ f s → IsOpen s → DifferentiableOn ℂ (deriv f) s","typeReferences":[["NormedSpace","toModule"],["PseudoMetricSpace","toUniformSpace"],["Complex","instDenselyNormedField"],["Set"],["SeminormedAddCommGroup","toPseudoMetricSpace"],["UniformSpace","toTopologicalSpace"],["DifferentiableOn"],["SeminormedCommRing","toSeminormedRing"],["NormedAddCommGroup","toAddCommGroup"],["Complex","instNormedField"],["Complex","addCommGroup"],["Complex"],["SeminormedRing","toPseudoMetricSpace"],["IsOpen"],["CompleteSpace"],["NormedSpace"],["DenselyNormedField","toNontriviallyNormedField"],["NormedAddCommGroup"],["deriv"],["NormedCommRing","toSeminormedCommRing"],["NormedField","toNormedCommRing"],["NormedAddCommGroup","toSeminormedAddCommGroup"],["Complex","instModuleSelf"]],"valueReferences":[["NormedField","toNormedSpace"],["NormedSpace","toModule"],["PseudoMetricSpace","toUniformSpace"],["Complex","instDenselyNormedField"],["SeminormedAddCommGroup","toPseudoMetricSpace"],["AnalyticOnNhd","deriv"],["UniformSpace","toTopologicalSpace"],["NormedAddCommGroup","toAddCommGroup"],["NonUnitalNormedRing","toNormedAddCommGroup"],["DifferentiableOn","analyticOnNhd"],["NontriviallyNormedField","toNormedField"],["Complex"],["DenselyNormedField","toNontriviallyNormedField"],["AnalyticOnNhd","differentiableOn"],["NonUnitalNormedCommRing","toNonUnitalNormedRing"],["deriv"],["NormedAddCommGroup","toSeminormedAddCommGroup"],["NormedField","toNormedCommRing"],["NormedCommRing","toNonUnitalNormedCommRing"]]},{"isProp":true,"kind":"theorem","name":["Complex","circleIntegral_sub_center_inv_smul_eq_of_differentiable_on_annulus_off_countable"],"typeFallback":"forall {E : Type.{u}} [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3836572986._hygCtx._hyg.3 : NormedAddCommGroup.{u} E] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3836572986._hygCtx._hyg.6 : NormedSpace.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3836572986._hygCtx._hyg.3)] {c : Complex} {r : Real} {R : Real}, (LT.lt.{0} Real Real.instLT (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)) r) -> (LE.le.{0} Real Real.instLE r R) -> (forall {f : Complex -> E} {s : Set.{0} Complex}, (Set.Countable.{0} Complex s) -> (ContinuousOn.{0, u} Complex E (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3836572986._hygCtx._hyg.3)))) f (SDiff.sdiff.{0} (Set.{0} Complex) (Set.instSDiff.{0} Complex) (Metric.closedBall.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))) c R) (Metric.ball.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))) c r))) -> (forall (z : Complex), (Membership.mem.{0, 0} Complex (Set.{0} Complex) (Set.instMembership.{0} Complex) (SDiff.sdiff.{0} (Set.{0} Complex) (Set.instSDiff.{0} Complex) (SDiff.sdiff.{0} (Set.{0} Complex) (Set.instSDiff.{0} Complex) (Metric.ball.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))) c R) (Metric.closedBall.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))) c r)) s) z) -> (DifferentiableAt.{0, 0, u} Complex (DenselyNormedField.toNontriviallyNormedField.{0} Complex Complex.instDenselyNormedField) Complex Complex.addCommGroup Complex.instModuleSelf (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) E (NormedAddCommGroup.toAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3836572986._hygCtx._hyg.3) (NormedSpace.toModule.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3836572986._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3836572986._hygCtx._hyg.6) (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3836572986._hygCtx._hyg.3)))) f z)) -> (Eq.{succ u} E (circleIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3836572986._hygCtx._hyg.3 inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3836572986._hygCtx._hyg.6 (fun (z : Complex) => HSMul.hSMul.{0, u, u} Complex E E (instHSMul.{0, u} Complex E (SMulZeroClass.toSMul.{0, u} Complex E (AddZero.toZero.{u} E (AddZeroClass.toAddZero.{u} E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3836572986._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3836572986._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E 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E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3836572986._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3836572986._hygCtx._hyg.3))))) (DistribMulAction.toDistribSMul.{0, u} Complex E (MonoidWithZero.toMonoid.{0} Complex (Semiring.toMonoidWithZero.{0} Complex Complex.instSemiring)) (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3836572986._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E 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inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.6))) (f' z) (OfNat.ofNat.{0} Complex 1 (One.toOfNat1.{0} Complex Complex.instOne)))) (DFunLike.coe.{succ u, 1, succ u} (ContinuousLinearMap.{0, 0, 0, u} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex 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(InnerProductSpace.toNormedSpace.{0, 0} Real Complex Real.instRCLike (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) instInnerProductSpaceRealComplex)) (NormedSpace.toModule.{0, u} Real E Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3) (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.6))) Complex (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Complex) => E) (ContinuousLinearMap.funLike.{0, 0, 0, u} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} 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(NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3))) (NormedSpace.toModule.{0, 0} Real Complex Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (InnerProductSpace.toNormedSpace.{0, 0} Real Complex Real.instRCLike (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) instInnerProductSpaceRealComplex)) (NormedSpace.toModule.{0, u} Real E Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3) (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.6))) (f' z) Complex.I)) (Complex.reProdIm (Set.uIcc.{0} Real Real.lattice (Complex.re z) (Complex.re w)) (Set.uIcc.{0} Real Real.lattice (Complex.im z) (Complex.im w))) (MeasureTheory.MeasureSpace.volume.{0} Complex (measureSpaceOfInnerProductSpace.{0} Complex Complex.instNormedAddCommGroup instInnerProductSpaceRealComplex (FiniteDimensional.rclike_to_real.{0} Complex Complex.instRCLike) Complex.measurableSpace Complex.borelSpace))) -> (Eq.{succ u} E (HSub.hSub.{u, u, u} E E E (instHSub.{u} E (SubNegMonoid.toSub.{u} E (AddGroup.toSubNegMonoid.{u} E (NormedAddGroup.toAddGroup.{u} E (NormedAddCommGroup.toNormedAddGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3))))) (HAdd.hAdd.{u, u, u} E E E (instHAdd.{u} E (AddCommMagma.toAdd.{u} E (AddCommSemigroup.toAddCommMagma.{u} E (AddCommMonoid.toAddCommSemigroup.{u} E (ESeminormedAddCommMonoid.toAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3))))))) (HSub.hSub.{u, u, u} E E E (instHSub.{u} E (SubNegMonoid.toSub.{u} E (AddGroup.toSubNegMonoid.{u} E (NormedAddGroup.toAddGroup.{u} E (NormedAddCommGroup.toNormedAddGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3))))) (intervalIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3 (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.6) (fun (x : Real) => f (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) (Complex.ofReal x) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.ofReal (Complex.im z)) Complex.I))) (Complex.re z) (Complex.re w) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace)) (intervalIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3 (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.6) (fun (x : Real) => f (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) (Complex.ofReal x) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.ofReal (Complex.im w)) Complex.I))) (Complex.re z) (Complex.re w) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace))) (HSMul.hSMul.{0, u, u} Complex E E (instHSMul.{0, u} Complex E (SMulZeroClass.toSMul.{0, u} Complex E (AddZero.toZero.{u} E (AddZeroClass.toAddZero.{u} E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3))))))) (DistribSMul.toSMulZeroClass.{0, u} Complex E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3))))) (DistribMulAction.toDistribSMul.{0, u} Complex E (MonoidWithZero.toMonoid.{0} Complex (Semiring.toMonoidWithZero.{0} Complex Complex.instSemiring)) (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (Module.toDistribMulAction.{0, u} Complex E Complex.instSemiring (ESeminormedAddCommMonoid.toAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3))) (NormedSpace.toModule.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.6)))))) Complex.I (intervalIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3 (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.6) (fun (y : Real) => f (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) (Complex.ofReal (Complex.re w)) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.ofReal y) Complex.I))) (Complex.im z) (Complex.im w) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace)))) (HSMul.hSMul.{0, u, u} Complex E E (instHSMul.{0, u} Complex E (SMulZeroClass.toSMul.{0, u} Complex E (AddZero.toZero.{u} E (AddZeroClass.toAddZero.{u} E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3))))))) (DistribSMul.toSMulZeroClass.{0, u} Complex E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3))))) (DistribMulAction.toDistribSMul.{0, u} Complex E (MonoidWithZero.toMonoid.{0} Complex (Semiring.toMonoidWithZero.{0} Complex Complex.instSemiring)) (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (Module.toDistribMulAction.{0, u} Complex E Complex.instSemiring (ESeminormedAddCommMonoid.toAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3))) (NormedSpace.toModule.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.6)))))) Complex.I (intervalIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3 (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.6) (fun (y : Real) => f (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) (Complex.ofReal (Complex.re z)) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.ofReal y) Complex.I))) (Complex.im z) (Complex.im w) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace)))) (intervalIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3 (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.6) (fun (x : Real) => intervalIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3 (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.6) (fun (y : Real) => HSub.hSub.{u, u, u} E E E (instHSub.{u} E (SubNegMonoid.toSub.{u} E (AddGroup.toSubNegMonoid.{u} E (NormedAddGroup.toAddGroup.{u} E (NormedAddCommGroup.toNormedAddGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3))))) (HSMul.hSMul.{0, u, u} Complex E E (instHSMul.{0, u} Complex E (SMulZeroClass.toSMul.{0, u} Complex E (AddZero.toZero.{u} E (AddZeroClass.toAddZero.{u} E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3))))))) (DistribSMul.toSMulZeroClass.{0, u} Complex E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3))))) (DistribMulAction.toDistribSMul.{0, u} Complex E (MonoidWithZero.toMonoid.{0} Complex (Semiring.toMonoidWithZero.{0} Complex Complex.instSemiring)) (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E 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(NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3280246299._hygCtx._hyg.6))) (f' (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) (Complex.ofReal x) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.ofReal y) Complex.I))) Complex.I)) (Complex.im z) (Complex.im w) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace)) (Complex.re z) (Complex.re w) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace)))","typeFull":"∀ {E : Type u} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] (f : ℂ → E) (f' : ℂ → ℂ →L[ℝ] E) (z w : ℂ)\n  (s : Set ℂ),\n  s.Countable →\n    ContinuousOn f (Set.uIcc z.re w.re ×ℂ Set.uIcc z.im w.im) →\n      (∀ x ∈ Set.Ioo (min z.re w.re) (max z.re w.re) ×ℂ Set.Ioo (min z.im w.im) (max z.im w.im) \\ s,\n          HasFDerivAt f (f' x) x) →\n        MeasureTheory.IntegrableOn (fun z => Complex.I • (f' z) 1 - (f' z) Complex.I)\n            (Set.uIcc z.re w.re ×ℂ Set.uIcc z.im w.im) MeasureTheory.volume →\n          (((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * Complex.I)) -\n                  ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * Complex.I)) +\n                Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * Complex.I)) -\n              Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * Complex.I) =\n            ∫ (x : ℝ) in z.re..w.re,\n              ∫ (y : ℝ) in z.im..w.im, Complex.I • (f' (↑x + ↑y * Complex.I)) 1 - (f' (↑x + ↑y * Complex.I)) Complex.I","typeReadable":"∀ {E : Type u} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] (f : ℂ → E) (f' : ℂ → ℂ →L[ℝ] E) (z w : ℂ)\n  (s : Set ℂ),\n  s.Countable →\n    ContinuousOn f (Set.uIcc z.re w.re ×ℂ Set.uIcc z.im w.im) →\n      (∀ x ∈ Set.Ioo (min z.re w.re) (max z.re w.re) ×ℂ Set.Ioo (min z.im w.im) (max z.im w.im) \\ s,\n          HasFDerivAt f (f' x) x) →\n        MeasureTheory.IntegrableOn (fun z => Complex.I • (f' z) 1 - (f' z) Complex.I)\n            (Set.uIcc z.re w.re ×ℂ Set.uIcc z.im w.im) MeasureTheory.volume →\n          (((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * Complex.I)) -\n                  ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * Complex.I)) +\n                Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * Complex.I)) -\n              Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * Complex.I) =\n            ∫ (x : ℝ) in z.re..w.re,\n              ∫ (y : ℝ) in z.im..w.im, Complex.I • (f' (↑x + ↑y * Complex.I)) 1 - (f' (↑x + ↑y * Complex.I)) 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{E : Type.{u}} [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3839512179._hygCtx._hyg.3 : NormedAddCommGroup.{u} E] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3839512179._hygCtx._hyg.6 : NormedSpace.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3839512179._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3839512179._hygCtx._hyg.12 : CompleteSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3839512179._hygCtx._hyg.3)))] {f : Complex -> E}, Iff (AnalyticOnNhd.{0, 0, u} Complex Complex E (DenselyNormedField.toNontriviallyNormedField.{0} Complex Complex.instDenselyNormedField) Complex.instNormedAddCommGroup (InnerProductSpace.toNormedSpace.{0, 0} Complex Complex Complex.instRCLike 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{E : Type.{u}} [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1356526000._hygCtx._hyg.3 : NormedAddCommGroup.{u} E] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1356526000._hygCtx._hyg.6 : NormedSpace.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1356526000._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1356526000._hygCtx._hyg.12 : CompleteSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1356526000._hygCtx._hyg.3)))] {R : NNReal} {c : Complex} {f : Complex -> E}, (DifferentiableOn.{0, 0, u} Complex (DenselyNormedField.toNontriviallyNormedField.{0} Complex Complex.instDenselyNormedField) Complex Complex.addCommGroup Complex.instModuleSelf (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} 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{E : Type.{u}} [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3241617489._hygCtx._hyg.3 : NormedAddCommGroup.{u} E] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3241617489._hygCtx._hyg.6 : NormedSpace.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3241617489._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3241617489._hygCtx._hyg.12 : CompleteSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3241617489._hygCtx._hyg.3)))] {R : Real} {f : Complex -> E} {c : Complex}, (LT.lt.{0} Real Real.instLT (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)) R) -> (forall (n : Nat), (DifferentiableOn.{0, 0, u} Complex (DenselyNormedField.toNontriviallyNormedField.{0} Complex Complex.instDenselyNormedField) Complex Complex.addCommGroup Complex.instModuleSelf (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) E (NormedAddCommGroup.toAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3241617489._hygCtx._hyg.3) (NormedSpace.toModule.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3241617489._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3241617489._hygCtx._hyg.6) (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3241617489._hygCtx._hyg.3)))) f (Metric.closedBall.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))) c R)) -> (Eq.{succ u} E (circleIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3241617489._hygCtx._hyg.3 inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3241617489._hygCtx._hyg.6 (fun (z : Complex) => HSMul.hSMul.{0, u, u} Complex E E (instHSMul.{0, u} Complex E (SMulZeroClass.toSMul.{0, u} Complex E (AddZero.toZero.{u} E (AddZeroClass.toAddZero.{u} E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.3241617489._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E 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(NormedAddCommGroup.toENormedAddCommMonoid.{u_5} E inst._@.Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic.404174004._hygCtx._hyg.9))))) (DistribMulAction.toDistribSMul.{u_2, u_5} 𝕜 E (MonoidWithZero.toMonoid.{u_2} 𝕜 (Semiring.toMonoidWithZero.{u_2} 𝕜 (DivisionSemiring.toSemiring.{u_2} 𝕜 (DivisionRing.toDivisionSemiring.{u_2} 𝕜 (NormedDivisionRing.toDivisionRing.{u_2} 𝕜 inst._@.Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic.404174004._hygCtx._hyg.50))))) (ESeminormedAddMonoid.toAddMonoid.{u_5} E (UniformSpace.toTopologicalSpace.{u_5} E (PseudoMetricSpace.toUniformSpace.{u_5} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_5} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_5} E inst._@.Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic.404174004._hygCtx._hyg.9)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u_5} E (UniformSpace.toTopologicalSpace.{u_5} E (PseudoMetricSpace.toUniformSpace.{u_5} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_5} E 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(DivisionSemiring.toSemiring.{u_2} 𝕜 (DivisionRing.toDivisionSemiring.{u_2} 𝕜 (NormedDivisionRing.toDivisionRing.{u_2} 𝕜 inst._@.Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic.404174004._hygCtx._hyg.50))) (ESeminormedAddCommMonoid.toAddCommMonoid.{u_5} E (UniformSpace.toTopologicalSpace.{u_5} E (PseudoMetricSpace.toUniformSpace.{u_5} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_5} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_5} E inst._@.Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic.404174004._hygCtx._hyg.9)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u_5} E (UniformSpace.toTopologicalSpace.{u_5} E (PseudoMetricSpace.toUniformSpace.{u_5} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_5} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_5} E inst._@.Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic.404174004._hygCtx._hyg.9)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u_5} E inst._@.Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic.404174004._hygCtx._hyg.9))) inst._@.Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic.404174004._hygCtx._hyg.53))))) r (intervalIntegral.{u_5} E inst._@.Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic.404174004._hygCtx._hyg.9 inst._@.Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic.404174004._hygCtx._hyg.24 (fun (x : Real) => f x) a b μ)) (intervalIntegral.{u_5} E inst._@.Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic.404174004._hygCtx._hyg.9 inst._@.Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic.404174004._hygCtx._hyg.24 (fun (x : Real) => HSMul.hSMul.{u_2, u_5, u_5} 𝕜 E E (instHSMul.{u_2, u_5} 𝕜 E (SMulZeroClass.toSMul.{u_2, u_5} 𝕜 E (AddZero.toZero.{u_5} E (AddZeroClass.toAddZero.{u_5} E (AddMonoid.toAddZeroClass.{u_5} E (ESeminormedAddMonoid.toAddMonoid.{u_5} E (UniformSpace.toTopologicalSpace.{u_5} E (PseudoMetricSpace.toUniformSpace.{u_5} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_5} 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u_5} 𝕜 E (AddMonoid.toAddZeroClass.{u_5} E (ESeminormedAddMonoid.toAddMonoid.{u_5} E (UniformSpace.toTopologicalSpace.{u_5} E (PseudoMetricSpace.toUniformSpace.{u_5} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_5} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_5} E inst._@.Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic.404174004._hygCtx._hyg.9)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u_5} E (UniformSpace.toTopologicalSpace.{u_5} E (PseudoMetricSpace.toUniformSpace.{u_5} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_5} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_5} E inst._@.Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic.404174004._hygCtx._hyg.9)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u_5} E (UniformSpace.toTopologicalSpace.{u_5} E (PseudoMetricSpace.toUniformSpace.{u_5} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_5} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_5} E 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(PseudoMetricSpace.toUniformSpace.{u_5} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_5} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_5} E inst._@.Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic.404174004._hygCtx._hyg.9)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u_5} E (UniformSpace.toTopologicalSpace.{u_5} E (PseudoMetricSpace.toUniformSpace.{u_5} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_5} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_5} E inst._@.Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic.404174004._hygCtx._hyg.9)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u_5} E inst._@.Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic.404174004._hygCtx._hyg.9)))) (Module.toDistribMulAction.{u_2, u_5} 𝕜 E (DivisionSemiring.toSemiring.{u_2} 𝕜 (DivisionRing.toDivisionSemiring.{u_2} 𝕜 (NormedDivisionRing.toDivisionRing.{u_2} 𝕜 inst._@.Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic.404174004._hygCtx._hyg.50))) (ESeminormedAddCommMonoid.toAddCommMonoid.{u_5} E (UniformSpace.toTopologicalSpace.{u_5} E (PseudoMetricSpace.toUniformSpace.{u_5} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_5} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_5} E inst._@.Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic.404174004._hygCtx._hyg.9)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u_5} E (UniformSpace.toTopologicalSpace.{u_5} E (PseudoMetricSpace.toUniformSpace.{u_5} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_5} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_5} E inst._@.Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic.404174004._hygCtx._hyg.9)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u_5} E inst._@.Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic.404174004._hygCtx._hyg.9))) inst._@.Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic.404174004._hygCtx._hyg.53))))) r (f x)) a b μ)","typeFull":"∀ {𝕜 : Type u_2} {E : Type u_5} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {a b : ℝ}\n  {μ : MeasureTheory.Measure ℝ} [inst_2 : NormedDivisionRing 𝕜] [inst_3 : Module 𝕜 E] [NormSMulClass 𝕜 E]\n  [SMulCommClass ℝ 𝕜 E] (r : 𝕜) (f : ℝ → E), r • ∫ (x : ℝ) in a..b, f x ∂μ = ∫ (x : ℝ) in a..b, r • f x ∂μ","typeReadable":"∀ {𝕜 : Type u_2} {E : Type u_5} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {a b : ℝ}\n  {μ : MeasureTheory.Measure ℝ} [inst_2 : NormedDivisionRing 𝕜] [inst_3 : Module 𝕜 E] [NormSMulClass 𝕜 E]\n  [SMulCommClass ℝ 𝕜 E] (r : 𝕜) (f : ℝ → E), r • ∫ (x : ℝ) in a..b, f x ∂μ = ∫ (x : ℝ) in a..b, r • f x ∂μ","typeReferences":[["NormedDivisionRing"],["PseudoMetricSpace","toUniformSpace"],["Module"],["ESeminormedAddCommMonoid","toESeminormedAddMonoid"],["UniformSpace","toTopologicalSpace"],["SMulZeroClass","toSMul"],["MeasureTheory","Measure"],["DivisionRing","toDivisionSemiring"],["NormedSpace"],["MonoidWithZero","toMonoid"],["instHSMul"],["NormedAddCommGroup"],["Eq"],["DistribSMul","toSMulZeroClass"],["ESeminormedAddMonoid","toAddMonoid"],["NormedAddCommGroup","toENormedAddCommMonoid"],["NormedSpace","toModule"],["Real"],["SeminormedAddCommGroup","toPseudoMetricSpace"],["DistribMulAction","toDistribSMul"],["intervalIntegral"],["Semiring","toMonoidWithZero"],["NormedDivisionRing","toNorm"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["Real","semiring"],["SMulCommClass"],["Module","toDistribMulAction"],["Real","normedField"],["Real","instMonoid"],["Real","measurableSpace"],["HSMul","hSMul"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["NormedDivisionRing","toDivisionRing"],["NormedAddCommGroup","toNorm"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["NormedAddCommGroup","toSeminormedAddCommGroup"],["NormSMulClass"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["PseudoMetricSpace","toUniformSpace"],["ESeminormedAddCommMonoid","toESeminormedAddMonoid"],["UniformSpace","toTopologicalSpace"],["SMulZeroClass","toSMul"],["DivisionRing","toDivisionSemiring"],["MonoidWithZero","toMonoid"],["Eq","symm"],["instHSMul"],["intervalIntegral","integral_smul"],["DistribSMul","toSMulZeroClass"],["ESeminormedAddMonoid","toAddMonoid"],["NormedAddCommGroup","toENormedAddCommMonoid"],["SeminormedAddCommGroup","toPseudoMetricSpace"],["DistribMulAction","toDistribSMul"],["intervalIntegral"],["Semiring","toMonoidWithZero"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["HSMul","hSMul"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["NormedDivisionRing","toDivisionRing"],["NormedAddCommGroup","toSeminormedAddCommGroup"],["AddZero","toZero"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Complex","integral_boundary_rect_of_continuousOn_of_hasFDerivAt_real"],"typeFallback":"forall {E : Type.{u}} [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3 : NormedAddCommGroup.{u} E] [inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.6 : NormedSpace.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)] (f : Complex -> E) (f' : Complex -> (ContinuousLinearMap.{0, 0, 0, u} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))) E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3))) (NormedSpace.toModule.{0, 0} Real Complex Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (InnerProductSpace.toNormedSpace.{0, 0} Real Complex Real.instRCLike (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) instInnerProductSpaceRealComplex)) (NormedSpace.toModule.{0, u} Real E Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3) (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.6)))) (z : Complex) (w : Complex), (ContinuousOn.{0, u} Complex E (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) f (Complex.reProdIm (Set.uIcc.{0} Real Real.lattice (Complex.re z) (Complex.re w)) (Set.uIcc.{0} Real Real.lattice (Complex.im z) (Complex.im w)))) -> (forall (x : Complex), (Membership.mem.{0, 0} Complex (Set.{0} Complex) (Set.instMembership.{0} Complex) (Complex.reProdIm (Set.Ioo.{0} Real Real.instPreorder (Min.min.{0} Real Real.instMin (Complex.re z) (Complex.re w)) (Max.max.{0} Real Real.instMax (Complex.re z) (Complex.re w))) (Set.Ioo.{0} Real Real.instPreorder (Min.min.{0} Real Real.instMin (Complex.im z) (Complex.im w)) (Max.max.{0} Real Real.instMax (Complex.im z) (Complex.im w)))) x) -> (HasFDerivAt.{0, 0, u} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Complex Complex.addCommGroup (NormedSpace.toModule.{0, 0} Real Complex Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (InnerProductSpace.toNormedSpace.{0, 0} Real Complex Real.instRCLike (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) instInnerProductSpaceRealComplex)) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) E (NormedAddCommGroup.toAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3) (NormedSpace.toModule.{0, u} Real E Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3) (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.6)) (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) f (f' x) x)) -> (MeasureTheory.IntegrableOn.{0, u} Complex E (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Complex (measureSpaceOfInnerProductSpace.{0} Complex Complex.instNormedAddCommGroup instInnerProductSpaceRealComplex (FiniteDimensional.rclike_to_real.{0} Complex Complex.instRCLike) Complex.measurableSpace Complex.borelSpace)) (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (SeminormedAddGroup.toContinuousENorm.{u} E (SeminormedAddCommGroup.toSeminormedAddGroup.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3))) (fun (z : Complex) => HSub.hSub.{u, u, u} E E E (instHSub.{u} E (SubNegMonoid.toSub.{u} E (AddGroup.toSubNegMonoid.{u} E (NormedAddGroup.toAddGroup.{u} E (NormedAddCommGroup.toNormedAddGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3))))) (HSMul.hSMul.{0, u, u} Complex E E (instHSMul.{0, u} Complex E (SMulZeroClass.toSMul.{0, u} Complex E (AddZero.toZero.{u} E (AddZeroClass.toAddZero.{u} E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E 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inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.6)))))) Complex.I (DFunLike.coe.{succ u, 1, succ u} (ContinuousLinearMap.{0, 0, 0, u} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) 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(NormedSpace.toModule.{0, u} Real E Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3) (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.6))) (f' z) (OfNat.ofNat.{0} Complex 1 (One.toOfNat1.{0} Complex Complex.instOne)))) (DFunLike.coe.{succ u, 1, succ u} (ContinuousLinearMap.{0, 0, 0, u} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) 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(NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3))) (NormedSpace.toModule.{0, 0} Real Complex Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (InnerProductSpace.toNormedSpace.{0, 0} Real Complex Real.instRCLike (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) instInnerProductSpaceRealComplex)) (NormedSpace.toModule.{0, u} Real E Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3) (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.6))) Complex (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Complex) => E) (ContinuousLinearMap.funLike.{0, 0, 0, u} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))) E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3))) (NormedSpace.toModule.{0, 0} Real Complex Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (InnerProductSpace.toNormedSpace.{0, 0} Real Complex Real.instRCLike (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) instInnerProductSpaceRealComplex)) (NormedSpace.toModule.{0, u} Real E Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3) (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.6))) (f' z) Complex.I)) (Complex.reProdIm (Set.uIcc.{0} Real Real.lattice (Complex.re z) (Complex.re w)) (Set.uIcc.{0} Real Real.lattice (Complex.im z) (Complex.im w))) (MeasureTheory.MeasureSpace.volume.{0} Complex (measureSpaceOfInnerProductSpace.{0} Complex Complex.instNormedAddCommGroup instInnerProductSpaceRealComplex (FiniteDimensional.rclike_to_real.{0} Complex Complex.instRCLike) Complex.measurableSpace Complex.borelSpace))) -> (Eq.{succ u} E (HSub.hSub.{u, u, u} E E E (instHSub.{u} E (SubNegMonoid.toSub.{u} E (AddGroup.toSubNegMonoid.{u} E (NormedAddGroup.toAddGroup.{u} E (NormedAddCommGroup.toNormedAddGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3))))) (HAdd.hAdd.{u, u, u} E E E (instHAdd.{u} E (AddCommMagma.toAdd.{u} E (AddCommSemigroup.toAddCommMagma.{u} E (AddCommMonoid.toAddCommSemigroup.{u} E (ESeminormedAddCommMonoid.toAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3))))))) (HSub.hSub.{u, u, u} E E E (instHSub.{u} E (SubNegMonoid.toSub.{u} E (AddGroup.toSubNegMonoid.{u} E (NormedAddGroup.toAddGroup.{u} E (NormedAddCommGroup.toNormedAddGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3))))) (intervalIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3 (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.6) (fun (x : Real) => f (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) (Complex.ofReal x) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.ofReal (Complex.im z)) Complex.I))) (Complex.re z) (Complex.re w) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace)) (intervalIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3 (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.6) (fun (x : Real) => f (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) (Complex.ofReal x) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.ofReal (Complex.im w)) Complex.I))) (Complex.re z) (Complex.re w) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace))) (HSMul.hSMul.{0, u, u} Complex E E (instHSMul.{0, u} Complex E (SMulZeroClass.toSMul.{0, u} Complex E (AddZero.toZero.{u} E (AddZeroClass.toAddZero.{u} E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3))))))) (DistribSMul.toSMulZeroClass.{0, u} Complex E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3))))) (DistribMulAction.toDistribSMul.{0, u} Complex E (MonoidWithZero.toMonoid.{0} Complex (Semiring.toMonoidWithZero.{0} Complex Complex.instSemiring)) (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (Module.toDistribMulAction.{0, u} Complex E Complex.instSemiring (ESeminormedAddCommMonoid.toAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3))) (NormedSpace.toModule.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.6)))))) Complex.I (intervalIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3 (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.6) (fun (y : Real) => f (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) (Complex.ofReal (Complex.re w)) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.ofReal y) Complex.I))) (Complex.im z) (Complex.im w) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace)))) (HSMul.hSMul.{0, u, u} Complex E E (instHSMul.{0, u} Complex E (SMulZeroClass.toSMul.{0, u} Complex E (AddZero.toZero.{u} E (AddZeroClass.toAddZero.{u} E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3))))))) (DistribSMul.toSMulZeroClass.{0, u} Complex E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3))))) (DistribMulAction.toDistribSMul.{0, u} Complex E (MonoidWithZero.toMonoid.{0} Complex (Semiring.toMonoidWithZero.{0} Complex Complex.instSemiring)) (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (Module.toDistribMulAction.{0, u} Complex E Complex.instSemiring (ESeminormedAddCommMonoid.toAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3))) (NormedSpace.toModule.{0, u} Complex E Complex.instNormedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.6)))))) Complex.I (intervalIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3 (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.6) (fun (y : Real) => f (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) (Complex.ofReal (Complex.re z)) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.ofReal y) Complex.I))) (Complex.im z) (Complex.im w) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace)))) (intervalIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3 (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.6) (fun (x : Real) => intervalIntegral.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3 (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.6) (fun (y : Real) => HSub.hSub.{u, u, u} E E E (instHSub.{u} E (SubNegMonoid.toSub.{u} E (AddGroup.toSubNegMonoid.{u} E (NormedAddGroup.toAddGroup.{u} E (NormedAddCommGroup.toNormedAddGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3))))) (HSMul.hSMul.{0, u, u} Complex E E (instHSMul.{0, u} Complex E (SMulZeroClass.toSMul.{0, u} Complex E (AddZero.toZero.{u} E (AddZeroClass.toAddZero.{u} E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3))))))) (DistribSMul.toSMulZeroClass.{0, u} Complex E (AddMonoid.toAddZeroClass.{u} E (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3))))) (DistribMulAction.toDistribSMul.{0, u} Complex E (MonoidWithZero.toMonoid.{0} Complex (Semiring.toMonoidWithZero.{0} Complex Complex.instSemiring)) (ESeminormedAddMonoid.toAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoMetricSpace.toUniformSpace.{u} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E 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(NormedSpace.toModule.{0, u} Real E Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3) (NormedSpace.complexToReal.{u} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u} E inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Complex.CauchyIntegral.1433764529._hygCtx._hyg.6))) (f' (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) (Complex.ofReal x) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.ofReal y) Complex.I))) Complex.I)) (Complex.im z) (Complex.im w) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace)) (Complex.re z) (Complex.re w) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace)))","typeFull":"∀ {E : Type u} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] (f : ℂ → E) (f' : ℂ → ℂ →L[ℝ] E) (z w : ℂ),\n  ContinuousOn f (Set.uIcc z.re 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diff --git a/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.CategoryTheory.Sites.Coverage.sym.json b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.CategoryTheory.Sites.Coverage.sym.json
new file mode 100644
index 0000000000000000000000000000000000000000..637548fed86521bc62d5952acf96932e1bd2a99a
--- /dev/null
+++ b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.CategoryTheory.Sites.Coverage.sym.json
@@ -0,0 +1 @@
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inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 X) (Lattice.toSemilatticeSup.{max v_1 u_1} (CategoryTheory.Sieve.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 X) (CompleteLattice.toLattice.{max v_1 u_1} (CategoryTheory.Sieve.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 X) (CategoryTheory.Sieve.instCompleteLattice.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 X)))))) (CompleteLattice.toBoundedOrder.{max v_1 u_1} (CategoryTheory.Sieve.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 X) (CategoryTheory.Sieve.instCompleteLattice.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 X))))) (CategoryTheory.Coverage.Saturate.top.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 K X) (CategoryTheory.Coverage.Saturate.below.top.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 K motive X)) -> (forall (X : C) (R : CategoryTheory.Sieve.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 X) (S : CategoryTheory.Sieve.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 X) (a._@._internal._hyg.0 : CategoryTheory.Coverage.Saturate.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 K X R) (a_1._@._internal._hyg.0 : forall {{Y : C}} {{f : Quiver.Hom.{v_1, u_1} C (CategoryTheory.CategoryStruct.toQuiver.{v_1, u_1} C (CategoryTheory.Category.toCategoryStruct.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12)) Y X}}, (CategoryTheory.Sieve.arrows.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 X R Y f) -> (CategoryTheory.Coverage.Saturate.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 K Y (CategoryTheory.Sieve.pullback.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 X Y f S))) (ih._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.86 : CategoryTheory.Coverage.Saturate.below.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 K motive X R a._@._internal._hyg.0) (a_ih._@._internal._hyg.0 : motive X R a._@._internal._hyg.0) (ih._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.87 : forall {{Y : C}} {{f : Quiver.Hom.{v_1, u_1} C (CategoryTheory.CategoryStruct.toQuiver.{v_1, u_1} C (CategoryTheory.Category.toCategoryStruct.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12)) Y X}} (a._@._internal._hyg.0 : CategoryTheory.Sieve.arrows.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 X R Y f), CategoryTheory.Coverage.Saturate.below.{v_1, u_1} C 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(X : C) (S : CategoryTheory.Presieve X) (hS : S ∈ K.coverings X), motive_1 ⋯ ⋯) →\n    (∀ (X : C), motive_1 ⋯ ⋯) →\n      (∀ (X : C) (R S : CategoryTheory.Sieve X) (a : K.Saturate X R)\n          (a_1 : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → K.Saturate Y (CategoryTheory.Sieve.pullback f S))\n          (ih : CategoryTheory.Coverage.Saturate.below a) (a_ih : motive X R a)\n          (ih_1 : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (a : R.arrows f), CategoryTheory.Coverage.Saturate.below ⋯)\n          (a_ih_1 : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (a : R.arrows f), motive Y (CategoryTheory.Sieve.pullback f S) ⋯),\n          motive_1 ⋯ ⋯) →\n        motive_1 t t_1","typeReadable":"∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] {K : CategoryTheory.Coverage C}\n  {motive : (X : C) → (a : CategoryTheory.Sieve X) → K.Saturate X a → Prop}\n  {motive_1 :\n    {X : C} → {a : CategoryTheory.Sieve X} → (t : K.Saturate X a) → CategoryTheory.Coverage.Saturate.below t → Prop}\n  {X : C} {a : CategoryTheory.Sieve X} {t : K.Saturate X a} (t_1 : CategoryTheory.Coverage.Saturate.below t),\n  (∀ (X : C) (S : CategoryTheory.Presieve X) (hS : S ∈ K.coverings X), motive_1 ⋯ ⋯) →\n    (∀ (X : C), motive_1 ⋯ ⋯) →\n      (∀ (X : C) (R S : CategoryTheory.Sieve X) (a : K.Saturate X R)\n          (a_1 : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → K.Saturate Y (CategoryTheory.Sieve.pullback f S))\n          (ih : CategoryTheory.Coverage.Saturate.below a) (a_ih : motive X R a)\n          (ih_1 : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (a : R.arrows f), CategoryTheory.Coverage.Saturate.below ⋯)\n          (a_ih_1 : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (a : R.arrows f), motive Y (CategoryTheory.Sieve.pullback f S) ⋯),\n          motive_1 ⋯ ⋯) →\n        motive_1 t t_1","typeReferences":[["Lattice","toSemilatticeSup"],["PartialOrder","toPreorder"],["CategoryTheory","Sieve","instCompleteLattice"],["CategoryTheory","Sieve","generate"],["Membership","mem"],["CategoryTheory","Category"],["SemilatticeSup","toPartialOrder"],["CategoryTheory","Coverage","Saturate","of"],["CategoryTheory","Coverage","Saturate","transitive"],["Quiver","Hom"],["BoundedOrder","toOrderTop"],["CategoryTheory","Coverage","Saturate","below"],["CategoryTheory","Coverage","Saturate","top"],["CategoryTheory","Coverage","Saturate","below","top"],["CategoryTheory","Sieve"],["Preorder","toLE"],["CategoryTheory","Precoverage","coverings"],["Set"],["CategoryTheory","Coverage","Saturate","below","transitive"],["CompleteLattice","toLattice"],["CategoryTheory","Sieve","pullback"],["CategoryTheory","Category","toCategoryStruct"],["CategoryTheory","Coverage","Saturate","below","of"],["Set","instMembership"],["CategoryTheory","CategoryStruct","toQuiver"],["Top","top"],["CategoryTheory","Coverage","toPrecoverage"],["CategoryTheory","Sieve","arrows"],["CategoryTheory","Presieve"],["CategoryTheory","Coverage","Saturate"],["CategoryTheory","Coverage"],["OrderTop","toTop"],["CompleteLattice","toBoundedOrder"]],"valueReferences":[["CategoryTheory","Coverage","Saturate","below","rec"]]},{"isProp":true,"kind":"theorem","name":["CategoryTheory","Coverage","instPartialOrder","_proof_2"],"typeFallback":"forall {C : Type.{u_2}} [inst._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.4 : CategoryTheory.Category.{u_1, u_2} C] (x._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.33 : CategoryTheory.Coverage.{u_1, u_2} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.4) (x._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.35 : C), LE.le.{max u_1 u_2} (Set.{max u_2 u_1} (CategoryTheory.Presieve.{u_1, u_2} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.4 x._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.35)) (Preorder.toLE.{max u_1 u_2} (Set.{max u_2 u_1} (CategoryTheory.Presieve.{u_1, u_2} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.4 x._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.35)) (PartialOrder.toPreorder.{max u_1 u_2} (Set.{max u_2 u_1} (CategoryTheory.Presieve.{u_1, u_2} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.4 x._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.35)) (ConditionallyCompletePartialOrderSup.toPartialOrder.{max u_1 u_2} (Set.{max u_2 u_1} (CategoryTheory.Presieve.{u_1, u_2} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.4 x._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.35)) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{max u_1 u_2} (Set.{max u_2 u_1} (CategoryTheory.Presieve.{u_1, u_2} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.4 x._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.35)) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{max u_1 u_2} (Set.{max u_2 u_1} (CategoryTheory.Presieve.{u_1, u_2} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.4 x._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.35)) (CompleteLattice.toConditionallyCompleteLattice.{max u_1 u_2} (Set.{max u_2 u_1} (CategoryTheory.Presieve.{u_1, u_2} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.4 x._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.35)) (CompleteBooleanAlgebra.toCompleteLattice.{max u_1 u_2} (Set.{max u_2 u_1} (CategoryTheory.Presieve.{u_1, u_2} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.4 x._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.35)) (CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra.{max u_1 u_2} (Set.{max u_2 u_1} (CategoryTheory.Presieve.{u_1, u_2} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.4 x._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.35)) (Set.instCompleteAtomicBooleanAlgebra.{max u_1 u_2} (CategoryTheory.Presieve.{u_1, u_2} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.4 x._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.35)))))))))) (CategoryTheory.Precoverage.coverings.{u_1, u_2} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.4 (CategoryTheory.Coverage.toPrecoverage.{u_1, u_2} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.4 x._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.33) x._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.35) (CategoryTheory.Precoverage.coverings.{u_1, u_2} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.4 (CategoryTheory.Coverage.toPrecoverage.{u_1, u_2} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.4 x._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.33) x._@.Mathlib.CategoryTheory.Sites.Coverage.3983714632._hygCtx._hyg.35)","typeFull":"∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] (x : CategoryTheory.Coverage C) (x_1 : C),\n  x.coverings x_1 ≤ x.coverings x_1","typeReadable":"∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] (x : CategoryTheory.Coverage C) (x_1 : C),\n  x.coverings x_1 ≤ x.coverings x_1","typeReferences":[["PartialOrder","toPreorder"],["CategoryTheory","Precoverage","coverings"],["CompleteAtomicBooleanAlgebra","toCompleteBooleanAlgebra"],["Set"],["CategoryTheory","Category"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["CompleteBooleanAlgebra","toCompleteLattice"],["ConditionallyCompletePartialOrderSup","toPartialOrder"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderSup"],["Set","instCompleteAtomicBooleanAlgebra"],["LE","le"],["CategoryTheory","Coverage","toPrecoverage"],["CategoryTheory","Coverage"],["CategoryTheory","Presieve"],["Preorder","toLE"],["CompleteLattice","toConditionallyCompleteLattice"]],"valueReferences":[["le_refl"],["PartialOrder","toPreorder"],["CompleteAtomicBooleanAlgebra","toCompleteBooleanAlgebra"],["CategoryTheory","Precoverage","coverings"],["Set"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["CompleteBooleanAlgebra","toCompleteLattice"],["ConditionallyCompletePartialOrderSup","toPartialOrder"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderSup"],["Set","instCompleteAtomicBooleanAlgebra"],["CategoryTheory","Coverage","toPrecoverage"],["CategoryTheory","Presieve"],["CompleteLattice","toConditionallyCompleteLattice"]]},{"isProp":true,"kind":"theorem","name":["CategoryTheory","Coverage","Saturate","hcongr_5"],"typeFallback":"forall (C : Type.{u_1}) (C' : Type.{u_1}), (Eq.{succ (succ u_1)} Type.{u_1} C C') -> (forall (inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 : CategoryTheory.Category.{v_1, u_1} C) (inst'._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 : CategoryTheory.Category.{v_1, u_1} C'), (HEq.{succ (max u_1 (succ v_1))} (CategoryTheory.Category.{v_1, u_1} C) inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 (CategoryTheory.Category.{v_1, u_1} C') inst'._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12) -> (forall (K : CategoryTheory.Coverage.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12) (K' : CategoryTheory.Coverage.{v_1, u_1} C' inst'._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12), (HEq.{max (succ u_1) (succ v_1)} (CategoryTheory.Coverage.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12) K (CategoryTheory.Coverage.{v_1, u_1} C' inst'._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12) K') -> (forall (X : C) (X' : C'), (HEq.{succ u_1} C X C' X') -> (forall (a._@._internal._hyg.0 : CategoryTheory.Sieve.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 X) (a'._@._internal._hyg.0 : CategoryTheory.Sieve.{v_1, u_1} C' inst'._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 X'), (HEq.{max (succ u_1) (succ v_1)} (CategoryTheory.Sieve.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 X) a._@._internal._hyg.0 (CategoryTheory.Sieve.{v_1, u_1} C' inst'._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 X') a'._@._internal._hyg.0) -> (HEq.{1} Prop (CategoryTheory.Coverage.Saturate.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 K X a._@._internal._hyg.0) Prop (CategoryTheory.Coverage.Saturate.{v_1, u_1} C' 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(h._@.Mathlib.CategoryTheory.Sites.Coverage.1055603703._hygCtx.34.Mathlib.CategoryTheory.Sites.Coverage.1055603703._hygCtx._hyg.47 : CategoryTheory.Presieve.pullbackArrows.{u_1, u_2} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.1055603703._hygCtx._hyg.4 Y X f R inst._@.Mathlib.CategoryTheory.Sites.Coverage.1055603703._hygCtx._hyg.19 Z._@.Mathlib.CategoryTheory.Sites.Coverage.1055603703._hygCtx._hyg.63 g._@.Mathlib.CategoryTheory.Sites.Coverage.1055603703._hygCtx._hyg.62), (forall (W : C) (b : Quiver.Hom.{u_1, u_2} C (CategoryTheory.CategoryStruct.toQuiver.{u_1, u_2} C (CategoryTheory.Category.toCategoryStruct.{u_1, u_2} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.1055603703._hygCtx._hyg.4)) W Y) (hb : R W b), motive ([mdata _inaccessible:1 CategoryTheory.Limits.pullback.{u_1, u_2} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.1055603703._hygCtx._hyg.4 W X Y b f (CategoryTheory.Presieve.hasPullback.{u_1, u_2} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.1055603703._hygCtx._hyg.4 Y R X f inst._@.Mathlib.CategoryTheory.Sites.Coverage.1055603703._hygCtx._hyg.19 W b hb)]) ([mdata _inaccessible:1 CategoryTheory.Limits.pullback.snd.{u_1, u_2} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.1055603703._hygCtx._hyg.4 W X Y b f (CategoryTheory.Presieve.hasPullback.{u_1, u_2} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.1055603703._hygCtx._hyg.4 Y R X f inst._@.Mathlib.CategoryTheory.Sites.Coverage.1055603703._hygCtx._hyg.19 W b hb)]) (CategoryTheory.Presieve.pullbackArrows.mk.{u_1, u_2} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.1055603703._hygCtx._hyg.4 Y X f R inst._@.Mathlib.CategoryTheory.Sites.Coverage.1055603703._hygCtx._hyg.19 W b hb)) -> (motive Z._@.Mathlib.CategoryTheory.Sites.Coverage.1055603703._hygCtx._hyg.63 g._@.Mathlib.CategoryTheory.Sites.Coverage.1055603703._hygCtx._hyg.62 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inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 K X R) (a_1._@._internal._hyg.0 : forall {{Y : C}} {{f : Quiver.Hom.{v_1, u_1} C (CategoryTheory.CategoryStruct.toQuiver.{v_1, u_1} C (CategoryTheory.Category.toCategoryStruct.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12)) Y X}}, (CategoryTheory.Sieve.arrows.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 X R Y f) -> (CategoryTheory.Coverage.Saturate.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 K Y (CategoryTheory.Sieve.pullback.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12 X Y f S))), (motive X R a._@._internal._hyg.0) -> (forall {{Y : C}} {{f : Quiver.Hom.{v_1, u_1} C (CategoryTheory.CategoryStruct.toQuiver.{v_1, u_1} C (CategoryTheory.Category.toCategoryStruct.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2161496067._hygCtx._hyg.12)) Y X}} 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(Opposite.{succ u_3} C) (CategoryTheory.Category.opposite.{u_2, u_3} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2830332328._hygCtx._hyg.4) Type.{u_1} CategoryTheory.types.{u_1} P (Opposite.op.{succ u_3} C X)) (fun (t : CategoryTheory.Functor.obj.{u_2, u_1, u_3, succ u_1} (Opposite.{succ u_3} C) (CategoryTheory.Category.opposite.{u_2, u_3} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2830332328._hygCtx._hyg.4) Type.{u_1} CategoryTheory.types.{u_1} P (Opposite.op.{succ u_3} C X)) => CategoryTheory.Presieve.FamilyOfElements.IsAmalgamation.{u_1, u_2, u_3} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2830332328._hygCtx._hyg.4 P X S x t)))), (forall (left._@.Mathlib.CategoryTheory.Sites.Coverage.2830332328._hygCtx._hyg.247 : CategoryTheory.Presieve.IsSeparatedFor.{u_1, u_2, u_3} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2830332328._hygCtx._hyg.4 X P S) (h2' : forall (x : CategoryTheory.Presieve.FamilyOfElements.{u_1, u_2, u_3} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2830332328._hygCtx._hyg.4 X P S), (CategoryTheory.Presieve.FamilyOfElements.Compatible.{u_1, u_2, u_3} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2830332328._hygCtx._hyg.4 P X S x) -> (Exists.{succ u_1} (CategoryTheory.Functor.obj.{u_2, u_1, u_3, succ u_1} (Opposite.{succ u_3} C) (CategoryTheory.Category.opposite.{u_2, u_3} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2830332328._hygCtx._hyg.4) Type.{u_1} CategoryTheory.types.{u_1} P (Opposite.op.{succ u_3} C X)) (fun (t : CategoryTheory.Functor.obj.{u_2, u_1, u_3, succ u_1} (Opposite.{succ u_3} C) (CategoryTheory.Category.opposite.{u_2, u_3} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2830332328._hygCtx._hyg.4) Type.{u_1} CategoryTheory.types.{u_1} P (Opposite.op.{succ u_3} C X)) => CategoryTheory.Presieve.FamilyOfElements.IsAmalgamation.{u_1, u_2, u_3} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2830332328._hygCtx._hyg.4 P X S x t))), motive (And.intro (CategoryTheory.Presieve.IsSeparatedFor.{u_1, u_2, u_3} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2830332328._hygCtx._hyg.4 X P S) (forall (x : CategoryTheory.Presieve.FamilyOfElements.{u_1, u_2, u_3} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2830332328._hygCtx._hyg.4 X P S), (CategoryTheory.Presieve.FamilyOfElements.Compatible.{u_1, u_2, u_3} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2830332328._hygCtx._hyg.4 P X S x) -> (Exists.{succ u_1} (CategoryTheory.Functor.obj.{u_2, u_1, u_3, succ u_1} (Opposite.{succ u_3} C) (CategoryTheory.Category.opposite.{u_2, u_3} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2830332328._hygCtx._hyg.4) Type.{u_1} CategoryTheory.types.{u_1} P (Opposite.op.{succ u_3} C X)) (fun (t : CategoryTheory.Functor.obj.{u_2, u_1, u_3, succ u_1} (Opposite.{succ u_3} C) (CategoryTheory.Category.opposite.{u_2, u_3} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2830332328._hygCtx._hyg.4) Type.{u_1} CategoryTheory.types.{u_1} P (Opposite.op.{succ u_3} C X)) => CategoryTheory.Presieve.FamilyOfElements.IsAmalgamation.{u_1, u_2, u_3} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2830332328._hygCtx._hyg.4 P X S x t))) left._@.Mathlib.CategoryTheory.Sites.Coverage.2830332328._hygCtx._hyg.247 h2')) -> (motive hS._@.Mathlib.CategoryTheory.Sites.Coverage.2830332328._hygCtx._hyg.238)","typeFull":"∀ {C : Type u_3} [inst : CategoryTheory.Category.{u_2, u_3} C] {X : C} {S : CategoryTheory.Presieve X}\n  (P : CategoryTheory.Functor Cᵒᵖ (Type u_1))\n  (motive :\n    (CategoryTheory.Presieve.IsSeparatedFor P S ∧\n        ∀ (x : CategoryTheory.Presieve.FamilyOfElements P S), x.Compatible → ∃ t, x.IsAmalgamation t) →\n      Prop)\n  (hS :\n    CategoryTheory.Presieve.IsSeparatedFor P S ∧\n      ∀ (x : CategoryTheory.Presieve.FamilyOfElements P S), x.Compatible → ∃ t, x.IsAmalgamation t),\n  (∀ (left : CategoryTheory.Presieve.IsSeparatedFor P S)\n      (h2' : ∀ (x : CategoryTheory.Presieve.FamilyOfElements P S), x.Compatible → ∃ t, x.IsAmalgamation t), motive ⋯) →\n    motive hS","typeReadable":"∀ {C : Type u_3} [inst : CategoryTheory.Category.{u_2, u_3} C] {X : C} {S : CategoryTheory.Presieve X}\n  (P : CategoryTheory.Functor Cᵒᵖ (Type u_1))\n  (motive :\n    (CategoryTheory.Presieve.IsSeparatedFor P S ∧\n        ∀ (x : CategoryTheory.Presieve.FamilyOfElements P S), x.Compatible → ∃ t, x.IsAmalgamation t) →\n      Prop)\n  (hS :\n    CategoryTheory.Presieve.IsSeparatedFor P S ∧\n      ∀ (x : CategoryTheory.Presieve.FamilyOfElements P S), x.Compatible → ∃ t, x.IsAmalgamation t),\n  (∀ (left : CategoryTheory.Presieve.IsSeparatedFor P S)\n      (h2' : ∀ (x : CategoryTheory.Presieve.FamilyOfElements P S), x.Compatible → ∃ t, x.IsAmalgamation t), motive ⋯) →\n    motive 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{C : Type.{u_1}} [inst._@.Mathlib.CategoryTheory.Sites.Coverage.1241641679._hygCtx._hyg.12 : CategoryTheory.Category.{v_1, u_1} C] [inst._@.Mathlib.CategoryTheory.Sites.Coverage.1241641679._hygCtx._hyg.64 : SizeOf.{succ u_1} C] (toPrecoverage : CategoryTheory.Precoverage.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.1241641679._hygCtx._hyg.12) (pullback : forall {{X : C}} {{Y : C}} (f : Quiver.Hom.{v_1, u_1} C (CategoryTheory.CategoryStruct.toQuiver.{v_1, u_1} C (CategoryTheory.Category.toCategoryStruct.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.1241641679._hygCtx._hyg.12)) Y X) (S : CategoryTheory.Presieve.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.1241641679._hygCtx._hyg.12 X), (Membership.mem.{max v_1 u_1, max v_1 u_1} (CategoryTheory.Presieve.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.1241641679._hygCtx._hyg.12 X) (Set.{max u_1 v_1} (CategoryTheory.Presieve.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.1241641679._hygCtx._hyg.12 X)) (Set.instMembership.{max v_1 u_1} (CategoryTheory.Presieve.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.1241641679._hygCtx._hyg.12 X)) (CategoryTheory.Precoverage.coverings.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.1241641679._hygCtx._hyg.12 toPrecoverage X) S) -> (Exists.{max (succ v_1) (succ u_1)} (CategoryTheory.Presieve.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.1241641679._hygCtx._hyg.12 Y) (fun (T : CategoryTheory.Presieve.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.1241641679._hygCtx._hyg.12 Y) => And (Membership.mem.{max v_1 u_1, max v_1 u_1} (CategoryTheory.Presieve.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.1241641679._hygCtx._hyg.12 Y) (Set.{max u_1 v_1} (CategoryTheory.Presieve.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.1241641679._hygCtx._hyg.12 Y)) (Set.instMembership.{max v_1 u_1} (CategoryTheory.Presieve.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.1241641679._hygCtx._hyg.12 Y)) (CategoryTheory.Precoverage.coverings.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.1241641679._hygCtx._hyg.12 toPrecoverage Y) T) (CategoryTheory.Presieve.FactorsThruAlong.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.1241641679._hygCtx._hyg.12 X Y T S f)))), Eq.{1} Nat (SizeOf.sizeOf.{max (succ u_1) (succ v_1)} (CategoryTheory.Coverage.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.1241641679._hygCtx._hyg.12) (CategoryTheory.Coverage._sizeOf_inst.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.1241641679._hygCtx._hyg.12 inst._@.Mathlib.CategoryTheory.Sites.Coverage.1241641679._hygCtx._hyg.64) (CategoryTheory.Coverage.mk.{v_1, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.1241641679._hygCtx._hyg.12 toPrecoverage pullback)) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) (OfNat.ofNat.{0} Nat 1 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∈ toPrecoverage.coverings Y, T.FactorsThruAlong S f),\n  sizeOf { toPrecoverage := toPrecoverage, pullback := pullback } = 1 + sizeOf 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{C : Type.{u}} [inst._@.Mathlib.CategoryTheory.Sites.Hypercover.Zero.3359080014._hygCtx._hyg.9 : CategoryTheory.Category.{v, u} C] {J : CategoryTheory.Precoverage.{v, u} C inst._@.Mathlib.CategoryTheory.Sites.Hypercover.Zero.3359080014._hygCtx._hyg.9} {S : C} (toPreZeroHypercover : CategoryTheory.PreZeroHypercover.{w, v, u} C inst._@.Mathlib.CategoryTheory.Sites.Hypercover.Zero.3359080014._hygCtx._hyg.9 S) (toPreZeroHypercover_1 : CategoryTheory.PreZeroHypercover.{w, v, u} C inst._@.Mathlib.CategoryTheory.Sites.Hypercover.Zero.3359080014._hygCtx._hyg.9 S) (e_toPreZeroHypercover : Eq.{max (max (succ u) (succ v)) (succ (succ w))} (CategoryTheory.PreZeroHypercover.{w, v, u} C inst._@.Mathlib.CategoryTheory.Sites.Hypercover.Zero.3359080014._hygCtx._hyg.9 S) toPreZeroHypercover toPreZeroHypercover_1) (mem₀ : Membership.mem.{max u v, max u v} (CategoryTheory.Presieve.{v, u} C inst._@.Mathlib.CategoryTheory.Sites.Hypercover.Zero.3359080014._hygCtx._hyg.9 S) (Set.{max u v} 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inst._@.Mathlib.CategoryTheory.Sites.Hypercover.Zero.3359080014._hygCtx._hyg.9 J S) (CategoryTheory.PreZeroHypercover.presieve₀.{w, v, u} C inst._@.Mathlib.CategoryTheory.Sites.Hypercover.Zero.3359080014._hygCtx._hyg.9 S toPreZeroHypercover)) mem₀ toPreZeroHypercover_1 e_toPreZeroHypercover))","typeFull":"∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {J : CategoryTheory.Precoverage C} {S : C}\n  (toPreZeroHypercover toPreZeroHypercover_1 : CategoryTheory.PreZeroHypercover S)\n  (e_toPreZeroHypercover : toPreZeroHypercover = toPreZeroHypercover_1)\n  (mem₀ : toPreZeroHypercover.presieve₀ ∈ J.coverings S),\n  { toPreZeroHypercover := toPreZeroHypercover, mem₀ := mem₀ } =\n    { toPreZeroHypercover := toPreZeroHypercover_1, mem₀ := ⋯ }","typeReadable":"∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {J : CategoryTheory.Precoverage C} {S : C}\n  (toPreZeroHypercover toPreZeroHypercover_1 : CategoryTheory.PreZeroHypercover S)\n  (e_toPreZeroHypercover : toPreZeroHypercover = toPreZeroHypercover_1)\n  (mem₀ : toPreZeroHypercover.presieve₀ ∈ J.coverings S),\n  { toPreZeroHypercover := toPreZeroHypercover, mem₀ := mem₀ } =\n    { toPreZeroHypercover := toPreZeroHypercover_1, mem₀ := ⋯ }","typeReferences":[["CategoryTheory","PreZeroHypercover"],["CategoryTheory","Precoverage","coverings"],["Set"],["CategoryTheory","Category"],["Membership","mem"],["Set","instMembership"],["CategoryTheory","PreZeroHypercover","presieve₀"],["CategoryTheory","Precoverage","ZeroHypercover","mk"],["CategoryTheory","Precoverage","ZeroHypercover"],["Eq"],["CategoryTheory","Presieve"],["Eq","ndrec"],["CategoryTheory","Precoverage"]],"valueReferences":[["CategoryTheory","PreZeroHypercover"],["CategoryTheory","Precoverage","coverings"],["Set"],["Membership","mem"],["Set","instMembership"],["CategoryTheory","PreZeroHypercover","presieve₀"],["CategoryTheory","Precoverage","ZeroHypercover","mk"],["Eq","refl"],["CategoryTheory","Precoverage","ZeroHypercover"],["Eq","rec"],["CategoryTheory","Presieve"],["Eq"],["Eq","ndrec"]]},{"isProp":true,"kind":"theorem","name":["CategoryTheory","Coverage","mem_toGrothendieck_sieves_of_superset"],"typeFallback":"forall {C : Type.{u_1}} 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(ConditionallyCompletePartialOrderSup.toPartialOrder.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (CompleteLattice.toConditionallyCompleteLattice.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (CategoryTheory.instCompleteLatticePresieve.{u_1, u_2} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T))))))) R (CategoryTheory.Sieve.arrows.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T S)))) -> Prop) (x._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx.64.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.72 : Exists.{succ (max u_1 u_2)} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (fun (R : CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) => And (Membership.mem.{max u_1 u_2, max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (Set.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T)) (Set.instMembership.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T)) (CategoryTheory.Precoverage.coverings.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 (CategoryTheory.Pretopology.toPrecoverage.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.10 J) T) R) (LE.le.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (Preorder.toLE.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (PartialOrder.toPreorder.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (ConditionallyCompletePartialOrderSup.toPartialOrder.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (CompleteLattice.toConditionallyCompleteLattice.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (CategoryTheory.instCompleteLatticePresieve.{u_1, u_2} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T))))))) R (CategoryTheory.Sieve.arrows.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T S)))), (forall (R : CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (hR : Membership.mem.{max u_1 u_2, max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C 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T))))))) R (CategoryTheory.Sieve.arrows.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T S)), motive (Exists.intro.{max (succ u_1) (succ u_2)} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (fun (R : CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) => And (Membership.mem.{max u_1 u_2, max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (Set.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T)) (Set.instMembership.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T)) (CategoryTheory.Precoverage.coverings.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 (CategoryTheory.Pretopology.toPrecoverage.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.10 J) T) R) (LE.le.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (Preorder.toLE.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (PartialOrder.toPreorder.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (ConditionallyCompletePartialOrderSup.toPartialOrder.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (CompleteLattice.toConditionallyCompleteLattice.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (CategoryTheory.instCompleteLatticePresieve.{u_1, u_2} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T))))))) R (CategoryTheory.Sieve.arrows.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T S))) R (And.intro (Membership.mem.{max u_1 u_2, max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (Set.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T)) (Set.instMembership.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T)) (CategoryTheory.Precoverage.coverings.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 (CategoryTheory.Pretopology.toPrecoverage.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.10 J) T) R) (LE.le.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (Preorder.toLE.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (PartialOrder.toPreorder.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (ConditionallyCompletePartialOrderSup.toPartialOrder.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (CompleteLattice.toConditionallyCompleteLattice.{max u_1 u_2} (CategoryTheory.Presieve.{u_2, u_1} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T) (CategoryTheory.instCompleteLatticePresieve.{u_1, u_2} C inst._@.Mathlib.CategoryTheory.Sites.Coverage.2639499054._hygCtx._hyg.4 T))))))) R (CategoryTheory.Sieve.arrows.{u_2, u_1} C 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inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.27859897._hygCtx._hyg.4) (CommSemiring.toSemiring.{v} R (CommRing.toCommSemiring.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.27859897._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{w} M inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.27859897._hygCtx._hyg.7) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.27859897._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.27859897._hygCtx._hyg.10 (smulCommClass_self.{v, w} R M (CommRing.toCommMonoid.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.27859897._hygCtx._hyg.4) (DistribMulAction.toMulAction.{v, w} R M (CommMonoid.toMonoid.{v} R (CommRing.toCommMonoid.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.27859897._hygCtx._hyg.4)) (AddCommMonoid.toAddMonoid.{w} M (AddCommGroup.toAddCommMonoid.{w} M inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.27859897._hygCtx._hyg.7)) (Module.toDistribMulAction.{v, w} R M (Ring.toSemiring.{v} R (CommRing.toRing.{v} 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inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.7 inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.7 inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.10 (RingHom.id.{v} R (Semiring.toNonAssocSemiring.{v} R (CommSemiring.toSemiring.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.4))) (DFunLike.coe.{max (succ w) (succ v), succ v, succ w} (AlgHom.{v, v, w} R (Polynomial.{v} R (CommSemiring.toSemiring.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.4)) (Module.End.{v, w} R M (CommSemiring.toSemiring.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.4) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.7 inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.10) 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inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.7) (Module.toDistribMulAction.{v, w} R M (CommSemiring.toSemiring.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.4) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.7 inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.10))) (Algebra.toSMul.{v, v} R R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.4 (CommSemiring.toSemiring.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.4) (Algebra.id.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.4)) (IsScalarTower.left.{v, w} R M (MonoidWithZero.toMonoid.{v} R (Semiring.toMonoidWithZero.{v} R (CommSemiring.toSemiring.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.4))) (DistribMulAction.toMulAction.{v, w} R M (MonoidWithZero.toMonoid.{v} R (Semiring.toMonoidWithZero.{v} R 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inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.4) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.7 inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.10) (Module.End.instAlgebra.{v, v, w} R R M inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.4 (CommSemiring.toSemiring.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.4) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.7 inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.10 (smulCommClass_self.{v, w} R M (CommSemiring.toCommMonoid.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.4) (DistribMulAction.toMulAction.{v, w} R M (CommMonoid.toMonoid.{v} R (CommSemiring.toCommMonoid.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.4)) (AddCommMonoid.toAddMonoid.{w} M inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.7) (Module.toDistribMulAction.{v, w} R M (CommSemiring.toSemiring.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.4) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.7 inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.10))) (Algebra.toSMul.{v, v} R R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.4 (CommSemiring.toSemiring.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.4) (Algebra.id.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.4)) (IsScalarTower.left.{v, w} R M (MonoidWithZero.toMonoid.{v} R (Semiring.toMonoidWithZero.{v} R (CommSemiring.toSemiring.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.4))) (DistribMulAction.toMulAction.{v, w} R M (MonoidWithZero.toMonoid.{v} R (Semiring.toMonoidWithZero.{v} R (CommSemiring.toSemiring.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.4))) (AddCommMonoid.toAddMonoid.{w} M inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.7) (Module.toDistribMulAction.{v, w} R M (CommSemiring.toSemiring.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.4) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.7 inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.10)))) f) (Units.val.{v} (Polynomial.{v} R (CommSemiring.toSemiring.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.4)) (MonoidWithZero.toMonoid.{v} (Polynomial.{v} R (CommSemiring.toSemiring.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.4)) (Semiring.toMonoidWithZero.{v} (Polynomial.{v} R (CommSemiring.toSemiring.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.4)) (Polynomial.semiring.{v} R (CommSemiring.toSemiring.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.4)))) c))) (Bot.bot.{w} (Submodule.{v, w} R M (CommSemiring.toSemiring.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.4) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.7 inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.10) (Submodule.instBot.{v, w} R M (CommSemiring.toSemiring.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.4) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.7 inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.2601675155._hygCtx._hyg.10))","typeFull":"∀ {R : Type v} {M : Type w} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  (f : Module.End R M) (c : (Polynomial R)ˣ), LinearMap.ker ((Polynomial.aeval f) ↑c) = ⊥","typeReadable":"∀ {R : Type v} {M : Type w} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  (f : Module.End R M) (c : (Polynomial R)ˣ), LinearMap.ker ((Polynomial.aeval f) ↑c) = 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{R : Type.{v}} {M : Type.{w}} [inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.829813063._hygCtx._hyg.4 : CommRing.{v} R] [inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.829813063._hygCtx._hyg.7 : AddCommGroup.{w} M] [inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.829813063._hygCtx._hyg.10 : Module.{v, w} R M (CommSemiring.toSemiring.{v} R (CommRing.toCommSemiring.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.829813063._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{w} M inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.829813063._hygCtx._hyg.7)] {f : Module.End.{v, w} R M (CommSemiring.toSemiring.{v} R (CommRing.toCommSemiring.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.829813063._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{w} M inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.829813063._hygCtx._hyg.7) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.829813063._hygCtx._hyg.10} {p : Polynomial.{v} R (CommSemiring.toSemiring.{v} R (CommRing.toCommSemiring.{v} 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inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.829813063._hygCtx._hyg.7) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.829813063._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.829813063._hygCtx._hyg.10 (smulCommClass_self.{v, w} R M (CommRing.toCommMonoid.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.829813063._hygCtx._hyg.4) (DistribMulAction.toMulAction.{v, w} R M (CommMonoid.toMonoid.{v} R (CommRing.toCommMonoid.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.829813063._hygCtx._hyg.4)) (AddCommMonoid.toAddMonoid.{w} M (AddCommGroup.toAddCommMonoid.{w} M inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.829813063._hygCtx._hyg.7)) (Module.toDistribMulAction.{v, w} R M (Ring.toSemiring.{v} R (CommRing.toRing.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.829813063._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{w} M inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.829813063._hygCtx._hyg.7) 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inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.1909391864._hygCtx._hyg.10) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.1909391864._hygCtx._hyg.4 (Module.End.instRing.{v, w} R M (CommSemiring.toSemiring.{v} R (CommRing.toCommSemiring.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.1909391864._hygCtx._hyg.4)) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.1909391864._hygCtx._hyg.7 inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.1909391864._hygCtx._hyg.10) (Module.End.instAlgebra.{v, v, w} R R M (CommRing.toCommSemiring.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.1909391864._hygCtx._hyg.4) (CommSemiring.toSemiring.{v} R (CommRing.toCommSemiring.{v} R inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.1909391864._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{w} M inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.1909391864._hygCtx._hyg.7) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.1909391864._hygCtx._hyg.10 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inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.10 (RingHom.id.{v} K (Semiring.toNonAssocSemiring.{v} K (DivisionSemiring.toSemiring.{v} K (Semifield.toDivisionSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4))))) (DFunLike.coe.{max (succ w) (succ v), succ v, succ w} (AlgHom.{v, v, w} K (Polynomial.{v} K (CommSemiring.toSemiring.{v} K (Semifield.toCommSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)))) (Module.End.{v, w} K V (DivisionSemiring.toSemiring.{v} K (Semifield.toDivisionSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{w} V inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.7) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.10) (Semifield.toCommSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)) (Polynomial.semiring.{v} K (CommSemiring.toSemiring.{v} K (Semifield.toCommSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)))) (Module.End.instSemiring.{v, w} K V (DivisionSemiring.toSemiring.{v} K (Semifield.toDivisionSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{w} V inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.7) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.10) (Polynomial.algebraOfAlgebra.{v, v} K K (Semifield.toCommSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)) (CommSemiring.toSemiring.{v} K (Semifield.toCommSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4))) (Algebra.id.{v} K (Semifield.toCommSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)))) (Module.End.instAlgebra.{v, v, w} K K V (Semifield.toCommSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)) (DivisionSemiring.toSemiring.{v} K (Semifield.toDivisionSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{w} V inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.7) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.10 (smulCommClass_self.{v, w} K V (CommRing.toCommMonoid.{v} K (EuclideanDomain.toCommRing.{v} K (Field.toEuclideanDomain.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4))) (DistribMulAction.toMulAction.{v, w} K V (CommMonoid.toMonoid.{v} K (CommRing.toCommMonoid.{v} K (EuclideanDomain.toCommRing.{v} K (Field.toEuclideanDomain.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{w} V (AddCommGroup.toAddCommMonoid.{w} V inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.7)) (Module.toDistribMulAction.{v, w} K V (Ring.toSemiring.{v} K (CommRing.toRing.{v} K (EuclideanDomain.toCommRing.{v} K (Field.toEuclideanDomain.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)))) (AddCommGroup.toAddCommMonoid.{w} V inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.7) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.10))) (Algebra.toSMul.{v, v} K K (Semifield.toCommSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)) (CommSemiring.toSemiring.{v} K (Semifield.toCommSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4))) (Algebra.id.{v} K (Semifield.toCommSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)))) (IsScalarTower.left.{v, w} K V (MonoidWithZero.toMonoid.{v} K (Semiring.toMonoidWithZero.{v} K (DivisionSemiring.toSemiring.{v} K (Semifield.toDivisionSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4))))) (DistribMulAction.toMulAction.{v, w} K V (MonoidWithZero.toMonoid.{v} K (Semiring.toMonoidWithZero.{v} K (DivisionSemiring.toSemiring.{v} K (Semifield.toDivisionSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4))))) (AddCommMonoid.toAddMonoid.{w} V (AddCommGroup.toAddCommMonoid.{w} V inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.7)) (Module.toDistribMulAction.{v, w} K V (DivisionSemiring.toSemiring.{v} K (Semifield.toDivisionSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{w} V inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.7) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.10))))) (Polynomial.{v} K (CommSemiring.toSemiring.{v} K (Semifield.toCommSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)))) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Polynomial.{v} K (CommSemiring.toSemiring.{v} K (Semifield.toCommSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)))) => Module.End.{v, w} K V (DivisionSemiring.toSemiring.{v} K (Semifield.toDivisionSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{w} V inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.7) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.10) (AlgHom.funLike.{v, v, w} K (Polynomial.{v} K (CommSemiring.toSemiring.{v} K (Semifield.toCommSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)))) (Module.End.{v, w} K V (DivisionSemiring.toSemiring.{v} K (Semifield.toDivisionSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{w} V inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.7) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.10) (Semifield.toCommSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)) (Polynomial.semiring.{v} K (CommSemiring.toSemiring.{v} K (Semifield.toCommSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)))) (Module.End.instSemiring.{v, w} K V (DivisionSemiring.toSemiring.{v} K (Semifield.toDivisionSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{w} V inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.7) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.10) (Polynomial.algebraOfAlgebra.{v, v} K K (Semifield.toCommSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)) (CommSemiring.toSemiring.{v} K (Semifield.toCommSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4))) (Algebra.id.{v} K (Semifield.toCommSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)))) (Module.End.instAlgebra.{v, v, w} K K V (Semifield.toCommSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)) (DivisionSemiring.toSemiring.{v} K (Semifield.toDivisionSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{w} V inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.7) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.10 (smulCommClass_self.{v, w} K V (CommRing.toCommMonoid.{v} K (EuclideanDomain.toCommRing.{v} K (Field.toEuclideanDomain.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4))) (DistribMulAction.toMulAction.{v, w} K V (CommMonoid.toMonoid.{v} K (CommRing.toCommMonoid.{v} K (EuclideanDomain.toCommRing.{v} K (Field.toEuclideanDomain.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{w} V (AddCommGroup.toAddCommMonoid.{w} V inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.7)) (Module.toDistribMulAction.{v, w} K V (Ring.toSemiring.{v} K (CommRing.toRing.{v} K (EuclideanDomain.toCommRing.{v} K (Field.toEuclideanDomain.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)))) (AddCommGroup.toAddCommMonoid.{w} V inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.7) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.10))) (Algebra.toSMul.{v, v} K K (Semifield.toCommSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)) (CommSemiring.toSemiring.{v} K (Semifield.toCommSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4))) (Algebra.id.{v} K (Semifield.toCommSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)))) (IsScalarTower.left.{v, w} K V (MonoidWithZero.toMonoid.{v} K (Semiring.toMonoidWithZero.{v} K (DivisionSemiring.toSemiring.{v} K (Semifield.toDivisionSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4))))) (DistribMulAction.toMulAction.{v, w} K V (MonoidWithZero.toMonoid.{v} K (Semiring.toMonoidWithZero.{v} K (DivisionSemiring.toSemiring.{v} K (Semifield.toDivisionSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4))))) (AddCommMonoid.toAddMonoid.{w} V (AddCommGroup.toAddCommMonoid.{w} V inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.7)) (Module.toDistribMulAction.{v, w} K V (DivisionSemiring.toSemiring.{v} K (Semifield.toDivisionSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{w} V inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.7) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.10))))) (Polynomial.aeval.{v, w} K (Module.End.{v, w} K V (DivisionSemiring.toSemiring.{v} K (Semifield.toDivisionSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{w} V inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.7) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.10) (Semifield.toCommSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)) (Module.End.instSemiring.{v, w} K V (DivisionSemiring.toSemiring.{v} K (Semifield.toDivisionSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{w} V inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.7) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.10) (Module.End.instAlgebra.{v, v, w} K K V (Semifield.toCommSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)) (DivisionSemiring.toSemiring.{v} K (Semifield.toDivisionSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{w} V inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.7) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.10 (smulCommClass_self.{v, w} K V (CommRing.toCommMonoid.{v} K (EuclideanDomain.toCommRing.{v} K (Field.toEuclideanDomain.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4))) (DistribMulAction.toMulAction.{v, w} K V (CommMonoid.toMonoid.{v} K (CommRing.toCommMonoid.{v} K (EuclideanDomain.toCommRing.{v} K (Field.toEuclideanDomain.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{w} V (AddCommGroup.toAddCommMonoid.{w} V inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.7)) (Module.toDistribMulAction.{v, w} K V (Ring.toSemiring.{v} K (CommRing.toRing.{v} K (EuclideanDomain.toCommRing.{v} K (Field.toEuclideanDomain.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)))) (AddCommGroup.toAddCommMonoid.{w} V inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.7) inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.10))) (Algebra.toSMul.{v, v} K K (Semifield.toCommSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)) (CommSemiring.toSemiring.{v} K (Semifield.toCommSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4))) (Algebra.id.{v} K (Semifield.toCommSemiring.{v} K (Field.toSemifield.{v} K inst._@.Mathlib.LinearAlgebra.Eigenspace.Minpoly.4252666845._hygCtx._hyg.4)))) (IsScalarTower.left.{v, w} K V (MonoidWithZero.toMonoid.{v} K (Semiring.toMonoidWithZero.{v} K (DivisionSemiring.toSemiring.{v} K (Semifield.toDivisionSemiring.{v} K (Field.toSemifield.{v} K 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V] [inst_2 : Module K V] (f : Module.End K V)\n  (q : Polynomial K), q.degree = 1 → f.eigenspace (-q.coeff 0 / q.leadingCoeff) = LinearMap.ker ((Polynomial.aeval f) q)","typeReadable":"∀ {K : Type v} {V : Type w} [inst : Field K] [inst_1 : AddCommGroup V] [inst_2 : Module K V] (f : Module.End K V)\n  (q : Polynomial K), q.degree = 1 → f.eigenspace (-q.coeff 0 / q.leadingCoeff) = LinearMap.ker ((Polynomial.aeval f) 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diff --git a/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo.sym.json b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo.sym.json
new file mode 100644
index 0000000000000000000000000000000000000000..0558fd27f04dd0b54163391694856df63286dc31
--- /dev/null
+++ b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo.sym.json
@@ -0,0 +1 @@
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m.discr","typeReferences":[["Fin"],["HMul","hMul"],["CommRing","toNonUnitalCommRing"],["Fin","fintype"],["Matrix"],["Matrix","inv"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["instOfNatNat"],["MonoidWithZero","toMonoid"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Eq"],["CommRing","toCommSemiring"],["Inv","inv"],["instDecidableEqFin"],["NonUnitalNonAssocSemiring","toDistrib"],["Matrix","instHMulOfFintypeOfMulOfAddCommMonoid"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["CommSemiring","toSemiring"],["Distrib","toMul"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Semiring","toMonoidWithZero"],["Matrix","discr"],["CommRing"],["OfNat","ofNat"],["Matrix","semiring"],["Nat"],["Units","val"],["Matrix","GeneralLinearGroup"]],"valueReferences":[["Matrix","discr_conj"],["Eq","trans"],["Eq","mp"],["Units","instInv"],["HMul","hMul"],["Fin"],["CommRing","toNonUnitalCommRing"],["Matrix","discr","congr_simp"],["Matrix"],["Fin","fintype"],["Matrix","inv"],["congrArg"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["instOfNatNat"],["congr"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["MonoidWithZero","toMonoid"],["congrFun'"],["Eq"],["Inv","inv"],["CommRing","toCommSemiring"],["instDecidableEqFin"],["NonUnitalNonAssocSemiring","toDistrib"],["Matrix","instHMulOfFintypeOfMulOfAddCommMonoid"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["Distrib","toMul"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["Matrix","discr"],["OfNat","ofNat"],["Nat"],["Matrix","semiring"],["Units","val"],["Units","invertible"],["Matrix","inv_inv_of_invertible"],["Matrix","GeneralLinearGroup"],["Matrix","coe_units_inv"]]},{"isProp":true,"kind":"theorem","name":["Matrix","isParabolic_iff_exists"],"typeFallback":"forall 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+[{"isProp":false,"kind":"definition","name":["Submodule","Quotient","instMeasurableSpaceQuotient"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.2202914794._hygCtx._hyg.4 : Ring.{u_1} R] [inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.2202914794._hygCtx._hyg.7 : AddCommGroup.{u_2} M] [inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.2202914794._hygCtx._hyg.10 : Module.{u_1, u_2} R M (Ring.toSemiring.{u_1} R inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.2202914794._hygCtx._hyg.4) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.2202914794._hygCtx._hyg.7)] {p : Submodule.{u_1, u_2} R M (Ring.toSemiring.{u_1} R inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.2202914794._hygCtx._hyg.4) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.2202914794._hygCtx._hyg.7) inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.2202914794._hygCtx._hyg.10} [inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.2202914794._hygCtx._hyg.17 : MeasurableSpace.{u_2} M], MeasurableSpace.{u_2} (HasQuotient.Quotient.{u_2, u_2} M (Submodule.{u_1, u_2} R M (Ring.toSemiring.{u_1} R inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.2202914794._hygCtx._hyg.4) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.2202914794._hygCtx._hyg.7) inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.2202914794._hygCtx._hyg.10) (Submodule.hasQuotient.{u_1, u_2} R M inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.2202914794._hygCtx._hyg.4 inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.2202914794._hygCtx._hyg.7 inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.2202914794._hygCtx._hyg.10) p)","typeFull":"{R : Type u_1} →\n  {M : Type u_2} →\n    [inst : Ring R] →\n      [inst_1 : AddCommGroup M] →\n        [inst_2 : Module R M] → {p : Submodule R M} → [MeasurableSpace M] → MeasurableSpace (M ⧸ p)","typeReadable":"{R : Type u_1} →\n  {M : Type u_2} →\n    [inst : Ring R] →\n      [inst_1 : AddCommGroup M] →\n        [inst_2 : Module R M] → {p : Submodule R M} → [MeasurableSpace M] → MeasurableSpace (M ⧸ p)","typeReferences":[["Submodule","hasQuotient"],["HasQuotient","Quotient"],["Module"],["AddCommGroup","toAddCommMonoid"],["MeasurableSpace"],["AddCommGroup"],["Submodule"],["Ring","toSemiring"],["Ring"]],"valueReferences":[["Quotient","instMeasurableSpace"],["Submodule","quotientRel"]]},{"isProp":true,"kind":"theorem","name":["Submodule","Quotient","instDiscreteMeasurableSpaceQuotient"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.383071636._hygCtx._hyg.4 : Ring.{u_1} R] [inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.383071636._hygCtx._hyg.7 : AddCommGroup.{u_2} M] [inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.383071636._hygCtx._hyg.10 : Module.{u_1, u_2} R M (Ring.toSemiring.{u_1} R inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.383071636._hygCtx._hyg.4) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.383071636._hygCtx._hyg.7)] {p : Submodule.{u_1, u_2} R M (Ring.toSemiring.{u_1} R inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.383071636._hygCtx._hyg.4) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.383071636._hygCtx._hyg.7) inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.383071636._hygCtx._hyg.10} [inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.383071636._hygCtx._hyg.17 : MeasurableSpace.{u_2} M] [inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.383071636._hygCtx._hyg.20 : DiscreteMeasurableSpace.{u_2} M inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.383071636._hygCtx._hyg.17], DiscreteMeasurableSpace.{u_2} (HasQuotient.Quotient.{u_2, u_2} M (Submodule.{u_1, u_2} R M (Ring.toSemiring.{u_1} R inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.383071636._hygCtx._hyg.4) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.383071636._hygCtx._hyg.7) inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.383071636._hygCtx._hyg.10) (Submodule.hasQuotient.{u_1, u_2} R M inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.383071636._hygCtx._hyg.4 inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.383071636._hygCtx._hyg.7 inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.383071636._hygCtx._hyg.10) p) (Submodule.Quotient.instMeasurableSpaceQuotient.{u_1, u_2} R M inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.383071636._hygCtx._hyg.4 inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.383071636._hygCtx._hyg.7 inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.383071636._hygCtx._hyg.10 p inst._@.Mathlib.MeasureTheory.Constructions.SubmoduleQuotient.383071636._hygCtx._hyg.17)","typeFull":"∀ {R : Type u_1} {M : Type u_2} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] {p : Submodule R M}\n  [inst_3 : MeasurableSpace M] [DiscreteMeasurableSpace M], DiscreteMeasurableSpace (M ⧸ p)","typeReadable":"∀ {R : Type u_1} {M : Type u_2} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] {p : Submodule R M}\n  [inst_3 : MeasurableSpace M] [DiscreteMeasurableSpace M], DiscreteMeasurableSpace (M ⧸ 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diff --git a/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.MeasureTheory.Measure.Complex.sym.json b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.MeasureTheory.Measure.Complex.sym.json
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sBoundedSMul"],["NormedSpace","complexToReal"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["NormedCommRing","toNonUnitalNormedCommRing"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","SignedMeasure","toComplexMeasure_apply_im"],"typeFallback":"forall 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(NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (MeasureTheory.SignedMeasure.toComplexMeasure.{u_1} α m s t) i)) (MeasureTheory.VectorMeasure.measureOf'.{u_1, 0} α m Real Real.instAddCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) t i)","typeFull":"∀ {α : Type u_1} {m : MeasurableSpace α} (s t : MeasureTheory.SignedMeasure α) (i : Set α),\n  (↑(s.toComplexMeasure t) i).im = ↑t i","typeReadable":"∀ {α : Type u_1} {m : MeasurableSpace α} (s t : MeasureTheory.SignedMeasure α) (i : Set α),\n  (↑(s.toComplexMeasure t) i).im = ↑t i","typeReferences":[["NormedAddCommGroup","toENormedAddCommMonoid"],["Real"],["PseudoMetricSpace","toUniformSpace"],["Set"],["UniformSpace","toTopologicalSpace"],["SeminormedCommRing","toSeminormedRing"],["MeasureTheory","SignedMeasure","toComplexMeasure"],["Complex","instNormedField"],["Complex"],["SeminormedRing","toPseudoMetricSpace"],["MeasureTheory","SignedMeasure"],["Real","pseudoMetricSpace"],["Complex","instNormedAddCommGroup"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["NormedCommRing","toSeminormedCommRing"],["MeasurableSpace"],["Real","instAddCommMonoid"],["Eq"],["Complex","im"],["MeasureTheory","VectorMeasure","measureOf'"],["NormedField","toNormedCommRing"],["ESeminormedAddCommMonoid","toAddCommMonoid"]],"valueReferences":[["NormedAddCommGroup","toENormedAddCommMonoid"],["Real"],["PseudoMetricSpace","toUniformSpace"],["UniformSpace","toTopologicalSpace"],["SeminormedCommRing","toSeminormedRing"],["MeasureTheory","SignedMeasure","toComplexMeasure"],["Complex","instNormedField"],["Complex"],["SeminormedRing","toPseudoMetricSpace"],["Complex","instNormedAddCommGroup"],["Eq","refl"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["NormedCommRing","toSeminormedCommRing"],["Complex","im"],["MeasureTheory","VectorMeasure","measureOf'"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["NormedField","toNormedCommRing"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","ComplexMeasure","equivSignedMeasureₗ","_proof_3"],"typeFallback":"forall {α : Type.{u_1}} {m : MeasurableSpace.{u_1} α}, Function.LeftInverse.{succ u_1, succ u_1} (MeasureTheory.ComplexMeasure.{u_1} α m) (Prod.{u_1, u_1} (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m)) (Equiv.invFun.{succ u_1, succ u_1} (MeasureTheory.ComplexMeasure.{u_1} α m) (Prod.{u_1, u_1} (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m)) (MeasureTheory.ComplexMeasure.equivSignedMeasure.{u_1} α m)) (Equiv.toFun.{succ u_1, succ u_1} (MeasureTheory.ComplexMeasure.{u_1} α m) (Prod.{u_1, u_1} (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m)) (MeasureTheory.ComplexMeasure.equivSignedMeasure.{u_1} α m))","typeFull":"∀ {α : Type u_1} {m : MeasurableSpace α},\n  Function.LeftInverse MeasureTheory.ComplexMeasure.equivSignedMeasure.invFun\n    MeasureTheory.ComplexMeasure.equivSignedMeasure.toFun","typeReadable":"∀ {α : Type u_1} {m : MeasurableSpace α},\n  Function.LeftInverse MeasureTheory.ComplexMeasure.equivSignedMeasure.invFun\n    MeasureTheory.ComplexMeasure.equivSignedMeasure.toFun","typeReferences":[["Prod"],["MeasureTheory","SignedMeasure"],["Equiv","invFun"],["Function","LeftInverse"],["MeasureTheory","ComplexMeasure"],["Equiv","toFun"],["MeasurableSpace"],["MeasureTheory","ComplexMeasure","equivSignedMeasure"]],"valueReferences":[["Prod"],["MeasureTheory","SignedMeasure"],["Equiv","left_inv"],["MeasureTheory","ComplexMeasure"],["MeasureTheory","ComplexMeasure","equivSignedMeasure"]]},{"isProp":false,"kind":"definition","name":["MeasureTheory","ComplexMeasure","equivSignedMeasureₗ"],"typeFallback":"forall {α : Type.{u_1}} {m : MeasurableSpace.{u_1} α} {R : Type.{u_2}} [inst._@.Mathlib.MeasureTheory.Measure.Complex.406428577._hygCtx._hyg.6 : Semiring.{u_2} R] [inst._@.Mathlib.MeasureTheory.Measure.Complex.406428577._hygCtx._hyg.9 : Module.{u_2, 0} R Real inst._@.Mathlib.MeasureTheory.Measure.Complex.406428577._hygCtx._hyg.6 Real.instAddCommMonoid] [inst._@.Mathlib.MeasureTheory.Measure.Complex.406428577._hygCtx._hyg.15 : ContinuousConstSMul.{u_2, 0} R Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (SMulZeroClass.toSMul.{u_2, 0} R Real (AddZero.toZero.{0} Real (AddZeroClass.toAddZero.{0} Real (AddMonoid.toAddZeroClass.{0} Real Real.instAddMonoid))) (DistribSMul.toSMulZeroClass.{u_2, 0} R Real (AddMonoid.toAddZeroClass.{0} Real Real.instAddMonoid) (DistribMulAction.toDistribSMul.{u_2, 0} R Real (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.MeasureTheory.Measure.Complex.406428577._hygCtx._hyg.6)) Real.instAddMonoid (Module.toDistribMulAction.{u_2, 0} R Real inst._@.Mathlib.MeasureTheory.Measure.Complex.406428577._hygCtx._hyg.6 Real.instAddCommMonoid inst._@.Mathlib.MeasureTheory.Measure.Complex.406428577._hygCtx._hyg.9))))] [inst._@.Mathlib.MeasureTheory.Measure.Complex.406428577._hygCtx._hyg.21 : ContinuousConstSMul.{u_2, 0} R 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Type u_2} →\n      [inst : Semiring R] →\n        [inst_1 : Module R ℝ] →\n          [inst_2 : ContinuousConstSMul R ℝ] →\n            [inst_3 : ContinuousConstSMul R ℂ] →\n              MeasureTheory.ComplexMeasure α ≃ₗ[R] MeasureTheory.SignedMeasure α × MeasureTheory.SignedMeasure α","typeReadable":"{α : Type u_1} →\n  {m : MeasurableSpace α} →\n    {R : Type u_2} →\n      [inst : Semiring R] →\n        [inst_1 : Module R ℝ] →\n          [inst_2 : ContinuousConstSMul R ℝ] →\n            [inst_3 : ContinuousConstSMul R ℂ] →\n              MeasureTheory.ComplexMeasure α ≃ₗ[R] MeasureTheory.SignedMeasure α × MeasureTheory.SignedMeasure α","typeReferences":[["PseudoMetricSpace","toUniformSpace"],["Real","normedCommRing"],["ContinuousConstSMul"],["SMulZeroClass","toSMul"],["Prod","instAddCommMonoid"],["Complex"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["IsSemitopologicalSemiring","toContinuousAdd"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["MeasureTheory","VectorMeasure","instAddCommMonoid"],["Real","instAddCommMonoid"],["NormedField","toNormedCommRing"],["DistribSMul","toSMulZeroClass"],["NormedAddCommGroup","toENormedAddCommMonoid"],["Real"],["SeminormedAddCommGroup","toPseudoMetricSpace"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["DistribMulAction","toDistribSMul"],["instIsTopologicalRingReal"],["SeminormedCommRing","toSeminormedRing"],["AddZeroClass","toAddZero"],["Complex","instNormedField"],["Prod"],["Real","pseudoMetricSpace"],["IsTopologicalRing","toIsSemitopologicalRing"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["RingHomInvPair","ids"],["MeasureTheory","ComplexMeasure"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Module"],["ESeminormedAddCommMonoid","toESeminormedAddMonoid"],["UniformSpace","toTopologicalSpace"],["LinearEquiv"],["SeminormedRing","toPseudoMetricSpace"],["MeasureTheory","SignedMeasure"],["Prod","instModule"],["Complex","instNormedAddCommGroup"],["MonoidWithZero","toMonoid"],["MeasurableSpace"],["ESeminormedAddMonoid","toAddMonoid"],["Semiring","toMonoidWithZero"],["NormedDivisionRing","to_isTopologicalDivisionRing"],["Complex","instModule"],["Module","toDistribMulAction"],["IsSemitopologicalRing","toIsSemitopologicalSemiring"],["MeasureTheory","VectorMeasure","instModule"],["IsTopologicalDivisionRing","toIsTopologicalRing"],["NormedField","toNormedDivisionRing"],["NormedDivisionRing","toDivisionRing"],["NormedCommRing","toSeminormedCommRing"],["Real","instAddMonoid"],["NormedAddCommGroup","toSeminormedAddCommGroup"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["NormedCommRing","toNonUnitalNormedCommRing"],["Semiring"]],"valueReferences":[["MeasureTheory","ComplexMeasure","equivSignedMeasureₗ","_proof_2"],["PseudoMetricSpace","toUniformSpace"],["LinearMap","mk"],["AddHom","mk"],["UniformSpace","toTopologicalSpace"],["MeasureTheory","ComplexMeasure","re","_proof_2"],["MeasureTheory","ComplexMeasure","equivSignedMeasure"],["MeasureTheory","ComplexMeasure","equivSignedMeasureₗ","_proof_4"],["Prod","instAddCommMonoid"],["Complex"],["SeminormedRing","toPseudoMetricSpace"],["MeasureTheory","SignedMeasure"],["Semiring","toNonAssocSemiring"],["Prod","instModule"],["RingHom","id"],["Complex","instNormedAddCommGroup"],["LinearEquiv","mk"],["Equiv","toFun"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["MeasureTheory","VectorMeasure","instAddCommMonoid"],["Real","instAddCommMonoid"],["NormedField","toNormedCommRing"],["NormedAddCommGroup","toENormedAddCommMonoid"],["Real"],["SeminormedCommRing","toSeminormedRing"],["Complex","instNormedField"],["Complex","instModule"],["Prod"],["AddCommMonoid","toAddCommSemigroup"],["Equiv","invFun"],["Real","pseudoMetricSpace"],["MeasureTheory","VectorMeasure","instModule"],["RingHomInvPair","ids"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["MeasureTheory","ComplexMeasure","re","_proof_1"],["MeasureTheory","ComplexMeasure"],["NormedCommRing","toSeminormedCommRing"],["MeasureTheory","ComplexMeasure","equivSignedMeasureₗ","_proof_3"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["MeasureTheory","ComplexMeasure","equivSignedMeasureₗ","_proof_1"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","ComplexMeasure","im_apply"],"typeFallback":"forall {α : Type.{u_1}} {m : MeasurableSpace.{u_1} α} (v : MeasureTheory.VectorMeasure.{u_1, 0} α m Complex (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))) (UniformSpace.toTopologicalSpace.{0} Complex 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Complex.continuous_im)","typeFull":"∀ {α : Type u_1} {m : MeasurableSpace α} (v : MeasureTheory.VectorMeasure α ℂ),\n  MeasureTheory.ComplexMeasure.im v = v.mapRange Complex.imLm.toAddMonoidHom Complex.continuous_im","typeReadable":"∀ {α : Type u_1} {m : MeasurableSpace α} (v : MeasureTheory.VectorMeasure α ℂ),\n  MeasureTheory.ComplexMeasure.im v = v.mapRange Complex.imLm.toAddMonoidHom Complex.continuous_im","typeReferences":[["IsSemitopologicalSemiring","toSeparatelyContinuousMul"],["PseudoMetricSpace","toUniformSpace"],["Real","normedCommRing"],["SMulZeroClass","toSMul"],["Complex"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["MeasureTheory","VectorMeasure","mapRange"],["RingHom","id"],["IsSemitopologicalSemiring","toContinuousAdd"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["MeasureTheory","VectorMeasure","instAddCommMonoid"],["Real","instAddCommMonoid"],["NormedField","toNormedCommRing"],["DistribSMul","toSMulZeroClass"],["NonUnitalSeminormedRing","toSeminormedAddCommGroup"],["NormedAddCommGroup","toENormedAddCommMonoid"],["LinearMap","toAddMonoidHom"],["NormedSpace","toModule"],["Real"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["IsBoundedSMul","toUniformContinuousConstSMul"],["DistribMulAction","toDistribSMul"],["instIsTopologicalRingReal"],["AddZeroClass","toAddZero"],["LinearMap"],["SeminormedCommRing","toSeminormedRing"],["Complex","instNormedField"],["Complex","instZero"],["Real","pseudoMetricSpace"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["IsTopologicalRing","toIsSemitopologicalRing"],["MeasureTheory","ComplexMeasure"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["MeasureTheory","ComplexMeasure","im"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Complex","continuous_im"],["NonUnitalSeminormedCommRing","toNonUnitalSeminormedRing"],["LinearMap","instFunLike"],["UniformSpace","toTopologicalSpace"],["Complex","imLm"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["UniformContinuousConstSMul","to_continuousConstSMul"],["SeminormedRing","toPseudoMetricSpace"],["MeasureTheory","SignedMeasure"],["SeminormedCommRing","toNonUnitalSeminormedCommRing"],["Complex","instNormedAddCommGroup"],["MonoidWithZero","toMonoid"],["MeasurableSpace"],["Eq"],["Real","instMul"],["SeparatelyContinuousMul","to_continuousSMul"],["NormedField","toNormedSpace"],["Semiring","toMonoidWithZero"],["Real","semiring"],["NormedDivisionRing","to_isTopologicalDivisionRing"],["Module","toDistribMulAction"],["Real","instZero"],["Real","normedField"],["IsSemitopologicalRing","toIsSemitopologicalSemiring"],["MeasureTheory","VectorMeasure","instModule"],["MeasureTheory","VectorMeasure"],["IsTopologicalDivisionRing","toIsTopologicalRing"],["NormedCommRing","toSeminormedCommRing"],["NormedDivisionRing","toDivisionRing"],["NormedField","toNormedDivisionRing"],["NormedSpace","toIsBoundedSMul"],["NormedSpace","complexToReal"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["NormedCommRing","toNonUnitalNormedCommRing"]],"valueReferences":[["IsSemitopologicalSemiring","toSeparatelyContinuousMul"],["PseudoMetricSpace","toUniformSpace"],["Real","normedCommRing"],["SMulZeroClass","toSMul"],["Complex"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["IsSemitopologicalSemiring","toContinuousAdd"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["MeasureTheory","VectorMeasure","instAddCommMonoid"],["Real","instAddCommMonoid"],["NormedField","toNormedCommRing"],["DistribSMul","toSMulZeroClass"],["NonUnitalSeminormedRing","toSeminormedAddCommGroup"],["NormedAddCommGroup","toENormedAddCommMonoid"],["NormedSpace","toModule"],["Real"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["IsBoundedSMul","toUniformContinuousConstSMul"],["DistribMulAction","toDistribSMul"],["instIsTopologicalRingReal"],["AddZeroClass","toAddZero"],["SeminormedCommRing","toSeminormedRing"],["LinearMap"],["Complex","instNormedField"],["Complex","instZero"],["Real","pseudoMetricSpace"],["Eq","refl"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["IsTopologicalRing","toIsSemitopologicalRing"],["MeasureTheory","ComplexMeasure"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["MeasureTheory","ComplexMeasure","im"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["NonUnitalSeminormedCommRing","toNonUnitalSeminormedRing"],["LinearMap","instFunLike"],["UniformSpace","toTopologicalSpace"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["UniformContinuousConstSMul","to_continuousConstSMul"],["SeminormedRing","toPseudoMetricSpace"],["MeasureTheory","SignedMeasure"],["SeminormedCommRing","toNonUnitalSeminormedCommRing"],["Complex","instNormedAddCommGroup"],["MonoidWithZero","toMonoid"],["Real","instMul"],["SeparatelyContinuousMul","to_continuousSMul"],["NormedField","toNormedSpace"],["Semiring","toMonoidWithZero"],["Real","semiring"],["NormedDivisionRing","to_isTopologicalDivisionRing"],["Module","toDistribMulAction"],["Real","instZero"],["Real","normedField"],["IsSemitopologicalRing","toIsSemitopologicalSemiring"],["MeasureTheory","VectorMeasure","instModule"],["MeasureTheory","VectorMeasure"],["IsTopologicalDivisionRing","toIsTopologicalRing"],["NormedCommRing","toSeminormedCommRing"],["NormedDivisionRing","toDivisionRing"],["NormedField","toNormedDivisionRing"],["NormedSpace","toIsBoundedSMul"],["NormedSpace","complexToReal"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["NormedCommRing","toNonUnitalNormedCommRing"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","ComplexMeasure","absolutelyContinuous_ennreal_iff"],"typeFallback":"forall {α : Type.{u_1}} {m : MeasurableSpace.{u_1} α} (c : MeasureTheory.ComplexMeasure.{u_1} α m) (μ : MeasureTheory.VectorMeasure.{u_1, 0} α m ENNReal ENNReal.instAddCommMonoid ENNReal.instTopologicalSpace), Iff (MeasureTheory.VectorMeasure.AbsolutelyContinuous.{u_1, 0, 0} α m Complex ENNReal (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) ENNReal.instAddCommMonoid ENNReal.instTopologicalSpace c μ) (And (MeasureTheory.VectorMeasure.AbsolutelyContinuous.{u_1, 0, 0} α m Real ENNReal Real.instAddCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) ENNReal.instAddCommMonoid ENNReal.instTopologicalSpace (DFunLike.coe.{succ u_1, succ u_1, succ u_1} (LinearMap.{0, 0, u_1, u_1} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (MeasureTheory.ComplexMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.VectorMeasure.instAddCommMonoid.{u_1, 0} α m Complex (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (IsSemitopologicalSemiring.toContinuousAdd.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{0} Complex (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{0} Complex (NonUnitalNormedCommRing.toNonUnitalCommRing.{0} Complex (NormedCommRing.toNonUnitalNormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (IsSemitopologicalRing.toIsSemitopologicalSemiring.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{0} Complex (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{0} Complex (NonUnitalNormedCommRing.toNonUnitalCommRing.{0} Complex (NormedCommRing.toNonUnitalNormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (IsTopologicalRing.toIsSemitopologicalRing.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex 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(MeasureTheory.VectorMeasure.instAddCommMonoid.{u_1, 0} α m Real Real.instAddCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (IsSemitopologicalSemiring.toContinuousAdd.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Real (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{0} Real (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{0} Real (NonUnitalNormedCommRing.toNonUnitalCommRing.{0} Real (NormedCommRing.toNonUnitalNormedCommRing.{0} Real Real.normedCommRing))))) (IsSemitopologicalRing.toIsSemitopologicalSemiring.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{0} Real (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{0} Real (NonUnitalNormedCommRing.toNonUnitalCommRing.{0} 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{α : Type.{u_1}} {m : MeasurableSpace.{u_1} α} (motive : (Prod.{u_1, u_1} (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m)) -> Sort.{u_2}) (x._@.Mathlib.MeasureTheory.Measure.Complex.2046710636._hygCtx.37.Mathlib.MeasureTheory.Measure.Complex.2046710636._hygCtx._hyg.45 : Prod.{u_1, u_1} (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m)), (forall (s : MeasureTheory.SignedMeasure.{u_1} α m) (t : MeasureTheory.SignedMeasure.{u_1} α m), motive (Prod.mk.{u_1, u_1} (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m) s t)) -> (motive x._@.Mathlib.MeasureTheory.Measure.Complex.2046710636._hygCtx.37.Mathlib.MeasureTheory.Measure.Complex.2046710636._hygCtx._hyg.45)","typeFull":"{α : Type u_1} →\n  {m : MeasurableSpace α} →\n    (motive : MeasureTheory.SignedMeasure α × MeasureTheory.SignedMeasure α → Sort u_2) →\n      (x : MeasureTheory.SignedMeasure α × MeasureTheory.SignedMeasure α) →\n        ((s t : MeasureTheory.SignedMeasure α) → motive (s, t)) → motive x","typeReadable":"{α : Type u_1} →\n  {m : MeasurableSpace α} →\n    (motive : MeasureTheory.SignedMeasure α × MeasureTheory.SignedMeasure α → Sort u_2) →\n      (x : MeasureTheory.SignedMeasure α × MeasureTheory.SignedMeasure α) →\n        ((s t : MeasureTheory.SignedMeasure α) → motive (s, t)) → motive x","typeReferences":[["Prod"],["MeasureTheory","SignedMeasure"],["Prod","mk"],["MeasurableSpace"]],"valueReferences":[["Prod","casesOn"],["MeasureTheory","SignedMeasure"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","SignedMeasure","re_toComplexMeasure"],"typeFallback":"forall {α : Type.{u_1}} {m : MeasurableSpace.{u_1} α} (s : MeasureTheory.SignedMeasure.{u_1} α m) (t : MeasureTheory.SignedMeasure.{u_1} α m), Eq.{succ u_1} (MeasureTheory.SignedMeasure.{u_1} α m) (DFunLike.coe.{succ u_1, succ u_1, succ u_1} (LinearMap.{0, 0, u_1, u_1} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real 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{α : Type.{u_1}} {m : MeasurableSpace.{u_1} α} {R : Type.{u_2}} [inst._@.Mathlib.MeasureTheory.Measure.Complex.406428577._hygCtx._hyg.6 : Semiring.{u_2} R] [inst._@.Mathlib.MeasureTheory.Measure.Complex.406428577._hygCtx._hyg.9 : Module.{u_2, 0} R Real inst._@.Mathlib.MeasureTheory.Measure.Complex.406428577._hygCtx._hyg.6 Real.instAddCommMonoid] [inst._@.Mathlib.MeasureTheory.Measure.Complex.406428577._hygCtx._hyg.15 : ContinuousConstSMul.{u_2, 0} R Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (SMulZeroClass.toSMul.{u_2, 0} R Real (AddZero.toZero.{0} Real (AddZeroClass.toAddZero.{0} Real (AddMonoid.toAddZeroClass.{0} Real Real.instAddMonoid))) (DistribSMul.toSMulZeroClass.{u_2, 0} R Real (AddMonoid.toAddZeroClass.{0} Real Real.instAddMonoid) (DistribMulAction.toDistribSMul.{u_2, 0} R Real (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R 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{α : Type.{u_1}} {m : MeasurableSpace.{u_1} α}, Equiv.{succ u_1, succ u_1} (MeasureTheory.ComplexMeasure.{u_1} α m) (Prod.{u_1, u_1} (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m))","typeFull":"{α : Type u_1} →\n  {m : MeasurableSpace α} →\n    MeasureTheory.ComplexMeasure α ≃ MeasureTheory.SignedMeasure α × MeasureTheory.SignedMeasure α","typeReadable":"{α : Type u_1} →\n  {m : MeasurableSpace α} →\n    MeasureTheory.ComplexMeasure α ≃ MeasureTheory.SignedMeasure α × MeasureTheory.SignedMeasure 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MeasureTheory.ComplexMeasure.{u_1} α m) x._@.Mathlib.MeasureTheory.Measure.Complex.2046710636._hygCtx._hyg.37 (fun (s : MeasureTheory.SignedMeasure.{u_1} α m) (t : MeasureTheory.SignedMeasure.{u_1} α m) => MeasureTheory.SignedMeasure.toComplexMeasure.{u_1} α m s t))","typeFull":"∀ {α : Type u_1} {m : MeasurableSpace α} (x : MeasureTheory.SignedMeasure α × MeasureTheory.SignedMeasure α),\n  MeasureTheory.ComplexMeasure.equivSignedMeasure.symm x =\n    match x with\n    | (s, t) => s.toComplexMeasure t","typeReadable":"∀ {α : Type u_1} {m : MeasurableSpace α} (x : MeasureTheory.SignedMeasure α × MeasureTheory.SignedMeasure α),\n  MeasureTheory.ComplexMeasure.equivSignedMeasure.symm x =\n    match x with\n    | (s, t) => s.toComplexMeasure 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Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring))) (MeasureTheory.ComplexMeasure.im.{u_1} α m) (MeasureTheory.SignedMeasure.toComplexMeasure.{u_1} α m s t)) t","typeFull":"∀ {α : Type u_1} {m : MeasurableSpace α} (s t : MeasureTheory.SignedMeasure α),\n  MeasureTheory.ComplexMeasure.im (s.toComplexMeasure t) = t","typeReadable":"∀ {α : Type u_1} {m : MeasurableSpace α} (s t : MeasureTheory.SignedMeasure α),\n  MeasureTheory.ComplexMeasure.im (s.toComplexMeasure t) = 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,"toNonUnitalNormedCommRing"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","SignedMeasure","toComplexMeasure","_proof_3"],"typeFallback":"forall {α : Type.{u_1}} {m : MeasurableSpace.{u_1} α} (s : MeasureTheory.SignedMeasure.{u_1} α m) (t : MeasureTheory.SignedMeasure.{u_1} α m) (x._@.Mathlib.MeasureTheory.Measure.Complex.3851699544._hygCtx._hyg.37 : Nat -> (Set.{u_1} α)), (forall (i : Nat), MeasurableSet.{u_1} α m (x._@.Mathlib.MeasureTheory.Measure.Complex.3851699544._hygCtx._hyg.37 i)) -> (Pairwise.{0} Nat (Function.onFun.{1, succ u_1, 1} Nat (Set.{u_1} α) Prop (Disjoint.{u_1} (Set.{u_1} α) (OmegaCompletePartialOrder.toPartialOrder.{u_1} (Set.{u_1} α) (CompleteLattice.instOmegaCompletePartialOrder.{u_1} (Set.{u_1} α) (CompleteBooleanAlgebra.toCompleteLattice.{u_1} (Set.{u_1} α) (CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra.{u_1} (Set.{u_1} α) (Set.instCompleteAtomicBooleanAlgebra.{u_1} α))))) (HeytingAlgebra.toOrderBot.{u_1} (Set.{u_1} α) (Order.Frame.toHeytingAlgebra.{u_1} (Set.{u_1} α) (CompleteDistribLattice.toFrame.{u_1} (Set.{u_1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u_1} (Set.{u_1} α) (CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra.{u_1} (Set.{u_1} α) (Set.instCompleteAtomicBooleanAlgebra.{u_1} α))))))) x._@.Mathlib.MeasureTheory.Measure.Complex.3851699544._hygCtx._hyg.37)) -> (HasSum.{0, 0} Complex Nat (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (fun (i : Nat) => Complex.mk (MeasureTheory.VectorMeasure.measureOf'.{u_1, 0} α m Real Real.instAddCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) s (x._@.Mathlib.MeasureTheory.Measure.Complex.3851699544._hygCtx._hyg.37 i)) (MeasureTheory.VectorMeasure.measureOf'.{u_1, 0} α m Real Real.instAddCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) t (x._@.Mathlib.MeasureTheory.Measure.Complex.3851699544._hygCtx._hyg.37 i))) (Complex.mk (MeasureTheory.VectorMeasure.measureOf'.{u_1, 0} α m Real Real.instAddCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) s (Set.iUnion.{u_1, 1} α Nat (fun (i : Nat) => x._@.Mathlib.MeasureTheory.Measure.Complex.3851699544._hygCtx._hyg.37 i))) (MeasureTheory.VectorMeasure.measureOf'.{u_1, 0} α m Real Real.instAddCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) t (Set.iUnion.{u_1, 1} α Nat (fun (i : Nat) => x._@.Mathlib.MeasureTheory.Measure.Complex.3851699544._hygCtx._hyg.37 i)))) (SummationFilter.unconditional.{0} Nat))","typeFull":"∀ {α : Type u_1} {m : MeasurableSpace α} (s t : MeasureTheory.SignedMeasure α) (x : ℕ → Set α),\n  (∀ (i : ℕ), MeasurableSet (x i)) →\n    Pairwise (Function.onFun Disjoint x) →\n      HasSum (fun i => { re := ↑s (x i), im := ↑t (x i) }) { re := ↑s (⋃ i, x i), im := ↑t (⋃ i, x i) }","typeReadable":"∀ {α : Type u_1} {m : MeasurableSpace α} (s t : MeasureTheory.SignedMeasure α) (x : ℕ → Set α),\n  (∀ (i : ℕ), MeasurableSet (x i)) →\n    Pairwise (Function.onFun Disjoint x) →\n      HasSum (fun i => { re := ↑s (x i), im := ↑t (x i) }) { re := ↑s (⋃ i, x i), im := ↑t (⋃ i, x i) }","typeReferences":[["CompleteLattice","instOmegaCompletePartialOrder"],["CompleteAtomicBooleanAlgebra","toCompleteBooleanAlgebra"],["PseudoMetricSpace","toUniformSpace"],["SummationFilter","unconditional"],["OmegaCompletePartialOrder","toPartialOrder"],["UniformSpace","toTopologicalSpace"],["Function","onFun"],["Set","iUnion"],["CompleteBooleanAlgebra","toCompleteLattice"],["CompleteBooleanAlgebra","toCompleteDistribLattice"],["Pairwise"],["Complex"],["SeminormedRing","toPseudoMetricSpace"],["MeasureTheory","SignedMeasure"],["Order","Frame","toHeytingAlgebra"],["Complex","instNormedAddCommGroup"],["Disjoint"],["MeasurableSpace"],["HeytingAlgebra","toOrderBot"],["Real","instAddCommMonoid"],["NormedField","toNormedCommRing"],["Complex","mk"],["CompleteDistribLattice","toFrame"],["NormedAddCommGroup","toENormedAddCommMonoid"],["Real"],["Set"],["SeminormedCommRing","toSeminormedRing"],["Complex","instNormedField"],["Nat"],["Real","pseudoMetricSpace"],["Set","instCompleteAtomicBooleanAlgebra"],["HasSum"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["MeasurableSet"],["NormedCommRing","toSeminormedCommRing"],["MeasureTheory","VectorMeasure","measureOf'"],["ESeminormedAddCommMonoid","toAddCommMonoid"]],"valueReferences":[["PseudoMetricSpace","toUniformSpace"],["UniformSpace","toTopologicalSpace"],["SummationFilter","unconditional"],["Set","iUnion"],["Complex"],["And","intro"],["SeminormedRing","toPseudoMetricSpace"],["Complex","instNormedAddCommGroup"],["Real","instAddCommMonoid"],["Complex","im"],["NormedField","toNormedCommRing"],["Complex","mk"],["NormedAddCommGroup","toENormedAddCommMonoid"],["Real"],["Complex","hasSum_iff"],["And"],["SeminormedCommRing","toSeminormedRing"],["Complex","re"],["Complex","instNormedField"],["Nat"],["MeasureTheory","VectorMeasure","m_iUnion"],["HasSum"],["Real","pseudoMetricSpace"],["Iff","mpr"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["NormedCommRing","toSeminormedCommRing"],["MeasureTheory","VectorMeasure","measureOf'"],["ESeminormedAddCommMonoid","toAddCommMonoid"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","ComplexMeasure","equivSignedMeasure_apply"],"typeFallback":"forall {α : Type.{u_1}} {m : MeasurableSpace.{u_1} α} (c : MeasureTheory.ComplexMeasure.{u_1} α m), Eq.{succ u_1} (Prod.{u_1, u_1} (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m)) (DFunLike.coe.{succ u_1, succ u_1, succ u_1} (Equiv.{succ u_1, succ u_1} (MeasureTheory.ComplexMeasure.{u_1} α m) (Prod.{u_1, u_1} (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m))) (MeasureTheory.ComplexMeasure.{u_1} α m) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : MeasureTheory.ComplexMeasure.{u_1} α m) => Prod.{u_1, u_1} (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m)) (EquivLike.toFunLike.{succ u_1, succ u_1, succ u_1} (Equiv.{succ u_1, succ u_1} (MeasureTheory.ComplexMeasure.{u_1} α m) (Prod.{u_1, u_1} (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m))) (MeasureTheory.ComplexMeasure.{u_1} α m) (Prod.{u_1, u_1} (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m)) (Equiv.instEquivLike.{succ u_1, succ u_1} (MeasureTheory.ComplexMeasure.{u_1} α m) (Prod.{u_1, u_1} (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m)))) (MeasureTheory.ComplexMeasure.equivSignedMeasure.{u_1} α m) c) (Prod.mk.{u_1, u_1} (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m) (DFunLike.coe.{succ u_1, succ u_1, succ u_1} (LinearMap.{0, 0, u_1, u_1} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (MeasureTheory.ComplexMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.VectorMeasure.instAddCommMonoid.{u_1, 0} α m Complex (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) MeasureTheory.ComplexMeasure.re._proof_1) (MeasureTheory.VectorMeasure.instAddCommMonoid.{u_1, 0} α m Real Real.instAddCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) MeasureTheory.ComplexMeasure.re._proof_2) (MeasureTheory.VectorMeasure.instModule.{u_1, 0, 0} α m Complex (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) Real Real.semiring (NormedSpace.toModule.{0, 0} Real Complex Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (NormedSpace.complexToReal.{0} Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (NormedField.toNormedSpace.{0} Complex Complex.instNormedField))) MeasureTheory.ComplexMeasure.re._proof_3 MeasureTheory.ComplexMeasure.re._proof_1) (MeasureTheory.VectorMeasure.instModule.{u_1, 0, 0} α m Real Real.instAddCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)) MeasureTheory.ComplexMeasure.re._proof_4 MeasureTheory.ComplexMeasure.re._proof_2)) (MeasureTheory.ComplexMeasure.{u_1} α m) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : MeasureTheory.ComplexMeasure.{u_1} α m) => MeasureTheory.SignedMeasure.{u_1} α m) (LinearMap.instFunLike.{0, 0, u_1, u_1} Real Real (MeasureTheory.ComplexMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m) Real.semiring Real.semiring (MeasureTheory.VectorMeasure.instAddCommMonoid.{u_1, 0} α m Complex (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) MeasureTheory.ComplexMeasure.re._proof_1) (MeasureTheory.VectorMeasure.instAddCommMonoid.{u_1, 0} α m Real Real.instAddCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) MeasureTheory.ComplexMeasure.re._proof_2) (MeasureTheory.VectorMeasure.instModule.{u_1, 0, 0} α m Complex (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) Real Real.semiring (NormedSpace.toModule.{0, 0} Real Complex Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (NormedSpace.complexToReal.{0} Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (NormedField.toNormedSpace.{0} Complex Complex.instNormedField))) MeasureTheory.ComplexMeasure.re._proof_3 MeasureTheory.ComplexMeasure.re._proof_1) (MeasureTheory.VectorMeasure.instModule.{u_1, 0, 0} α m Real Real.instAddCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)) MeasureTheory.ComplexMeasure.re._proof_4 MeasureTheory.ComplexMeasure.re._proof_2) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring))) (MeasureTheory.ComplexMeasure.re.{u_1} α m) c) (DFunLike.coe.{succ u_1, succ u_1, succ u_1} (LinearMap.{0, 0, u_1, u_1} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (MeasureTheory.ComplexMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.VectorMeasure.instAddCommMonoid.{u_1, 0} α m Complex (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) MeasureTheory.ComplexMeasure.re._proof_1) (MeasureTheory.VectorMeasure.instAddCommMonoid.{u_1, 0} α m Real Real.instAddCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) MeasureTheory.ComplexMeasure.re._proof_2) (MeasureTheory.VectorMeasure.instModule.{u_1, 0, 0} α m Complex (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) Real Real.semiring (NormedSpace.toModule.{0, 0} Real Complex Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (NormedSpace.complexToReal.{0} Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (NormedField.toNormedSpace.{0} Complex Complex.instNormedField))) MeasureTheory.ComplexMeasure.re._proof_3 MeasureTheory.ComplexMeasure.re._proof_1) (MeasureTheory.VectorMeasure.instModule.{u_1, 0, 0} α m Real Real.instAddCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)) MeasureTheory.ComplexMeasure.re._proof_4 MeasureTheory.ComplexMeasure.re._proof_2)) (MeasureTheory.ComplexMeasure.{u_1} α m) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : MeasureTheory.ComplexMeasure.{u_1} α m) => MeasureTheory.SignedMeasure.{u_1} α m) (LinearMap.instFunLike.{0, 0, u_1, u_1} Real Real (MeasureTheory.ComplexMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m) Real.semiring Real.semiring (MeasureTheory.VectorMeasure.instAddCommMonoid.{u_1, 0} α m Complex (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) MeasureTheory.ComplexMeasure.re._proof_1) (MeasureTheory.VectorMeasure.instAddCommMonoid.{u_1, 0} α m Real Real.instAddCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) MeasureTheory.ComplexMeasure.re._proof_2) (MeasureTheory.VectorMeasure.instModule.{u_1, 0, 0} α m Complex (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) Real Real.semiring (NormedSpace.toModule.{0, 0} Real Complex Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (NormedSpace.complexToReal.{0} Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (NormedField.toNormedSpace.{0} Complex Complex.instNormedField))) MeasureTheory.ComplexMeasure.re._proof_3 MeasureTheory.ComplexMeasure.re._proof_1) (MeasureTheory.VectorMeasure.instModule.{u_1, 0, 0} α m Real Real.instAddCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)) MeasureTheory.ComplexMeasure.re._proof_4 MeasureTheory.ComplexMeasure.re._proof_2) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring))) (MeasureTheory.ComplexMeasure.im.{u_1} α m) c))","typeFull":"∀ {α : Type u_1} {m : MeasurableSpace α} (c : MeasureTheory.ComplexMeasure α),\n  MeasureTheory.ComplexMeasure.equivSignedMeasure c =\n    (MeasureTheory.ComplexMeasure.re c, MeasureTheory.ComplexMeasure.im c)","typeReadable":"∀ {α : Type u_1} {m : MeasurableSpace α} (c : MeasureTheory.ComplexMeasure α),\n  MeasureTheory.ComplexMeasure.equivSignedMeasure c =\n    (MeasureTheory.ComplexMeasure.re c, MeasureTheory.ComplexMeasure.im c)","typeReferences":[["NonUnitalSeminormedCommRing","toNonUnitalSeminormedRing"],["Equiv","instEquivLike"],["LinearMap","instFunLike"],["PseudoMetricSpace","toUniformSpace"],["Prod","mk"],["Real","normedCommRing"],["UniformSpace","toTopologicalSpace"],["MeasureTheory","ComplexMeasure","re","_proof_2"],["MeasureTheory","ComplexMeasure","equivSignedMeasure"],["DFunLike","coe"],["Equiv"],["Complex"],["SeminormedRing","toPseudoMetricSpace"],["MeasureTheory","ComplexMeasure","re","_proof_3"],["SeminormedCommRing","toNonUnitalSeminormedCommRing"],["MeasureTheory","SignedMeasure"],["Semiring","toNonAssocSemiring"],["Complex","instNormedAddCommGroup"],["RingHom","id"],["EquivLike","toFunLike"],["MeasurableSpace"],["MeasureTheory","VectorMeasure","instAddCommMonoid"],["Real","instAddCommMonoid"],["Eq"],["NormedField","toNormedCommRing"],["NormedField","toNormedSpace"],["NonUnitalSeminormedRing","toSeminormedAddCommGroup"],["NormedAddCommGroup","toENormedAddCommMonoid"],["NormedSpace","toModule"],["Real"],["LinearMap"],["SeminormedCommRing","toSeminormedRing"],["Complex","instNormedField"],["Real","semiring"],["Prod"],["MeasureTheory","ComplexMeasure","re","_proof_4"],["Real","normedField"],["Real","pseudoMetricSpace"],["MeasureTheory","VectorMeasure","instModule"],["MeasureTheory","ComplexMeasure","re","_proof_1"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["MeasureTheory","ComplexMeasure","re"],["MeasureTheory","ComplexMeasure"],["NormedCommRing","toSeminormedCommRing"],["MeasureTheory","ComplexMeasure","im"],["NormedSpace","complexToReal"],["ESeminormedAddCommMonoid","toAddCommMonoid"]],"valueReferences":[["Prod"],["MeasureTheory","SignedMeasure"],["Equiv","instEquivLike"],["Eq","refl"],["EquivLike","toFunLike"],["MeasureTheory","ComplexMeasure"],["MeasureTheory","ComplexMeasure","equivSignedMeasure"],["DFunLike","coe"],["Equiv"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","ComplexMeasure","toComplexMeasure_to_signedMeasure"],"typeFallback":"forall {α : Type.{u_1}} {m : MeasurableSpace.{u_1} α} (c : MeasureTheory.ComplexMeasure.{u_1} α m), Eq.{succ u_1} (MeasureTheory.ComplexMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.toComplexMeasure.{u_1} α m (DFunLike.coe.{succ u_1, succ u_1, succ u_1} (LinearMap.{0, 0, u_1, u_1} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (MeasureTheory.ComplexMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.VectorMeasure.instAddCommMonoid.{u_1, 0} α m Complex (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (IsSemitopologicalSemiring.toContinuousAdd.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{0} Complex (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{0} Complex (NonUnitalNormedCommRing.toNonUnitalCommRing.{0} Complex (NormedCommRing.toNonUnitalNormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (IsSemitopologicalRing.toIsSemitopologicalSemiring.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{0} Complex (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{0} Complex (NonUnitalNormedCommRing.toNonUnitalCommRing.{0} Complex (NormedCommRing.toNonUnitalNormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (IsTopologicalRing.toIsSemitopologicalRing.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{0} Complex (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{0} Complex (NonUnitalNormedCommRing.toNonUnitalCommRing.{0} Complex (NormedCommRing.toNonUnitalNormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (IsTopologicalDivisionRing.toIsTopologicalRing.{0} Complex (NormedDivisionRing.toDivisionRing.{0} Complex (NormedField.toNormedDivisionRing.{0} Complex Complex.instNormedField)) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedDivisionRing.to_isTopologicalDivisionRing.{0} Complex (NormedField.toNormedDivisionRing.{0} Complex Complex.instNormedField))))))) (MeasureTheory.VectorMeasure.instAddCommMonoid.{u_1, 0} α m Real Real.instAddCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (IsSemitopologicalSemiring.toContinuousAdd.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Real (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{0} Real (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{0} Real (NonUnitalNormedCommRing.toNonUnitalCommRing.{0} Real (NormedCommRing.toNonUnitalNormedCommRing.{0} Real Real.normedCommRing))))) (IsSemitopologicalRing.toIsSemitopologicalSemiring.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{0} Real (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{0} Real (NonUnitalNormedCommRing.toNonUnitalCommRing.{0} Real (NormedCommRing.toNonUnitalNormedCommRing.{0} Real Real.normedCommRing)))) (IsTopologicalRing.toIsSemitopologicalRing.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{0} Real (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{0} Real (NonUnitalNormedCommRing.toNonUnitalCommRing.{0} Real (NormedCommRing.toNonUnitalNormedCommRing.{0} Real Real.normedCommRing)))) instIsTopologicalRingReal)))) (MeasureTheory.VectorMeasure.instModule.{u_1, 0, 0} α m Complex (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) Real Real.semiring (NormedSpace.toModule.{0, 0} Real Complex Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (NormedSpace.complexToReal.{0} Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (NormedField.toNormedSpace.{0} Complex Complex.instNormedField))) (UniformContinuousConstSMul.to_continuousConstSMul.{0, 0} Real Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (SMulZeroClass.toSMul.{0, 0} Real Complex (AddZero.toZero.{0} Complex (AddZeroClass.toAddZero.{0} Complex (AddMonoid.toAddZeroClass.{0} Complex (AddCommMonoid.toAddMonoid.{0} Complex (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))))))) (DistribSMul.toSMulZeroClass.{0, 0} Real Complex (AddMonoid.toAddZeroClass.{0} Complex (AddCommMonoid.toAddMonoid.{0} Complex (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))))) (DistribMulAction.toDistribSMul.{0, 0} Real Complex (MonoidWithZero.toMonoid.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)) (AddCommMonoid.toAddMonoid.{0} Complex (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup)))) (Module.toDistribMulAction.{0, 0} Real Complex Real.semiring (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))) (NormedSpace.toModule.{0, 0} Real Complex Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (NormedSpace.complexToReal.{0} Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (NormedField.toNormedSpace.{0} Complex Complex.instNormedField))))))) (IsBoundedSMul.toUniformContinuousConstSMul.{0, 0} Real Complex Real.pseudoMetricSpace (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))) Real.instZero Complex.instZero (SMulZeroClass.toSMul.{0, 0} Real Complex (AddZero.toZero.{0} Complex (AddZeroClass.toAddZero.{0} Complex (AddMonoid.toAddZeroClass.{0} Complex (AddCommMonoid.toAddMonoid.{0} Complex (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) 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(NonUnitalCommRing.toNonUnitalNonAssocCommRing.{0} Real (NonUnitalNormedCommRing.toNonUnitalCommRing.{0} Real (NormedCommRing.toNonUnitalNormedCommRing.{0} Real Real.normedCommRing)))) instIsTopologicalRingReal)))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring))) (MeasureTheory.ComplexMeasure.im.{u_1} α m) c)) c","typeFull":"∀ {α : Type u_1} {m : MeasurableSpace α} (c : MeasureTheory.ComplexMeasure α),\n  (MeasureTheory.ComplexMeasure.re c).toComplexMeasure (MeasureTheory.ComplexMeasure.im c) = c","typeReadable":"∀ {α : Type u_1} {m : MeasurableSpace α} (c : MeasureTheory.ComplexMeasure α),\n  (MeasureTheory.ComplexMeasure.re c).toComplexMeasure (MeasureTheory.ComplexMeasure.im c) = c","typeReferences":[["IsSemitopologicalSemiring","toSeparatelyContinuousMul"],["PseudoMetricSpace","toUniformSpace"],["Real","normedCommRing"],["SMulZeroClass","toSMul"],["Complex"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["IsSemitopologicalSemiring","toContinuousAdd"],["RingHom","id"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["MeasureTheory","VectorMeasure","instAddCommMonoid"],["Real","instAddCommMonoid"],["NormedField","toNormedCommRing"],["DistribSMul","toSMulZeroClass"],["NonUnitalSeminormedRing","toSeminormedAddCommGroup"],["NormedAddCommGroup","toENormedAddCommMonoid"],["NormedSpace","toModule"],["Real"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["IsBoundedSMul","toUniformContinuousConstSMul"],["DistribMulAction","toDistribSMul"],["instIsTopologicalRingReal"],["AddZeroClass","toAddZero"],["SeminormedCommRing","toSeminormedRing"],["LinearMap"],["Complex","instNormedField"],["Complex","instZero"],["Real","pseudoMetricSpace"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["IsTopologicalRing","toIsSemitopologicalRing"],["MeasureTheory","ComplexMeasure"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["MeasureTheory","ComplexMeasure","im"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["NonUnitalSeminormedCommRing","toNonUnitalSeminormedRing"],["LinearMap","instFunLike"],["UniformSpace","toTopologicalSpace"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["UniformContinuousConstSMul","to_continuousConstSMul"],["MeasureTheory","SignedMeasure","toComplexMeasure"],["SeminormedRing","toPseudoMetricSpace"],["MeasureTheory","SignedMeasure"],["SeminormedCommRing","toNonUnitalSeminormedCommRing"],["Complex","instNormedAddCommGroup"],["MonoidWithZero","toMonoid"],["MeasurableSpace"],["Eq"],["Real","instMul"],["SeparatelyContinuousMul","to_continuousSMul"],["NormedField","toNormedSpace"],["Semiring","toMonoidWithZero"],["Real","semiring"],["NormedDivisionRing","to_isTopologicalDivisionRing"],["Module","toDistribMulAction"],["Real","instZero"],["Real","normedField"],["IsSemitopologicalRing","toIsSemitopologicalSemiring"],["MeasureTheory","VectorMeasure","instModule"],["MeasureTheory","ComplexMeasure","re"],["IsTopologicalDivisionRing","toIsTopologicalRing"],["NormedCommRing","toSeminormedCommRing"],["NormedDivisionRing","toDivisionRing"],["NormedField","toNormedDivisionRing"],["NormedSpace","toIsBoundedSMul"],["NormedSpace","complexToReal"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["NormedCommRing","toNonUnitalNormedCommRing"]],"valueReferences":[["IsSemitopologicalSemiring","toSeparatelyContinuousMul"],["PseudoMetricSpace","toUniformSpace"],["Real","normedCommRing"],["SMulZeroClass","toSMul"],["Complex"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["IsSemitopologicalSemiring","toContinuousAdd"],["RingHom","id"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["MeasureTheory","VectorMeasure","instAddCommMonoid"],["Real","instAddCommMonoid"],["NormedField","toNormedCommRing"],["DistribSMul","toSMulZeroClass"],["rfl"],["NonUnitalSeminormedRing","toSeminormedAddCommGroup"],["NormedAddCommGroup","toENormedAddCommMonoid"],["NormedSpace","toModule"],["Real"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["IsBoundedSMul","toUniformContinuousConstSMul"],["DistribMulAction","toDistribSMul"],["instIsTopologicalRingReal"],["AddZeroClass","toAddZero"],["SeminormedCommRing","toSeminormedRing"],["LinearMap"],["Complex","instNormedField"],["Complex","instZero"],["Real","pseudoMetricSpace"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["IsTopologicalRing","toIsSemitopologicalRing"],["MeasureTheory","ComplexMeasure"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["MeasureTheory","ComplexMeasure","im"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["NonUnitalSeminormedCommRing","toNonUnitalSeminormedRing"],["LinearMap","instFunLike"],["UniformSpace","toTopologicalSpace"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["UniformContinuousConstSMul","to_continuousConstSMul"],["MeasureTheory","SignedMeasure","toComplexMeasure"],["SeminormedRing","toPseudoMetricSpace"],["MeasureTheory","SignedMeasure"],["SeminormedCommRing","toNonUnitalSeminormedCommRing"],["Complex","instNormedAddCommGroup"],["MonoidWithZero","toMonoid"],["Real","instMul"],["SeparatelyContinuousMul","to_continuousSMul"],["NormedField","toNormedSpace"],["Semiring","toMonoidWithZero"],["Real","semiring"],["NormedDivisionRing","to_isTopologicalDivisionRing"],["Module","toDistribMulAction"],["Real","instZero"],["Real","normedField"],["IsSemitopologicalRing","toIsSemitopologicalSemiring"],["MeasureTheory","VectorMeasure","instModule"],["MeasureTheory","ComplexMeasure","re"],["IsTopologicalDivisionRing","toIsTopologicalRing"],["NormedCommRing","toSeminormedCommRing"],["NormedDivisionRing","toDivisionRing"],["NormedField","toNormedDivisionRing"],["NormedSpace","toIsBoundedSMul"],["NormedSpace","complexToReal"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["NormedCommRing","toNonUnitalNormedCommRing"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","ComplexMeasure","equivSignedMeasure","_proof_1"],"typeFallback":"forall {α : Type.{u_1}} {m : MeasurableSpace.{u_1} α} (x._@.Mathlib.MeasureTheory.Measure.Complex.2046710636._hygCtx._hyg.64 : Prod.{u_1, u_1} (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m)), Eq.{succ u_1} (Prod.{u_1, u_1} (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m)) (Prod.mk.{u_1, u_1} (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m) (DFunLike.coe.{succ u_1, succ u_1, succ u_1} (LinearMap.{0, 0, u_1, u_1} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (MeasureTheory.ComplexMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.VectorMeasure.instAddCommMonoid.{u_1, 0} α m Complex (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) MeasureTheory.ComplexMeasure.re._proof_1) (MeasureTheory.VectorMeasure.instAddCommMonoid.{u_1, 0} α m Real Real.instAddCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) MeasureTheory.ComplexMeasure.re._proof_2) (MeasureTheory.VectorMeasure.instModule.{u_1, 0, 0} α m Complex (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) Real Real.semiring (NormedSpace.toModule.{0, 0} Real Complex Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (NormedSpace.complexToReal.{0} Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (NormedField.toNormedSpace.{0} Complex Complex.instNormedField))) MeasureTheory.ComplexMeasure.re._proof_3 MeasureTheory.ComplexMeasure.re._proof_1) (MeasureTheory.VectorMeasure.instModule.{u_1, 0, 0} α m Real Real.instAddCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)) MeasureTheory.ComplexMeasure.re._proof_4 MeasureTheory.ComplexMeasure.re._proof_2)) (MeasureTheory.ComplexMeasure.{u_1} α m) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : MeasureTheory.ComplexMeasure.{u_1} α m) => MeasureTheory.SignedMeasure.{u_1} α m) (LinearMap.instFunLike.{0, 0, u_1, u_1} Real Real (MeasureTheory.ComplexMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m) Real.semiring Real.semiring (MeasureTheory.VectorMeasure.instAddCommMonoid.{u_1, 0} α m Complex (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) MeasureTheory.ComplexMeasure.re._proof_1) (MeasureTheory.VectorMeasure.instAddCommMonoid.{u_1, 0} α m Real Real.instAddCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) MeasureTheory.ComplexMeasure.re._proof_2) (MeasureTheory.VectorMeasure.instModule.{u_1, 0, 0} α m Complex (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) Real Real.semiring (NormedSpace.toModule.{0, 0} Real Complex Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (NormedSpace.complexToReal.{0} Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (NormedField.toNormedSpace.{0} Complex Complex.instNormedField))) MeasureTheory.ComplexMeasure.re._proof_3 MeasureTheory.ComplexMeasure.re._proof_1) (MeasureTheory.VectorMeasure.instModule.{u_1, 0, 0} α m Real Real.instAddCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)) MeasureTheory.ComplexMeasure.re._proof_4 MeasureTheory.ComplexMeasure.re._proof_2) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring))) (MeasureTheory.ComplexMeasure.re.{u_1} α m) (MeasureTheory.ComplexMeasure.equivSignedMeasure.match_1.{u_1, succ u_1} α m (fun (x._@.Mathlib.MeasureTheory.Measure.Complex.2046710636._hygCtx.37.Mathlib.MeasureTheory.Measure.Complex.2046710636._hygCtx._hyg.45 : Prod.{u_1, u_1} (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m)) => MeasureTheory.ComplexMeasure.{u_1} α m) x._@.Mathlib.MeasureTheory.Measure.Complex.2046710636._hygCtx._hyg.64 (fun (s : MeasureTheory.SignedMeasure.{u_1} α m) (t : MeasureTheory.SignedMeasure.{u_1} α m) => MeasureTheory.SignedMeasure.toComplexMeasure.{u_1} α m s t))) (DFunLike.coe.{succ u_1, succ u_1, succ u_1} (LinearMap.{0, 0, u_1, u_1} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (MeasureTheory.ComplexMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.VectorMeasure.instAddCommMonoid.{u_1, 0} α m Complex (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) MeasureTheory.ComplexMeasure.re._proof_1) (MeasureTheory.VectorMeasure.instAddCommMonoid.{u_1, 0} α m Real Real.instAddCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) MeasureTheory.ComplexMeasure.re._proof_2) (MeasureTheory.VectorMeasure.instModule.{u_1, 0, 0} α m Complex (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) Real Real.semiring (NormedSpace.toModule.{0, 0} Real Complex Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (NormedSpace.complexToReal.{0} Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (NormedField.toNormedSpace.{0} Complex Complex.instNormedField))) MeasureTheory.ComplexMeasure.re._proof_3 MeasureTheory.ComplexMeasure.re._proof_1) (MeasureTheory.VectorMeasure.instModule.{u_1, 0, 0} α m Real Real.instAddCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)) MeasureTheory.ComplexMeasure.re._proof_4 MeasureTheory.ComplexMeasure.re._proof_2)) (MeasureTheory.ComplexMeasure.{u_1} α m) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : MeasureTheory.ComplexMeasure.{u_1} α m) => MeasureTheory.SignedMeasure.{u_1} α m) (LinearMap.instFunLike.{0, 0, u_1, u_1} Real Real (MeasureTheory.ComplexMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m) Real.semiring Real.semiring (MeasureTheory.VectorMeasure.instAddCommMonoid.{u_1, 0} α m Complex (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) MeasureTheory.ComplexMeasure.re._proof_1) (MeasureTheory.VectorMeasure.instAddCommMonoid.{u_1, 0} α m Real Real.instAddCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) MeasureTheory.ComplexMeasure.re._proof_2) (MeasureTheory.VectorMeasure.instModule.{u_1, 0, 0} α m Complex (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex 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Complex Complex.instNormedField)))))) Real Real.semiring (NormedSpace.toModule.{0, 0} Real Complex Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (NormedSpace.complexToReal.{0} Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Complex (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField))))) (NormedField.toNormedSpace.{0} Complex Complex.instNormedField))) MeasureTheory.ComplexMeasure.re._proof_3 MeasureTheory.ComplexMeasure.re._proof_1) (MeasureTheory.VectorMeasure.instModule.{u_1, 0, 0} α m Real Real.instAddCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)) MeasureTheory.ComplexMeasure.re._proof_4 MeasureTheory.ComplexMeasure.re._proof_2) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring))) (MeasureTheory.ComplexMeasure.im.{u_1} α m) (MeasureTheory.ComplexMeasure.equivSignedMeasure.match_1.{u_1, succ u_1} α m (fun (x._@.Mathlib.MeasureTheory.Measure.Complex.2046710636._hygCtx.37.Mathlib.MeasureTheory.Measure.Complex.2046710636._hygCtx._hyg.45 : Prod.{u_1, u_1} (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m)) => MeasureTheory.ComplexMeasure.{u_1} α m) x._@.Mathlib.MeasureTheory.Measure.Complex.2046710636._hygCtx._hyg.64 (fun (s : MeasureTheory.SignedMeasure.{u_1} α m) (t : MeasureTheory.SignedMeasure.{u_1} α m) => MeasureTheory.SignedMeasure.toComplexMeasure.{u_1} α m s t)))) x._@.Mathlib.MeasureTheory.Measure.Complex.2046710636._hygCtx._hyg.64","typeFull":"∀ {α : Type u_1} {m : MeasurableSpace α} (x : MeasureTheory.SignedMeasure α × MeasureTheory.SignedMeasure α),\n  (MeasureTheory.ComplexMeasure.re\n        (match x with\n        | (s, t) => s.toComplexMeasure t),\n      MeasureTheory.ComplexMeasure.im\n        (match x with\n        | (s, t) => s.toComplexMeasure t)) =\n    x","typeReadable":"∀ {α : Type u_1} {m : MeasurableSpace α} (x : MeasureTheory.SignedMeasure α × MeasureTheory.SignedMeasure α),\n  (MeasureTheory.ComplexMeasure.re\n        (match x with\n        | (s, t) => s.toComplexMeasure t),\n      MeasureTheory.ComplexMeasure.im\n        (match x with\n        | (s, t) => s.toComplexMeasure t)) =\n    x","typeReferences":[["NonUnitalSeminormedCommRing","toNonUnitalSeminormedRing"],["LinearMap","instFunLike"],["PseudoMetricSpace","toUniformSpace"],["Prod","mk"],["Real","normedCommRing"],["UniformSpace","toTopologicalSpace"],["MeasureTheory","ComplexMeasure","re","_proof_2"],["DFunLike","coe"],["MeasureTheory","SignedMeasure","toComplexMeasure"],["Complex"],["SeminormedRing","toPseudoMetricSpace"],["MeasureTheory","ComplexMeasure","re","_proof_3"],["SeminormedCommRing","toNonUnitalSeminormedCommRing"],["MeasureTheory","SignedMeasure"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Complex","instNormedAddCommGroup"],["MeasureTheory","VectorMeasure","instAddCommMonoid"],["MeasurableSpace"],["Real","instAddCommMonoid"],["Eq"],["NormedField","toNormedCommRing"],["NormedField","toNormedSpace"],["NonUnitalSeminormedRing","toSeminormedAddCommGroup"],["NormedAddCommGroup","toENormedAddCommMonoid"],["NormedSpace","toModule"],["Real"],["MeasureTheory","ComplexMeasure","equivSignedMeasure","match_1"],["LinearMap"],["SeminormedCommRing","toSeminormedRing"],["Real","semiring"],["Complex","instNormedField"],["Prod"],["MeasureTheory","ComplexMeasure","re","_proof_4"],["Real","normedField"],["Real","pseudoMetricSpace"],["MeasureTheory","VectorMeasure","instModule"],["MeasureTheory","ComplexMeasure","re","_proof_1"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["MeasureTheory","ComplexMeasure","re"],["MeasureTheory","ComplexMeasure"],["NormedCommRing","toSeminormedCommRing"],["MeasureTheory","ComplexMeasure","im"],["NormedSpace","complexToReal"],["ESeminormedAddCommMonoid","toAddCommMonoid"]],"valueReferences":[["IsSemitopologicalSemiring","toSeparatelyContinuousMul"],["PseudoMetricSpace","toUniformSpace"],["Prod","mk"],["Real","normedCommRing"],["SMulZeroClass","toSMul"],["Complex"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["IsSemitopologicalSemiring","toContinuousAdd"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["MeasureTheory","VectorMeasure","instAddCommMonoid"],["Real","instAddCommMonoid"],["NormedField","toNormedCommRing"],["MeasureTheory","SignedMeasure","re_toComplexMeasure"],["DistribSMul","toSMulZeroClass"],["NonUnitalSeminormedRing","toSeminormedAddCommGroup"],["NormedAddCommGroup","toENormedAddCommMonoid"],["NormedSpace","toModule"],["Real"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["IsBoundedSMul","toUniformContinuousConstSMul"],["DistribMulAction","toDistribSMul"],["instIsTopologicalRingReal"],["AddZeroClass","toAddZero"],["SeminormedCommRing","toSeminormedRing"],["LinearMap"],["Complex","instNormedField"],["Prod"],["Complex","instZero"],["Real","pseudoMetricSpace"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["IsTopologicalRing","toIsSemitopologicalRing"],["MeasureTheory","ComplexMeasure"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["MeasureTheory","ComplexMeasure","im"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["MeasureTheory","SignedMeasure","im_toComplexMeasure"],["NonUnitalSeminormedCommRing","toNonUnitalSeminormedRing"],["LinearMap","instFunLike"],["UniformSpace","toTopologicalSpace"],["AddCommMonoid","toAddMonoid"],["UniformContinuousConstSMul","to_continuousConstSMul"],["DFunLike","coe"],["MeasureTheory","SignedMeasure","toComplexMeasure"],["SeminormedRing","toPseudoMetricSpace"],["MeasureTheory","SignedMeasure"],["SeminormedCommRing","toNonUnitalSeminormedCommRing"],["Complex","instNormedAddCommGroup"],["MonoidWithZero","toMonoid"],["Eq"],["Real","instMul"],["SeparatelyContinuousMul","to_continuousSMul"],["NormedField","toNormedSpace"],["MeasureTheory","ComplexMeasure","equivSignedMeasure","match_1"],["Prod","ext"],["Semiring","toMonoidWithZero"],["Real","semiring"],["NormedDivisionRing","to_isTopologicalDivisionRing"],["Module","toDistribMulAction"],["Real","instZero"],["Real","normedField"],["IsSemitopologicalRing","toIsSemitopologicalSemiring"],["MeasureTheory","VectorMeasure","instModule"],["MeasureTheory","ComplexMeasure","re"],["IsTopologicalDivisionRing","toIsTopologicalRing"],["NormedCommRing","toSeminormedCommRing"],["NormedDivisionRing","toDivisionRing"],["NormedField","toNormedDivisionRing"],["NormedSpace","toIsBoundedSMul"],["NormedSpace","complexToReal"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["MeasureTheory","ComplexMeasure","equivSignedMeasure","match_3"],["NormedCommRing","toNonUnitalNormedCommRing"]]},{"isProp":true,"kind":"definition","name":["MeasureTheory","ComplexMeasure","equivSignedMeasure","match_3"],"typeFallback":"forall {α : Type.{u_1}} {m : MeasurableSpace.{u_1} α} (motive : (Prod.{u_1, u_1} (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m)) -> Prop) (x._@.Mathlib.MeasureTheory.Measure.Complex.2046710636._hygCtx.64.Mathlib.MeasureTheory.Measure.Complex.2046710636._hygCtx._hyg.72 : Prod.{u_1, u_1} (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m)), (forall (s : MeasureTheory.SignedMeasure.{u_1} α m) (t : MeasureTheory.SignedMeasure.{u_1} α m), motive (Prod.mk.{u_1, u_1} (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m) s t)) -> (motive x._@.Mathlib.MeasureTheory.Measure.Complex.2046710636._hygCtx.64.Mathlib.MeasureTheory.Measure.Complex.2046710636._hygCtx._hyg.72)","typeFull":"∀ {α : Type u_1} {m : MeasurableSpace α} (motive : MeasureTheory.SignedMeasure α × MeasureTheory.SignedMeasure α → Prop)\n  (x : MeasureTheory.SignedMeasure α × MeasureTheory.SignedMeasure α),\n  (∀ (s t : MeasureTheory.SignedMeasure α), motive (s, t)) → motive x","typeReadable":"∀ {α : Type u_1} {m : MeasurableSpace α} (motive : MeasureTheory.SignedMeasure α × MeasureTheory.SignedMeasure α → Prop)\n  (x : MeasureTheory.SignedMeasure α × MeasureTheory.SignedMeasure α),\n  (∀ (s t : MeasureTheory.SignedMeasure α), motive (s, t)) → motive x","typeReferences":[["Prod"],["MeasureTheory","SignedMeasure"],["Prod","mk"],["MeasurableSpace"]],"valueReferences":[["Prod","casesOn"],["MeasureTheory","SignedMeasure"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","ComplexMeasure","equivSignedMeasureₗ","_proof_1"],"typeFallback":"forall {α : Type.{u_1}} {m : MeasurableSpace.{u_1} α} (c : MeasureTheory.ComplexMeasure.{u_1} α m) (d : MeasureTheory.ComplexMeasure.{u_1} α m), Eq.{succ u_1} (Prod.{u_1, u_1} (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m)) (Equiv.toFun.{succ u_1, succ u_1} (MeasureTheory.ComplexMeasure.{u_1} α m) (Prod.{u_1, u_1} (MeasureTheory.SignedMeasure.{u_1} α m) (MeasureTheory.SignedMeasure.{u_1} α m)) (MeasureTheory.ComplexMeasure.equivSignedMeasure.{u_1} α m) (HAdd.hAdd.{u_1, u_1, u_1} (MeasureTheory.ComplexMeasure.{u_1} α m) (MeasureTheory.ComplexMeasure.{u_1} α m) (MeasureTheory.ComplexMeasure.{u_1} α m) (instHAdd.{u_1} (MeasureTheory.ComplexMeasure.{u_1} α m) (AddCommMagma.toAdd.{u_1} (MeasureTheory.ComplexMeasure.{u_1} α m) (AddCommSemigroup.toAddCommMagma.{u_1} (MeasureTheory.ComplexMeasure.{u_1} α m) (AddCommMonoid.toAddCommSemigroup.{u_1} (MeasureTheory.ComplexMeasure.{u_1} α m) (MeasureTheory.VectorMeasure.instAddCommMonoid.{u_1, 0} α m Complex (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) 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(NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{0} Real (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{0} Real (NonUnitalNormedCommRing.toNonUnitalCommRing.{0} Real (NormedCommRing.toNonUnitalNormedCommRing.{0} Real Real.normedCommRing))))) (IsSemitopologicalRing.toIsSemitopologicalSemiring.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{0} Real (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{0} Real (NonUnitalNormedCommRing.toNonUnitalCommRing.{0} Real (NormedCommRing.toNonUnitalNormedCommRing.{0} Real Real.normedCommRing)))) (IsTopologicalRing.toIsSemitopologicalRing.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{0} Real (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{0} Real (NonUnitalNormedCommRing.toNonUnitalCommRing.{0} Real (NormedCommRing.toNonUnitalNormedCommRing.{0} Real Real.normedCommRing)))) instIsTopologicalRingReal)))) (MeasureTheory.VectorMeasure.instAddCommMonoid.{u_1, 0} α m Real Real.instAddCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (IsSemitopologicalSemiring.toContinuousAdd.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Real (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{0} Real (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{0} Real (NonUnitalNormedCommRing.toNonUnitalCommRing.{0} Real (NormedCommRing.toNonUnitalNormedCommRing.{0} Real Real.normedCommRing))))) (IsSemitopologicalRing.toIsSemitopologicalSemiring.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{0} Real 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(MeasureTheory.SignedMeasure.{u_1} α m)) (MeasureTheory.ComplexMeasure.equivSignedMeasure.{u_1} α m) d))","typeFull":"∀ {α : Type u_1} {m : MeasurableSpace α} (c d : MeasureTheory.ComplexMeasure α),\n  MeasureTheory.ComplexMeasure.equivSignedMeasure.toFun (c + d) =\n    MeasureTheory.ComplexMeasure.equivSignedMeasure.toFun c + MeasureTheory.ComplexMeasure.equivSignedMeasure.toFun d","typeReadable":"∀ {α : Type u_1} {m : MeasurableSpace α} (c d : MeasureTheory.ComplexMeasure α),\n  MeasureTheory.ComplexMeasure.equivSignedMeasure.toFun (c + d) =\n    MeasureTheory.ComplexMeasure.equivSignedMeasure.toFun c + MeasureTheory.ComplexMeasure.equivSignedMeasure.toFun d","typeReferences":[["PseudoMetricSpace","toUniformSpace"],["Real","normedCommRing"],["UniformSpace","toTopologicalSpace"],["MeasureTheory","ComplexMeasure","equivSignedMeasure"],["Prod","instAddCommMonoid"],["Complex"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["SeminormedRing","toPseudoMetricSpace"],["MeasureTheory","SignedMeasure"],["IsSemitopologicalSemiring","toContinuousAdd"],["Complex","instNormedAddCommGroup"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["MeasureTheory","VectorMeasure","instAddCommMonoid"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Equiv","toFun"],["MeasurableSpace"],["Real","instAddCommMonoid"],["Eq"],["NormedField","toNormedCommRing"],["NormedAddCommGroup","toENormedAddCommMonoid"],["Real"],["instHAdd"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["instIsTopologicalRingReal"],["SeminormedCommRing","toSeminormedRing"],["NormedDivisionRing","to_isTopologicalDivisionRing"],["Complex","instNormedField"],["HAdd","hAdd"],["Prod"],["AddCommMonoid","toAddCommSemigroup"],["Real","pseudoMetricSpace"],["IsSemitopologicalRing","toIsSemitopologicalSemiring"],["IsTopologicalRing","toIsSemitopologicalRing"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["IsTopologicalDivisionRing","toIsTopologicalRing"],["MeasureTheory","ComplexMeasure"],["NormedField","toNormedDivisionRing"],["NormedDivisionRing","toDivisionRing"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["NormedCommRing","toSeminormedCommRing"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["NormedCommRing","toNonUnitalNormedCommRing"]],"valueReferences":[["PseudoMetricSpace","toUniformSpace"],["UniformSpace","toTopologicalSpace"],["MeasureTheory","ComplexMeasure","equivSignedMeasure"],["Complex"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["SeminormedRing","toPseudoMetricSpace"],["MeasureTheory","SignedMeasure"],["Complex","instNormedAddCommGroup"],["IsSemitopologicalSemiring","toContinuousAdd"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["MeasureTheory","VectorMeasure","instAddCommMonoid"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Equiv","toFun"],["NormedField","toNormedCommRing"],["NormedAddCommGroup","toENormedAddCommMonoid"],["instHAdd"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["SeminormedCommRing","toSeminormedRing"],["NormedDivisionRing","to_isTopologicalDivisionRing"],["Complex","instNormedField"],["HAdd","hAdd"],["Prod"],["AddCommMonoid","toAddCommSemigroup"],["IsSemitopologicalRing","toIsSemitopologicalSemiring"],["Eq","refl"],["IsTopologicalRing","toIsSemitopologicalRing"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["IsTopologicalDivisionRing","toIsTopologicalRing"],["MeasureTheory","ComplexMeasure"],["NormedField","toNormedDivisionRing"],["NormedDivisionRing","toDivisionRing"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["NormedCommRing","toSeminormedCommRing"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["NormedCommRing","toNonUnitalNormedCommRing"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","ComplexMeasure","equivSignedMeasureₗ_apply"],"typeFallback":"forall {α : Type.{u_1}} {m : MeasurableSpace.{u_1} α} {R : Type.{u_2}} [inst._@.Mathlib.MeasureTheory.Measure.Complex.406428577._hygCtx._hyg.6 : Semiring.{u_2} R] [inst._@.Mathlib.MeasureTheory.Measure.Complex.406428577._hygCtx._hyg.9 : Module.{u_2, 0} R Real inst._@.Mathlib.MeasureTheory.Measure.Complex.406428577._hygCtx._hyg.6 Real.instAddCommMonoid] [inst._@.Mathlib.MeasureTheory.Measure.Complex.406428577._hygCtx._hyg.15 : ContinuousConstSMul.{u_2, 0} R Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (SMulZeroClass.toSMul.{u_2, 0} R Real (AddZero.toZero.{0} Real (AddZeroClass.toAddZero.{0} Real (AddMonoid.toAddZeroClass.{0} Real Real.instAddMonoid))) (DistribSMul.toSMulZeroClass.{u_2, 0} R Real (AddMonoid.toAddZeroClass.{0} Real Real.instAddMonoid) (DistribMulAction.toDistribSMul.{u_2, 0} R Real (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.MeasureTheory.Measure.Complex.406428577._hygCtx._hyg.6)) Real.instAddMonoid (Module.toDistribMulAction.{u_2, 0} R Real inst._@.Mathlib.MeasureTheory.Measure.Complex.406428577._hygCtx._hyg.6 Real.instAddCommMonoid inst._@.Mathlib.MeasureTheory.Measure.Complex.406428577._hygCtx._hyg.9))))] [inst._@.Mathlib.MeasureTheory.Measure.Complex.406428577._hygCtx._hyg.21 : ContinuousConstSMul.{u_2, 0} R Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (SMulZeroClass.toSMul.{u_2, 0} R Complex (AddZero.toZero.{0} Complex (AddZeroClass.toAddZero.{0} Complex (AddMonoid.toAddZeroClass.{0} Complex (ESeminormedAddMonoid.toAddMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} 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Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (Distrib.toAdd.{0} Real (NonUnitalNonAssocSemiring.toDistrib.{0} Real (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Real (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{0} Real (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{0} Real (NonUnitalNormedCommRing.toNonUnitalCommRing.{0} Real (NormedCommRing.toNonUnitalNormedCommRing.{0} Real Real.normedCommRing)))))))","typeFull":"ContinuousAdd ℝ","typeReadable":"ContinuousAdd 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= { re := ↑s i, im := ↑t i }","typeReadable":"∀ {α : Type u_1} {m : MeasurableSpace α} {s t : MeasureTheory.SignedMeasure α} {i : Set α},\n  ↑(s.toComplexMeasure t) i = { re := ↑s i, im := ↑t i 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{α : Type.{u_1}} {m : MeasurableSpace.{u_1} α} {R : Type.{u_2}} [inst._@.Mathlib.MeasureTheory.Measure.Complex.406428577._hygCtx._hyg.6 : Semiring.{u_2} R] [inst._@.Mathlib.MeasureTheory.Measure.Complex.406428577._hygCtx._hyg.9 : Module.{u_2, 0} R Real inst._@.Mathlib.MeasureTheory.Measure.Complex.406428577._hygCtx._hyg.6 Real.instAddCommMonoid] [inst._@.Mathlib.MeasureTheory.Measure.Complex.406428577._hygCtx._hyg.15 : ContinuousConstSMul.{u_2, 0} R Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (SMulZeroClass.toSMul.{u_2, 0} R Real (AddZero.toZero.{0} Real (AddZeroClass.toAddZero.{0} Real (AddMonoid.toAddZeroClass.{0} Real Real.instAddMonoid))) (DistribSMul.toSMulZeroClass.{u_2, 0} R Real (AddMonoid.toAddZeroClass.{0} Real Real.instAddMonoid) (DistribMulAction.toDistribSMul.{u_2, 0} R Real (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R 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+[{"isProp":true,"kind":"theorem","name":["MeasureTheory","MemLp","isProbabilityMeasure_of_indepFun"],"typeFallback":"forall {Ω : Type.{u_1}} {E : Type.{u_2}} {F : Type.{u_3}} [inst._@.Mathlib.Probability.Independence.Integrable.3458462314._hygCtx._hyg.5 : MeasurableSpace.{u_1} Ω] {μ : MeasureTheory.Measure.{u_1} Ω inst._@.Mathlib.Probability.Independence.Integrable.3458462314._hygCtx._hyg.5} [inst._@.Mathlib.Probability.Independence.Integrable.3458462314._hygCtx._hyg.10 : NormedAddCommGroup.{u_2} E] [inst._@.Mathlib.Probability.Independence.Integrable.3458462314._hygCtx._hyg.13 : MeasurableSpace.{u_2} E] [inst._@.Mathlib.Probability.Independence.Integrable.3458462314._hygCtx._hyg.16 : OpensMeasurableSpace.{u_2} E (UniformSpace.toTopologicalSpace.{u_2} E (PseudoMetricSpace.toUniformSpace.{u_2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_2} E inst._@.Mathlib.Probability.Independence.Integrable.3458462314._hygCtx._hyg.10)))) inst._@.Mathlib.Probability.Independence.Integrable.3458462314._hygCtx._hyg.13] [inst._@.Mathlib.Probability.Independence.Integrable.3458462314._hygCtx._hyg.19 : MeasurableSpace.{u_3} F] (f : Ω -> E) (g : Ω -> F) {p : ENNReal}, (Ne.{1} ENNReal p (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instZeroENNReal))) -> (Ne.{1} ENNReal p (Top.top.{0} ENNReal instTopENNReal)) -> (MeasureTheory.MemLp.{u_1, u_2} Ω E inst._@.Mathlib.Probability.Independence.Integrable.3458462314._hygCtx._hyg.5 (ContinuousENorm.toENorm.{u_2} E (UniformSpace.toTopologicalSpace.{u_2} E (PseudoMetricSpace.toUniformSpace.{u_2} E (SeminormedAddGroup.toPseudoMetricSpace.{u_2} E (SeminormedAddCommGroup.toSeminormedAddGroup.{u_2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_2} E inst._@.Mathlib.Probability.Independence.Integrable.3458462314._hygCtx._hyg.10))))) (SeminormedAddGroup.toContinuousENorm.{u_2} E (SeminormedAddCommGroup.toSeminormedAddGroup.{u_2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_2} E 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: Type u_1} {F : Type u_3} [inst : FunLike F (Set α) ENNReal] [inst_1 : MeasureTheory.OuterMeasureClass F α]\n  {μ : F} {p : α → Prop}, (∀ᵐ (a : α) ∂μ, p a) = (μ {a | ¬p a} = 0)","typeReadable":"∀ {α : Type u_1} {F : Type u_3} [inst : FunLike F (Set α) ENNReal] [inst_1 : MeasureTheory.OuterMeasureClass F α]\n  {μ : F} {p : α → Prop}, (∀ᵐ (a : α) ∂μ, p a) = (μ {a | ¬p a} = 0)","typeReferences":[["Filter","Eventually"],["Not"],["FunLike"],["Set"],["instZeroENNReal"],["DFunLike","coe"],["OfNat","ofNat"],["ENNReal"],["MeasureTheory","OuterMeasureClass"],["Zero","toOfNat0"],["Eq"],["MeasureTheory","ae"],["setOf"]],"valueReferences":[["Filter","Eventually"],["Not"],["MeasureTheory","ae_iff"],["Set"],["instZeroENNReal"],["DFunLike","coe"],["OfNat","ofNat"],["ENNReal"],["Zero","toOfNat0"],["Eq"],["MeasureTheory","ae"],["propext"],["setOf"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","Integrable","isProbabilityMeasure_of_indepFun"],"typeFallback":"forall {Ω : Type.{u_1}} {E : Type.{u_2}} {F : Type.{u_3}} [inst._@.Mathlib.Probability.Independence.Integrable.3458462315._hygCtx._hyg.5 : MeasurableSpace.{u_1} Ω] {μ : MeasureTheory.Measure.{u_1} Ω inst._@.Mathlib.Probability.Independence.Integrable.3458462315._hygCtx._hyg.5} [inst._@.Mathlib.Probability.Independence.Integrable.3458462315._hygCtx._hyg.10 : NormedAddCommGroup.{u_2} E] [inst._@.Mathlib.Probability.Independence.Integrable.3458462315._hygCtx._hyg.13 : MeasurableSpace.{u_2} E] [inst._@.Mathlib.Probability.Independence.Integrable.3458462315._hygCtx._hyg.16 : OpensMeasurableSpace.{u_2} E (UniformSpace.toTopologicalSpace.{u_2} E (PseudoMetricSpace.toUniformSpace.{u_2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_2} E inst._@.Mathlib.Probability.Independence.Integrable.3458462315._hygCtx._hyg.10)))) inst._@.Mathlib.Probability.Independence.Integrable.3458462315._hygCtx._hyg.13] [inst._@.Mathlib.Probability.Independence.Integrable.3458462315._hygCtx._hyg.19 : MeasurableSpace.{u_3} F] (f : Ω -> E) (g : Ω -> F), (MeasureTheory.Integrable.{u_2, u_1} E (UniformSpace.toTopologicalSpace.{u_2} E (PseudoMetricSpace.toUniformSpace.{u_2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_2} E inst._@.Mathlib.Probability.Independence.Integrable.3458462315._hygCtx._hyg.10)))) (SeminormedAddGroup.toContinuousENorm.{u_2} E (SeminormedAddCommGroup.toSeminormedAddGroup.{u_2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_2} E inst._@.Mathlib.Probability.Independence.Integrable.3458462315._hygCtx._hyg.10))) Ω inst._@.Mathlib.Probability.Independence.Integrable.3458462315._hygCtx._hyg.5 f μ) -> (Not (Filter.Eventually.{u_1} Ω (fun (ω : Ω) => Eq.{succ u_2} E (f ω) (OfNat.ofNat.{u_2} E 0 (Zero.toOfNat0.{u_2} E (NegZeroClass.toZero.{u_2} E (SubNegZeroMonoid.toNegZeroClass.{u_2} E (SubtractionMonoid.toSubNegZeroMonoid.{u_2} E 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inst._@.Mathlib.Probability.Independence.Integrable.3458462315._hygCtx._hyg.5 μ)","typeFull":"∀ {Ω : Type u_1} {E : Type u_2} {F : Type u_3} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω}\n  [inst_1 : NormedAddCommGroup E] [inst_2 : MeasurableSpace E] [OpensMeasurableSpace E] [inst_4 : MeasurableSpace F]\n  (f : Ω → E) (g : Ω → F),\n  MeasureTheory.Integrable f μ →\n    (¬∀ᵐ (ω : Ω) ∂μ, f ω = 0) → ProbabilityTheory.IndepFun f g μ → MeasureTheory.IsProbabilityMeasure μ","typeReadable":"∀ {Ω : Type u_1} {E : Type u_2} {F : Type u_3} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω}\n  [inst_1 : NormedAddCommGroup E] [inst_2 : MeasurableSpace E] [OpensMeasurableSpace E] [inst_4 : MeasurableSpace F]\n  (f : Ω → E) (g : Ω → F),\n  MeasureTheory.Integrable f μ →\n    (¬∀ᵐ (ω : Ω) ∂μ, f ω = 0) → ProbabilityTheory.IndepFun f g μ → MeasureTheory.IsProbabilityMeasure 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diff --git a/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Probability.ProbabilityMassFunction.Monad.sym.json b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Probability.ProbabilityMassFunction.Monad.sym.json
new file mode 100644
index 0000000000000000000000000000000000000000..dec30aef45c4768d3ae40e3b27cf46f3c1090fd2
--- /dev/null
+++ b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Probability.ProbabilityMassFunction.Monad.sym.json
@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Probability","ProbabilityMassFunction","Monad",0,"PMF","bindOnSupport_bindOnSupport","_simp_1_1"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Topology.Algebra.InfiniteSum.Basic.3507231003._hygCtx._hyg.5 : AddCommMonoid.{u_1} α] [inst._@.Mathlib.Topology.Algebra.InfiniteSum.Basic.3507231003._hygCtx._hyg.8 : TopologicalSpace.{u_1} α] {L : SummationFilter.{u_2} β} (P : Prop) [inst._@.Mathlib.Topology.Algebra.InfiniteSum.Basic.3507231003._hygCtx._hyg.20 : Decidable P] (x : β -> (Not P) -> α), Eq.{succ u_1} α (dite.{succ u_1} α P inst._@.Mathlib.Topology.Algebra.InfiniteSum.Basic.3507231003._hygCtx._hyg.20 (fun (h : P) => OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (AddZero.toZero.{u_1} α (AddZeroClass.toAddZero.{u_1} α (AddMonoid.toAddZeroClass.{u_1} α (AddCommMonoid.toAddMonoid.{u_1} α inst._@.Mathlib.Topology.Algebra.InfiniteSum.Basic.3507231003._hygCtx._hyg.5)))))) (fun (h : Not P) => tsum.{u_1, u_2} α β inst._@.Mathlib.Topology.Algebra.InfiniteSum.Basic.3507231003._hygCtx._hyg.5 inst._@.Mathlib.Topology.Algebra.InfiniteSum.Basic.3507231003._hygCtx._hyg.8 (fun (b : β) => x b h) L)) (tsum.{u_1, u_2} α β inst._@.Mathlib.Topology.Algebra.InfiniteSum.Basic.3507231003._hygCtx._hyg.5 inst._@.Mathlib.Topology.Algebra.InfiniteSum.Basic.3507231003._hygCtx._hyg.8 (fun (b : β) => dite.{succ u_1} α P inst._@.Mathlib.Topology.Algebra.InfiniteSum.Basic.3507231003._hygCtx._hyg.20 (fun (h : P) => OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (AddZero.toZero.{u_1} α (AddZeroClass.toAddZero.{u_1} α (AddMonoid.toAddZeroClass.{u_1} α (AddCommMonoid.toAddMonoid.{u_1} α inst._@.Mathlib.Topology.Algebra.InfiniteSum.Basic.3507231003._hygCtx._hyg.5)))))) (fun (h : Not P) => x b h)) L)","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst_1 : TopologicalSpace α] {L : SummationFilter β}\n  (P : Prop) [inst_2 : Decidable P] (x : β → ¬P → α),\n  (if h : P then 0 else ∑'[L] (b : β), x b h) = ∑'[L] (b : β), if h : P then 0 else x b h","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst_1 : TopologicalSpace α] {L : SummationFilter β}\n  (P : Prop) [inst_2 : Decidable P] (x : β → ¬P → α),\n  (if h : P then 0 else ∑'[L] (b : β), x b h) = ∑'[L] (b : β), if h : P then 0 else x b h","typeReferences":[["Not"],["Decidable"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["TopologicalSpace"],["AddCommMonoid"],["tsum"],["Zero","toOfNat0"],["Eq"],["dite"],["SummationFilter"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["tsum"],["Eq","symm"],["Zero","toOfNat0"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["dite"],["AddZero","toZero"],["tsum_dite_right"],["OfNat","ofNat"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["PMF","bindOnSupport_comm"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} {γ : Type.{u_3}} (p : PMF.{u_1} α) (q : 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+[{"isProp":true,"kind":"theorem","name":["Algebra","exists_notMem_and_isIntegral_forall_mem_of_ne_of_liesOver"],"typeFallback":"forall {R : Type.{u_2}} {S : Type.{u_3}} [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 : CommRing.{u_2} R] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 : CommRing.{u_3} S] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 : Algebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27))] (p : Ideal.{u_2} R (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24))) [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.36 : Ideal.IsPrime.{u_2} R (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24)) p] (q : Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27))) [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.40 : Ideal.IsPrime.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) q] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.42 : Ideal.LiesOver.{u_2, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 q p] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.45 : Algebra.FiniteType.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.49 : Algebra.QuasiFiniteAt.{u_2, u_3} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.40], Exists.{succ u_3} S (fun (s : S) => And (Not (Membership.mem.{u_3, u_3} S (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27))) (SetLike.instMembership.{u_3, u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27))) S (Submodule.setLike.{u_3, u_3} S S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27))))) (Semiring.toModule.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27))))) q s)) (Exists.{0} (IsIntegral.{u_2, u_3} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 (CommRing.toRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 s) (fun (hs : IsIntegral.{u_2, u_3} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 (CommRing.toRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 s) => And (forall (q' : Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27))), (Ideal.IsPrime.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) q') -> (Ne.{succ u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27))) q' q) -> (Ideal.LiesOver.{u_2, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 q' p) -> (Membership.mem.{u_3, u_3} S (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27))) (SetLike.instMembership.{u_3, u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27))) S (Submodule.setLike.{u_3, u_3} S S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27))))) (Semiring.toModule.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27))))) q' s)) (forall (q' : Ideal.{u_3} (Subtype.{succ u_3} S (fun (x : S) => Membership.mem.{u_3, u_3} S (Subalgebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_3, u_3} (Subalgebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_2, u_3} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toSemiring.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_2, u_3} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))), (Ideal.IsPrime.{u_3} (Subtype.{succ u_3} S (fun (x : S) => Membership.mem.{u_3, u_3} S (Subalgebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_3, u_3} (Subalgebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_2, u_3} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toSemiring.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_2, u_3} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) q') -> (Ne.{succ u_3} (Ideal.{u_3} (Subtype.{succ u_3} S (fun (x : S) => Membership.mem.{u_3, u_3} S (Subalgebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_3, u_3} (Subalgebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_2, u_3} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toSemiring.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_2, u_3} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) q' (Ideal.under.{u_3, u_3} (Subtype.{succ u_3} S (fun (x : S) => Membership.mem.{u_3, u_3} S (Subalgebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_3, u_3} (Subalgebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_2, u_3} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_2, u_3} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (Subalgebra.toAlgebra.{u_3, u_2, u_3} S R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (Algebra.id.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (integralClosure.{u_2, u_3} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) q)) -> (Ideal.LiesOver.{u_2, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (Subtype.{succ u_3} S (fun (x : S) => Membership.mem.{u_3, u_3} S (Subalgebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_3, u_3} (Subalgebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_2, u_3} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toSemiring.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_2, u_3} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (Subalgebra.algebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_2, u_3} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) q' p) -> (Membership.mem.{u_3, u_3} (Subtype.{succ u_3} S (fun (x : S) => Membership.mem.{u_3, u_3} S (Subalgebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_3, u_3} (Subalgebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_2, u_3} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Ideal.{u_3} (Subtype.{succ u_3} S (fun (x : S) => Membership.mem.{u_3, u_3} S (Subalgebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_3, u_3} (Subalgebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_2, u_3} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toSemiring.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_2, u_3} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) (SetLike.instMembership.{u_3, u_3} (Ideal.{u_3} (Subtype.{succ u_3} S (fun (x : S) => Membership.mem.{u_3, u_3} S (Subalgebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_3, u_3} (Subalgebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_2, u_3} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toSemiring.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_2, u_3} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) (Subtype.{succ u_3} S (fun (x : S) => Membership.mem.{u_3, u_3} S (Subalgebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_3, u_3} (Subalgebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_2, u_3} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Submodule.setLike.{u_3, u_3} (Subtype.{succ u_3} S (fun (x : S) => Membership.mem.{u_3, u_3} S (Subalgebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_3, u_3} (Subalgebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_2, u_3} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subtype.{succ u_3} S (fun (x : S) => Membership.mem.{u_3, u_3} S (Subalgebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_3, u_3} (Subalgebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_2, u_3} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toSemiring.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_2, u_3} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} (Subtype.{succ u_3} S (fun (x : S) => Membership.mem.{u_3, u_3} S (Subalgebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_3, u_3} (Subalgebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_2, u_3} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} (Subtype.{succ u_3} S (fun (x : S) => Membership.mem.{u_3, u_3} S (Subalgebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_3, u_3} (Subalgebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_2, u_3} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 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(R : Type.{u_3}) (S : Type.{u_4}) [inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 : CommRing.{u_3} R] [inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23 : CommRing.{u_4} S] [inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26 : Algebra.{u_3, u_4} R S (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))] (p : Ideal.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20))) [inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32 : Ideal.IsPrime.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26) (Algebra.TensorProduct.instCommSemiring.{u_3, u_3, u_4} R (Ideal.ResidueField.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) S (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) (Semifield.toCommSemiring.{u_3} (Ideal.ResidueField.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (Field.toSemifield.{u_3} (Ideal.ResidueField.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (IsLocalRing.ResidueField.field.{u_3} (Localization.AtPrime.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (OreLocalization.instCommRing.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (OreLocalization.oreSetComm.{u_3} R (CommSemiring.toCommMonoid.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32))) (Ideal.ResidueField._proof_1.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32)))) (IsLocalRing.ResidueField.algebra.{u_3, u_3} (Localization.AtPrime.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (OreLocalization.instCommRing.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (OreLocalization.oreSetComm.{u_3} R (CommSemiring.toCommMonoid.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32))) (Ideal.ResidueField._proof_1.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 (OreLocalization.instAlgebra.{u_3, u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (OreLocalization.oreSetComm.{u_3} R (CommSemiring.toCommMonoid.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32)) R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) (Algebra.id.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)))) (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26)) (Subtype.instLE.{u_4} (Ideal.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))) (Preorder.toLE.{u_4} (Ideal.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))) (PartialOrder.toPreorder.{u_4} (Ideal.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))) (Submodule.instPartialOrder.{u_4, u_4} S S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))))) (Semiring.toModule.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)))))) (fun (x : Ideal.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))) => Membership.mem.{u_4, u_4} (Ideal.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))) (Set.{u_4} (Ideal.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)))) (Set.instMembership.{u_4} (Ideal.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)))) (Ideal.primesOver.{u_3, u_4} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) p S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26) x)) (Preorder.toLE.{max u_3 u_4} (PrimeSpectrum.{max u_3 u_4} (Ideal.Fiber.{u_3, u_4} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32 S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26) (Algebra.TensorProduct.instCommSemiring.{u_3, u_3, u_4} R (Ideal.ResidueField.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) S (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) (Semifield.toCommSemiring.{u_3} (Ideal.ResidueField.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (Field.toSemifield.{u_3} (Ideal.ResidueField.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (IsLocalRing.ResidueField.field.{u_3} (Localization.AtPrime.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (OreLocalization.instCommRing.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (OreLocalization.oreSetComm.{u_3} R (CommSemiring.toCommMonoid.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32))) (Ideal.ResidueField._proof_1.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32)))) (IsLocalRing.ResidueField.algebra.{u_3, u_3} (Localization.AtPrime.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (OreLocalization.instCommRing.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (OreLocalization.oreSetComm.{u_3} R (CommSemiring.toCommMonoid.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32))) (Ideal.ResidueField._proof_1.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 (OreLocalization.instAlgebra.{u_3, u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (OreLocalization.oreSetComm.{u_3} R (CommSemiring.toCommMonoid.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32)) R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) (Algebra.id.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)))) (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26)) (PartialOrder.toPreorder.{max u_3 u_4} (PrimeSpectrum.{max u_3 u_4} (Ideal.Fiber.{u_3, u_4} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32 S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26) (Algebra.TensorProduct.instCommSemiring.{u_3, u_3, u_4} R (Ideal.ResidueField.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) S (CommRing.toCommSemiring.{u_3} R 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) (Algebra.id.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)))) (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26)) (Subtype.instLE.{u_4} (Ideal.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))) (Preorder.toLE.{u_4} (Ideal.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))) (PartialOrder.toPreorder.{u_4} (Ideal.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))) (Submodule.instPartialOrder.{u_4, u_4} S S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))))) (Semiring.toModule.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)))))) (fun (x : Ideal.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))) => Membership.mem.{u_4, u_4} (Ideal.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))) (Set.{u_4} (Ideal.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)))) (Set.instMembership.{u_4} (Ideal.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)))) (Ideal.primesOver.{u_3, u_4} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) p S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26) x)) (Subtype.instLE.{u_4} (PrimeSpectrum.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) (Preorder.toLE.{u_4} (PrimeSpectrum.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) (PartialOrder.toPreorder.{u_4} (PrimeSpectrum.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) (PrimeSpectrum.instPartialOrder.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)))) (fun (x : PrimeSpectrum.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) => Membership.mem.{u_4, u_4} (PrimeSpectrum.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) (Set.{u_4} (PrimeSpectrum.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))) (Set.instMembership.{u_4} (PrimeSpectrum.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))) (Set.preimage.{u_4, u_3} (PrimeSpectrum.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) (PrimeSpectrum.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (PrimeSpectrum.comap.{u_3, u_4} R S (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23) (algebraMap.{u_3, u_4} R S (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26)) (Singleton.singleton.{u_3, u_3} (PrimeSpectrum.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (Set.{u_3} (PrimeSpectrum.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20))) (Set.instSingletonSet.{u_3} (PrimeSpectrum.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20))) (PrimeSpectrum.mk.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32))) x)) (Preorder.toLE.{max u_3 u_4} (PrimeSpectrum.{max u_3 u_4} (Ideal.Fiber.{u_3, u_4} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32 S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26) (Algebra.TensorProduct.instCommSemiring.{u_3, u_3, u_4} R (Ideal.ResidueField.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) S (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) (Semifield.toCommSemiring.{u_3} (Ideal.ResidueField.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (Field.toSemifield.{u_3} (Ideal.ResidueField.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (IsLocalRing.ResidueField.field.{u_3} (Localization.AtPrime.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (OreLocalization.instCommRing.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (OreLocalization.oreSetComm.{u_3} R (CommSemiring.toCommMonoid.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32))) (Ideal.ResidueField._proof_1.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32)))) (IsLocalRing.ResidueField.algebra.{u_3, u_3} (Localization.AtPrime.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (OreLocalization.instCommRing.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (OreLocalization.oreSetComm.{u_3} R (CommSemiring.toCommMonoid.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32))) (Ideal.ResidueField._proof_1.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 (OreLocalization.instAlgebra.{u_3, u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (OreLocalization.oreSetComm.{u_3} R (CommSemiring.toCommMonoid.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32)) R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) (Algebra.id.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)))) (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26)) (PartialOrder.toPreorder.{max u_3 u_4} (PrimeSpectrum.{max u_3 u_4} (Ideal.Fiber.{u_3, u_4} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32 S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26) (Algebra.TensorProduct.instCommSemiring.{u_3, u_3, u_4} R (Ideal.ResidueField.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) S (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) (Semifield.toCommSemiring.{u_3} (Ideal.ResidueField.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (Field.toSemifield.{u_3} (Ideal.ResidueField.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (IsLocalRing.ResidueField.field.{u_3} (Localization.AtPrime.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (OreLocalization.instCommRing.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (OreLocalization.oreSetComm.{u_3} R (CommSemiring.toCommMonoid.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32))) (Ideal.ResidueField._proof_1.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32)))) (IsLocalRing.ResidueField.algebra.{u_3, u_3} (Localization.AtPrime.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (OreLocalization.instCommRing.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (OreLocalization.oreSetComm.{u_3} R (CommSemiring.toCommMonoid.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32))) (Ideal.ResidueField._proof_1.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 (OreLocalization.instAlgebra.{u_3, u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (OreLocalization.oreSetComm.{u_3} R (CommSemiring.toCommMonoid.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32)) R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) (Algebra.id.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)))) (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26)) (PrimeSpectrum.instPartialOrder.{max u_3 u_4} (Ideal.Fiber.{u_3, u_4} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32 S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26) (Algebra.TensorProduct.instCommSemiring.{u_3, u_3, u_4} R (Ideal.ResidueField.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) S (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) (Semifield.toCommSemiring.{u_3} (Ideal.ResidueField.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (Field.toSemifield.{u_3} (Ideal.ResidueField.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (IsLocalRing.ResidueField.field.{u_3} (Localization.AtPrime.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (OreLocalization.instCommRing.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (OreLocalization.oreSetComm.{u_3} R (CommSemiring.toCommMonoid.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32))) (Ideal.ResidueField._proof_1.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32)))) (IsLocalRing.ResidueField.algebra.{u_3, u_3} (Localization.AtPrime.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (OreLocalization.instCommRing.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (OreLocalization.oreSetComm.{u_3} R (CommSemiring.toCommMonoid.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32))) (Ideal.ResidueField._proof_1.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 (OreLocalization.instAlgebra.{u_3, u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (OreLocalization.oreSetComm.{u_3} R (CommSemiring.toCommMonoid.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (Ideal.primeCompl.{u_3} R (CommSemiring.toSemiring.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32)) R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) (Algebra.id.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)))) (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26)))) (RelIso.mk.{u_4, u_4} (Set.Elem.{u_4} (Ideal.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))) (Ideal.primesOver.{u_3, u_4} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) p S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26)) (Set.Elem.{u_4} (PrimeSpectrum.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) (Set.preimage.{u_4, u_3} (PrimeSpectrum.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) (PrimeSpectrum.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (PrimeSpectrum.comap.{u_3, u_4} R S (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23) (algebraMap.{u_3, u_4} R S (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26)) (Singleton.singleton.{u_3, u_3} (PrimeSpectrum.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (Set.{u_3} (PrimeSpectrum.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20))) (Set.instSingletonSet.{u_3} (PrimeSpectrum.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20))) (PrimeSpectrum.mk.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32)))) (fun (x1._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21 : Set.Elem.{u_4} (Ideal.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))) (Ideal.primesOver.{u_3, u_4} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) p S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26)) (x2._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21 : Set.Elem.{u_4} (Ideal.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))) (Ideal.primesOver.{u_3, u_4} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) p S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26)) => LE.le.{u_4} (Set.Elem.{u_4} (Ideal.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))) (Ideal.primesOver.{u_3, u_4} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) p S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26)) (Subtype.instLE.{u_4} (Ideal.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))) (Preorder.toLE.{u_4} (Ideal.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))) (PartialOrder.toPreorder.{u_4} (Ideal.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))) (Submodule.instPartialOrder.{u_4, u_4} S S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))))) (Semiring.toModule.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)))))) (fun (x : Ideal.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))) => Membership.mem.{u_4, u_4} (Ideal.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))) (Set.{u_4} (Ideal.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)))) (Set.instMembership.{u_4} (Ideal.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)))) (Ideal.primesOver.{u_3, u_4} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) p S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26) x)) x1._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21 x2._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21) (fun (x1._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.33 : Set.Elem.{u_4} (PrimeSpectrum.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) (Set.preimage.{u_4, u_3} (PrimeSpectrum.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) (PrimeSpectrum.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (PrimeSpectrum.comap.{u_3, u_4} R S (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23) (algebraMap.{u_3, u_4} R S (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26)) (Singleton.singleton.{u_3, u_3} (PrimeSpectrum.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (Set.{u_3} (PrimeSpectrum.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20))) (Set.instSingletonSet.{u_3} (PrimeSpectrum.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20))) (PrimeSpectrum.mk.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32)))) (x2._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.33 : Set.Elem.{u_4} (PrimeSpectrum.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) (Set.preimage.{u_4, u_3} (PrimeSpectrum.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) (PrimeSpectrum.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (PrimeSpectrum.comap.{u_3, u_4} R S (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) 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S (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26)) (Singleton.singleton.{u_3, u_3} (PrimeSpectrum.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (Set.{u_3} (PrimeSpectrum.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20))) (Set.instSingletonSet.{u_3} (PrimeSpectrum.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20))) (PrimeSpectrum.mk.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32))) x)) x1._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.33 x2._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.33) (Equiv.mk.{succ u_4, succ u_4} (Set.Elem.{u_4} (Ideal.{u_4} S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))) (Ideal.primesOver.{u_3, u_4} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) p S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26)) (Set.Elem.{u_4} (PrimeSpectrum.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) (Set.preimage.{u_4, u_3} 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(PrimeSpectrum.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))) (Set.preimage.{u_4, u_3} (PrimeSpectrum.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) (PrimeSpectrum.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (PrimeSpectrum.comap.{u_3, u_4} R S (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23) (algebraMap.{u_3, u_4} R S (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)))) (Ideal.primesOver.{u_3, u_4} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) p S (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26) x) q) (PrimeSpectrum.primesOverOrderIsoFiber._proof_1.{u_4, u_3} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26 p q)) (PrimeSpectrum.primesOverOrderIsoFiber._proof_2.{u_3, u_4} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32 q)) (fun (q : Set.Elem.{u_4} (PrimeSpectrum.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) (Set.preimage.{u_4, u_3} (PrimeSpectrum.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) (PrimeSpectrum.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (PrimeSpectrum.comap.{u_3, u_4} R S (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23) (algebraMap.{u_3, u_4} R S (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) (CommSemiring.toSemiring.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26)) (Singleton.singleton.{u_3, u_3} (PrimeSpectrum.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (Set.{u_3} (PrimeSpectrum.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20))) (Set.instSingletonSet.{u_3} (PrimeSpectrum.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20))) (PrimeSpectrum.mk.{u_3} R (CommRing.toCommSemiring.{u_3} R 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(PrimeSpectrum.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) (Set.{u_4} (PrimeSpectrum.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))) (Set.instMembership.{u_4} (PrimeSpectrum.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23))) (Set.preimage.{u_4, u_3} (PrimeSpectrum.{u_4} S (CommRing.toCommSemiring.{u_4} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23)) (PrimeSpectrum.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20)) (PrimeSpectrum.comap.{u_3, u_4} R S (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) (CommRing.toCommSemiring.{u_4} S 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(CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32))) x) q)) (PrimeSpectrum.primesOverOrderIsoFiber._proof_3.{u_4, u_3} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32 q)) (PrimeSpectrum.primesOverOrderIsoFiber._proof_5.{u_4, u_3} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32) (PrimeSpectrum.primesOverOrderIsoFiber._proof_8.{u_4, u_3} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32)) (PrimeSpectrum.primesOverOrderIsoFiber._proof_9.{u_4, u_3} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26 p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32)) (PrimeSpectrum.preimageOrderIsoFiber.{u_3, u_4} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.23 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.26 (PrimeSpectrum.mk.{u_3} R (CommRing.toCommSemiring.{u_3} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.20) p inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.849802863._hygCtx._hyg.32)))","typeFull":"∀ (R : Type u_3) (S : Type u_4) [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal R)\n  [inst_3 : p.IsPrime],\n  PrimeSpectrum.primesOverOrderIsoFiber R S p =\n    OrderIso.trans\n      { toFun := fun q => ⟨{ asIdeal := ↑q, isPrime := ⋯ }, ⋯⟩, invFun := fun q => ⟨(↑q).asIdeal, ⋯⟩, left_inv := ⋯,\n        right_inv := ⋯, map_rel_iff' := ⋯ }\n      (PrimeSpectrum.preimageOrderIsoFiber R S { asIdeal := p, isPrime := inst_3 })","typeReadable":"∀ (R : Type u_3) (S : Type u_4) [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal R)\n  [inst_3 : p.IsPrime],\n  PrimeSpectrum.primesOverOrderIsoFiber R S p =\n    OrderIso.trans\n      { toFun := fun q => ⟨{ asIdeal := ↑q, isPrime := ⋯ }, ⋯⟩, invFun := fun q => ⟨(↑q).asIdeal, ⋯⟩, left_inv := ⋯,\n        right_inv := ⋯, map_rel_iff' := ⋯ }\n      (PrimeSpectrum.preimageOrderIsoFiber R S { asIdeal := p, isPrime := inst_3 })","typeReferences":[["RelIso","mk"],["PartialOrder","toPreorder"],["Equiv","mk"],["Ideal","IsPrime"],["Subtype","instLE"],["PrimeSpectrum","primesOverOrderIsoFiber","_proof_8"],["Singleton","singleton"],["OrderIso"],["Membership","mem"],["Ideal","ResidueField"],["Subtype","val"],["Algebra","id"],["Set","Elem"],["Semiring","toNonAssocSemiring"],["OreLocalization","instAlgebra"],["PrimeSpectrum","primesOverOrderIsoFiber"],["OrderIso","trans"],["Algebra","TensorProduct","instCommSemiring"],["PrimeSpectrum","instPartialOrder"],["PrimeSpectrum","primesOverOrderIsoFiber","_proof_1"],["Semiring","toModule"],["Localization","AtPrime"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["PrimeSpectrum","mk"],["CommSemiring","toCommMonoid"],["Algebra"],["Set","instMembership"],["Ideal","primesOver"],["PrimeSpectrum","preimageOrderIsoFiber"],["algebraMap"],["Semifield","toCommSemiring"],["IsLocalRing","ResidueField","algebra"],["OreLocalization","oreSetComm"],["PrimeSpectrum","primesOverOrderIsoFiber","_proof_2"],["PrimeSpectrum","primesOverOrderIsoFiber","_proof_3"],["OreLocalization","instCommRing"],["Ideal"],["Ideal","primeCompl"],["PrimeSpectrum","asIdeal"],["Eq"],["Preorder","toLE"],["PrimeSpectrum","comap"],["CommRing","toCommSemiring"],["Set","preimage"],["Set"],["CommSemiring","toSemiring"],["Submodule","instPartialOrder"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommRing"],["Set","instSingletonSet"],["PrimeSpectrum"],["Ideal","Fiber"],["Ideal","ResidueField","_proof_1"],["LE","le"],["Field","toSemifield"],["PrimeSpectrum","primesOverOrderIsoFiber","_proof_5"],["Subtype","mk"],["IsLocalRing","ResidueField","field"],["PrimeSpectrum","primesOverOrderIsoFiber","_proof_9"]],"valueReferences":[["Semifield","toCommSemiring"],["IsLocalRing","ResidueField","algebra"],["PartialOrder","toPreorder"],["OreLocalization","oreSetComm"],["Subtype","instLE"],["Membership","mem"],["OrderIso"],["Ideal","ResidueField"],["Algebra","id"],["Set","Elem"],["Semiring","toNonAssocSemiring"],["OreLocalization","instCommRing"],["PrimeSpectrum","primesOverOrderIsoFiber"],["Ideal"],["OreLocalization","instAlgebra"],["PrimeSpectrum","instPartialOrder"],["Algebra","TensorProduct","instCommSemiring"],["Ideal","primeCompl"],["Semiring","toModule"],["Preorder","toLE"],["Localization","AtPrime"],["CommRing","toCommSemiring"],["Set"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Submodule","instPartialOrder"],["CommSemiring","toSemiring"],["CommSemiring","toCommMonoid"],["Set","instMembership"],["Ideal","Fiber"],["PrimeSpectrum"],["Ideal","ResidueField","_proof_1"],["Eq","refl"],["Field","toSemifield"],["Ideal","primesOver"],["IsLocalRing","ResidueField","field"]]},{"isProp":true,"kind":"theorem","name":["Ideal","fiberIsoOfBijectiveResidueField","_proof_3"],"typeFallback":"forall {R : Type.{u_2}} {R' : Type.{u_1}} [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 : CommRing.{u_2} R] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 : CommRing.{u_1} R'] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 : Algebra.{u_2, u_1} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))] {p : Ideal.{u_2} R (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5))} {q : Ideal.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))} [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26 : Ideal.IsPrime.{u_2} R (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28 : Ideal.IsPrime.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.30 : Ideal.LiesOver.{u_2, u_1} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 q p], SMulCommClass.{u_1, u_2, u_1} R' (Ideal.ResidueField.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Algebra.toSMul.{u_1, u_1} R' (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) (DivisionSemiring.toSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} (Localization.AtPrime.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_1} R' (CommSemiring.toCommMonoid.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) (Semifield.toDivisionSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} (Localization.AtPrime.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_1} R' (CommSemiring.toCommMonoid.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) (Field.toSemifield.{u_1} (IsLocalRing.ResidueField.{u_1} (Localization.AtPrime.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_1} R' (CommSemiring.toCommMonoid.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_1} R' (CommSemiring.toCommMonoid.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (OreLocalization.instAlgebra.{u_1, u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' 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(Ideal.ResidueField.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semifield.toCommSemiring.{u_2} (IsLocalRing.ResidueField.{u_2} (Localization.AtPrime.{u_2} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (OreLocalization.instCommRing.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 (Ideal.primeCompl.{u_2} R (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (OreLocalization.oreSetComm.{u_2} R 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(CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (OreLocalization.oreSetComm.{u_2} R (CommSemiring.toCommMonoid.{u_2} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) (Ideal.primeCompl.{u_2} R (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26))) (Ideal.ResidueField._proof_1.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26)) (IsLocalRing.ResidueField.field.{u_2} (Localization.AtPrime.{u_2} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (OreLocalization.instCommRing.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 (Ideal.primeCompl.{u_2} R (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (OreLocalization.oreSetComm.{u_2} R (CommSemiring.toCommMonoid.{u_2} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) (Ideal.primeCompl.{u_2} R (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26))) (Ideal.ResidueField._proof_1.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26)))) (DivisionSemiring.toSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} (Localization.AtPrime.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_1} R' (CommSemiring.toCommMonoid.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) (Semifield.toDivisionSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} (Localization.AtPrime.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_1} R' (CommSemiring.toCommMonoid.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) (Field.toSemifield.{u_1} (IsLocalRing.ResidueField.{u_1} (Localization.AtPrime.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_1} R' (CommSemiring.toCommMonoid.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) (IsLocalRing.ResidueField.field.{u_1} (Localization.AtPrime.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_1} R' (CommSemiring.toCommMonoid.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) => Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (AlgHom.funLike.{u_1, u_1, u_2} R (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 (OreLocalization.instAlgebra.{u_1, u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26)) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)))) (IsLocalRing.ResidueField.algebra.{u_2, u_1} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' 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S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Semiring.toNonAssocSemiring.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)))) (Semiring.toModule.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18))))) (fun (x : Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) => Membership.mem.{max u_2 u_3, max u_2 u_3} (Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Set.{max u_2 u_3} (Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18))) (Set.instMembership.{max u_2 u_3} (Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18))) (Ideal.primesOver.{u_2, max u_3 u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S 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{R : Type.{u_1}} {R' : Type.{u_2}} {S : Type.{u_3}} [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 : CommRing.{u_1} R] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 : CommRing.{u_2} R'] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11 : CommRing.{u_3} S] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 : Algebra.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18 : Algebra.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))] {p : Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5))} {q : Ideal.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))} [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26 : Ideal.IsPrime.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28 : Ideal.IsPrime.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.30 : Ideal.LiesOver.{u_1, u_2} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))))) (IsLocalRing.ResidueField.algebra.{u_1, u_1} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (OreLocalization.instCommRing.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26))) (Ideal.ResidueField._proof_1.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 (OreLocalization.instAlgebra.{u_1, u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26)) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)))) (IsLocalRing.ResidueField.algebra.{u_2, u_1} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 (OreLocalization.instAlgebra.{u_2, u_1} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) => Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (AlgHom.funLike.{u_1, u_1, u_2} R (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (DivisionSemiring.toSemiring.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (Semifield.toDivisionSemiring.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (Field.toSemifield.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (IsLocalRing.ResidueField.field.{u_1} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (OreLocalization.instCommRing.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26))) (Ideal.ResidueField._proof_1.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26))))) (DivisionSemiring.toSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semifield.toDivisionSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Field.toSemifield.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (IsLocalRing.ResidueField.field.{u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))))) (IsLocalRing.ResidueField.algebra.{u_1, u_1} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (OreLocalization.instCommRing.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26))) (Ideal.ResidueField._proof_1.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 (OreLocalization.instAlgebra.{u_1, u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26)) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)))) (IsLocalRing.ResidueField.algebra.{u_2, u_1} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 (OreLocalization.instAlgebra.{u_2, u_1} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14))) (Ideal.ResidueField.mapₐ.{u_1, u_1, u_2} R R R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28 (Algebra.ofId.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Ideal.over_def.{u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 q p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.30)))) -> (OrderIso.{max u_2 u_3, u_3} (Set.Elem.{max u_2 u_3} (Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Ideal.primesOver.{u_2, max u_3 u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18) (Algebra.TensorProduct.leftAlgebra.{u_1, u_2, u_2, u_3} R R' R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18 (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) (Algebra.id.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Algebra.to_smulCommClass.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14)))) (Set.Elem.{u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))) (Ideal.primesOver.{u_1, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) p S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Subtype.instLE.{max u_2 u_3} (Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Preorder.toLE.{max u_2 u_3} (Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (PartialOrder.toPreorder.{max u_2 u_3} (Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Submodule.instPartialOrder.{max u_2 u_3, max u_2 u_3} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' 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u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18))))) (fun (x : Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S 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(NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (hs₂ : Function.Bijective.{succ u_2, succ u_2} (Localization.Away.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (Semiring.toNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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(Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) 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(Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Semiring.toNonAssocSemiring.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)))) (AlgHom.toRingHom.{u_1, u_2, u_2} R (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (Semiring.toNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Semiring.toNonAssocSemiring.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)))) (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) => S) (RingHom.instFunLike.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (Semiring.toNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Semiring.toNonAssocSemiring.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)))) (AlgHom.toRingHom.{u_1, u_2, u_2} R (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (Subalgebra.toSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (Subalgebra.algebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (Subalgebra.val.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) s₂)) 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(CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S 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(SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂) (OreLocalization.oreSetComm.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂)))) (Semiring.toNonAssocSemiring.{u_2} (Localization.Away.{u_2} S (CommSemiring.toCommMonoid.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (DFunLike.coe.{succ u_2, succ u_2, succ u_2} (RingHom.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (Semiring.toNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Semiring.toNonAssocSemiring.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)))) (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) => S) (RingHom.instFunLike.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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(CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Semiring.toNonAssocSemiring.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)))) (AlgHom.toRingHom.{u_1, u_2, u_2} R (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S 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(Semiring.toNonAssocSemiring.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)))) (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) => S) (RingHom.instFunLike.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (Semiring.toNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Semiring.toNonAssocSemiring.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)))) (AlgHom.toRingHom.{u_1, u_2, u_2} R (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (Subalgebra.toSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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s₂)))))) (Localization.Away.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) s₂) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Localization.Away.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) s₂) => Localization.Away.{u_2} S (CommSemiring.toCommMonoid.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (DFunLike.coe.{succ u_2, succ u_2, succ u_2} (RingHom.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (Semiring.toNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Semiring.toNonAssocSemiring.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)))) (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) => S) (RingHom.instFunLike.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (Semiring.toNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Semiring.toNonAssocSemiring.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)))) (AlgHom.toRingHom.{u_1, u_2, u_2} R (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S 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(Localization.Away.{u_2} S (CommSemiring.toCommMonoid.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (DFunLike.coe.{succ u_2, succ u_2, succ u_2} (RingHom.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (Semiring.toNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Semiring.toNonAssocSemiring.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)))) (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) => S) (RingHom.instFunLike.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (Semiring.toNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Semiring.toNonAssocSemiring.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)))) (AlgHom.toRingHom.{u_1, u_2, u_2} R (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (Subalgebra.toSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (Subalgebra.algebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) s₂)) (Semiring.toNonAssocSemiring.{u_2} (Localization.Away.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) s₂) (OreLocalization.instSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂) (OreLocalization.oreSetComm.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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(Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂)))) (Semiring.toNonAssocSemiring.{u_2} (Localization.Away.{u_2} S (CommSemiring.toCommMonoid.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (DFunLike.coe.{succ u_2, succ u_2, succ u_2} (RingHom.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (Semiring.toNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Semiring.toNonAssocSemiring.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)))) (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) => S) (RingHom.instFunLike.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (Semiring.toNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Semiring.toNonAssocSemiring.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)))) (AlgHom.toRingHom.{u_1, u_2, u_2} R (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) s₂)) (OreLocalization.instSemiring.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (Submonoid.powers.{u_2} S (CommMonoid.toMonoid.{u_2} S (CommSemiring.toCommMonoid.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27))) (DFunLike.coe.{succ u_2, succ u_2, succ u_2} (RingHom.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (Semiring.toNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Semiring.toNonAssocSemiring.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)))) (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) => S) (RingHom.instFunLike.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (Semiring.toNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Semiring.toNonAssocSemiring.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)))) (AlgHom.toRingHom.{u_1, u_2, u_2} R (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (Subalgebra.toSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (Subalgebra.algebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (Subalgebra.val.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) s₂)) (OreLocalization.oreSetComm.{u_2} S (CommSemiring.toCommMonoid.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (Submonoid.powers.{u_2} S (CommMonoid.toMonoid.{u_2} S 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S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (Semiring.toNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Semiring.toNonAssocSemiring.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)))) (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) => S) (RingHom.instFunLike.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Semiring.toNonAssocSemiring.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)))) (AlgHom.toRingHom.{u_1, u_2, u_2} R (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) s₂)))))) (Localization.awayMap.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Ring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 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=> Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂)) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (Subalgebra.algebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) (Ideal.map.{u_2, u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) s₂) (RingHom.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 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(SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Semiring.toNonAssocSemiring.{u_2} (Localization.Away.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Ring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toRing.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂) (OreLocalization.oreSetComm.{u_2} (Subtype.{succ u_2} S (fun (x : S) => 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂))))) (Subalgebra.toSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (OreLocalization.instSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toRing.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) 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S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂) (OreLocalization.oreSetComm.{u_2} (Subtype.{succ u_2} 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂))) (RingHom.instFunLike.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) s₂) (Semiring.toNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Semiring.toNonAssocSemiring.{u_2} (Localization.Away.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) s₂) (OreLocalization.instSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Ring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toRing.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂) (OreLocalization.oreSetComm.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂) (OreLocalization.oreSetComm.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toRing.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S 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(Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂)) (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (Algebra.id.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))))) q') p) -> (IsScalarTower.{u_1, u_2, u_2} R (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (Algebra.toSMul.{u_1, u_2} R (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) (Subalgebra.algebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) (Algebra.toSMul.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (Subalgebra.toAlgebra.{u_2, u_1, u_2} S R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (Algebra.id.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) (Algebra.toSMul.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) -> (Ideal.LiesOver.{u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (Ideal.comap.{u_2, u_2, u_2} S (Localization.Away.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) s₂) (RingHom.{u_2, u_2} S (Localization.Away.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) s₂) (Semiring.toNonAssocSemiring.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27))) (Semiring.toNonAssocSemiring.{u_2} (Localization.Away.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) s₂) (OreLocalization.instSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Ring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂))))) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Ring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) 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(CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂) (OreLocalization.oreSetComm.{u_2} (Subtype.{succ u_2} S (fun (x : S) => 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂))) (RingHom.instFunLike.{u_2, u_2} S (Localization.Away.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) s₂) (OreLocalization.instSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Ring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂) (OreLocalization.oreSetComm.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) s₂) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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(Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂) (OreLocalization.oreSetComm.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂))) (Subalgebra.toAlgebra.{u_2, u_1, u_2} S R S (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Ring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toRing.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) 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(CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂) (OreLocalization.oreSetComm.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 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(Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x) s₂)) (Localization.Away.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toRing.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂) (OreLocalization.oreSetComm.{u_2} (Subtype.{succ u_2} S (fun (x : S) => 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂))) (Subalgebra.toAlgebra.{u_2, u_1, u_2} S R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (Algebra.id.{u_2} S (CommRing.toCommSemiring.{u_2} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 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(CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x) s₂))) (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) (Ring.toSemiring.{u_2} S (CommRing.toRing.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (Algebra.id.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) (OreLocalization.instAlgebra.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S 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(CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Localization.Away.{u_2} S (CommRing.toCommMonoid.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) (Subtype.val.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 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(Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) s₂) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (OreLocalization.instSemiring.{u_2} S (Ring.toSemiring.{u_2} S (CommRing.toRing.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (Submonoid.powers.{u_2} S (CommMonoid.toMonoid.{u_2} S (CommRing.toCommMonoid.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (Subtype.val.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x) s₂)))) (OreLocalization.instSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Ring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) 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(Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂) (OreLocalization.oreSetComm.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} 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(CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S 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(CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Ring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toRing.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S 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(CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂) (OreLocalization.oreSetComm.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂)) (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (Algebra.id.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) s₂) (Localization.Away.{u_2} S (CommRing.toCommMonoid.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) (Subtype.val.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x) s₂)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (OreLocalization.instSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Ring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toRing.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂) (OreLocalization.oreSetComm.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂) (OreLocalization.oreSetComm.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) s₂) (Localization.Away.{u_2} S (CommRing.toCommMonoid.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) (Subtype.val.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x) s₂)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂) (OreLocalization.oreSetComm.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S 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(Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂))) (OreLocalization.instSemiring.{u_2} S (Ring.toSemiring.{u_2} S (CommRing.toRing.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (Submonoid.powers.{u_2} S (CommMonoid.toMonoid.{u_2} S (CommRing.toCommMonoid.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (Subtype.val.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x) s₂)) (OreLocalization.oreSetComm.{u_2} S (CommRing.toCommMonoid.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) (Submonoid.powers.{u_2} S (CommMonoid.toMonoid.{u_2} S (CommRing.toCommMonoid.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (Subtype.val.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x) s₂)))) (OreLocalization.instAlgebra.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Ring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toRing.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂) (OreLocalization.oreSetComm.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂)) (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x) s₂)))) (Subalgebra.toAlgebra.{u_2, u_1, u_2} S R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S 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(CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S 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(Subalgebra.toAlgebra.{u_2, u_1, u_2} S R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) (Ring.toSemiring.{u_2} S (CommRing.toRing.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (Algebra.id.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) (OreLocalization.instIsScalarTower.{u_2, u_2, u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S S S (CommMonoid.toMonoid.{u_2} S (CommRing.toCommMonoid.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (Submonoid.powers.{u_2} S (CommMonoid.toMonoid.{u_2} S (CommRing.toCommMonoid.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (Subtype.val.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x) s₂)) (OreLocalization.oreSetComm.{u_2} S (CommRing.toCommMonoid.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) (Submonoid.powers.{u_2} S (CommMonoid.toMonoid.{u_2} S (CommRing.toCommMonoid.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (Subtype.val.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x) s₂))) (Monoid.toMulAction.{u_2} S (CommMonoid.toMonoid.{u_2} S (CommRing.toCommMonoid.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27))) (SemigroupAction.toSMul.{u_2, u_2} 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(CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (Monoid.toSemigroup.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (MonoidWithZero.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Semiring.toMonoidWithZero.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))))) (MulAction.toSemigroupAction.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (MonoidWithZero.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Semiring.toMonoidWithZero.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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(Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (MonoidWithZero.toMonoid.{u_2} 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(CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Semiring.toMonoidWithZero.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))))) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (CommSemiring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} S (Semiring.toNonAssocSemiring.{u_2} S (Ring.toSemiring.{u_2} S (CommRing.toRing.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27))))) (Algebra.toModule.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (Ring.toSemiring.{u_2} S (CommRing.toRing.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (Subalgebra.toAlgebra.{u_2, u_1, u_2} S R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) (Ring.toSemiring.{u_2} S (CommRing.toRing.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (Algebra.id.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))))))) (SemigroupAction.toSMul.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (Monoid.toSemigroup.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (MonoidWithZero.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Semiring.toMonoidWithZero.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))))) (MulAction.toSemigroupAction.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (MonoidWithZero.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Semiring.toMonoidWithZero.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))))) (DistribMulAction.toMulAction.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (MonoidWithZero.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Semiring.toMonoidWithZero.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))))) (AddCommMonoid.toAddMonoid.{u_2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} S (Semiring.toNonAssocSemiring.{u_2} S (Ring.toSemiring.{u_2} S (CommRing.toRing.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)))))) (Module.toDistribMulAction.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (CommSemiring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} S (Semiring.toNonAssocSemiring.{u_2} S (Ring.toSemiring.{u_2} S (CommRing.toRing.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27))))) (Algebra.toModule.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (Ring.toSemiring.{u_2} S (CommRing.toRing.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (Subalgebra.toAlgebra.{u_2, u_1, u_2} S R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) (Ring.toSemiring.{u_2} S (CommRing.toRing.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (Algebra.id.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))))))) (IsScalarTower.right.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (Ring.toSemiring.{u_2} S (CommRing.toRing.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (Subalgebra.toAlgebra.{u_2, u_1, u_2} S R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) (Ring.toSemiring.{u_2} S (CommRing.toRing.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (Algebra.id.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (integralClosure.{u_1, u_2} R S 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(Algebra.id.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27))) (Algebra.toSMul.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) S (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (Subalgebra.toAlgebra.{u_2, u_1, u_2} S R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (Algebra.id.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) (IsScalarTower.right.{u_2, u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (Subalgebra.toAlgebra.{u_2, u_1, u_2} S R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (Algebra.id.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))))))) (RingHom.instRingHomClass.{u_2, u_2} S (Localization.Away.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) s₂) (Semiring.toNonAssocSemiring.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S 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(Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) s₂) (OreLocalization.instSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Ring.toSemiring.{u_2} (Subtype.{succ 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toRing.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂) (OreLocalization.oreSetComm.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S 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(Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) s₂) (Semiring.toNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Semiring.toNonAssocSemiring.{u_2} (Localization.Away.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) s₂) (OreLocalization.instSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Ring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toRing.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂))))) (Subalgebra.toSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (OreLocalization.instSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Ring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) s₂) (Semiring.toNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommSemiring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Semiring.toNonAssocSemiring.{u_2} (Localization.Away.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) s₂) (OreLocalization.instSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Ring.toSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toRing.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂) (OreLocalization.oreSetComm.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S 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(Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) s₂) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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(Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂) (OreLocalization.oreSetComm.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂) (OreLocalization.oreSetComm.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))) (Submonoid.powers.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommMonoid.toMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (CommRing.toCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)))) s₂)) (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (Algebra.id.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30) x)) (Subalgebra.toCommSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1283313180._hygCtx._hyg.30))))) q')) p)","typeFull":"∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal R)\n  (s₂ : ↥(integralClosure R S))\n  (hs₂ : Function.Bijective ⇑(Localization.awayMap (integralClosure R S).val.toRingHom s₂))\n  (q' : Ideal ↥(integralClosure R S)),\n  (Ideal.map (algebraMap (↥(integralClosure R S)) (Localization.Away s₂)) q').LiesOver p →\n    IsScalarTower R (↥(integralClosure R S)) S →\n      (Ideal.comap\n            ((↑(AlgEquiv.ofBijective (Localization.awayMapₐ (Algebra.ofId (↥(integralClosure R S)) S) s₂)\n                        hs₂).symm).comp\n                (IsScalarTower.toAlgHom (↥(integralClosure R S)) S (Localization.Away ↑s₂))).toRingHom\n            (Ideal.map (algebraMap (↥(integralClosure R S)) (Localization.Away s₂)) q')).LiesOver\n        p","typeReadable":"∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal R)\n  (s₂ : ↥(integralClosure R S))\n  (hs₂ : Function.Bijective ⇑(Localization.awayMap (integralClosure R S).val.toRingHom s₂))\n  (q' : Ideal ↥(integralClosure R S)),\n  (Ideal.map (algebraMap (↥(integralClosure R S)) (Localization.Away s₂)) q').LiesOver p →\n    IsScalarTower R (↥(integralClosure R S)) S →\n      (Ideal.comap\n            ((↑(AlgEquiv.ofBijective (Localization.awayMapₐ (Algebra.ofId (↥(integralClosure R S)) S) s₂)\n                        hs₂).symm).comp\n                (IsScalarTower.toAlgHom (↥(integralClosure R S)) S (Localization.Away ↑s₂))).toRingHom\n            (Ideal.map (algebraMap (↥(integralClosure R S)) (Localization.Away s₂)) q')).LiesOver\n        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{R : Type.{u_1}} {R' : Type.{u_2}} {S : Type.{u_3}} [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 : CommRing.{u_1} R] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 : CommRing.{u_2} R'] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11 : CommRing.{u_3} S] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 : Algebra.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18 : Algebra.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))] {p : Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5))} {q : Ideal.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))} [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26 : Ideal.IsPrime.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28 : Ideal.IsPrime.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.30 : Ideal.LiesOver.{u_1, u_2} R 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R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)))) (Semiring.toModule.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18))))) (fun (x : Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) => Membership.mem.{max u_2 u_3, max u_2 u_3} (Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Set.{max u_2 u_3} (Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18))) (Set.instMembership.{max u_2 u_3} (Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18))) (Ideal.primesOver.{u_2, max u_3 u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18) (Algebra.TensorProduct.leftAlgebra.{u_1, u_2, u_2, u_3} R R' R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18 (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) (Algebra.id.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Algebra.to_smulCommClass.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14))) x)) (Subtype.instLE.{u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))) (Preorder.toLE.{u_3} (Ideal.{u_3} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18) x))) (Ideal.fiberIsoOfBijectiveResidueField.{u_1, u_2, u_3} R R' S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18 p q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.30 H) (OrderIso.trans.{max u_2 u_3, max u_2 u_3, u_3} (Set.Elem.{max u_2 u_3} (Ideal.{max u_2 u_3} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (CommSemiring.toSemiring.{max u_2 u_3} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (CommRing.toCommSemiring.{max u_2 u_3} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.instCommRing.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)))) (Ideal.primesOver.{u_2, max u_2 u_3} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (CommSemiring.toSemiring.{max u_2 u_3} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (CommRing.toCommSemiring.{max u_2 u_3} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18))) (Algebra.TensorProduct.leftAlgebra.{u_1, u_2, u_2, u_3} R R' R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18 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(CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) (Algebra.id.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)))) (CommRing.toCommSemiring.{max u_2 u_3} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14)))) (Set.Elem.{u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))) (Ideal.primesOver.{u_1, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) p S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Subtype.instLE.{max u_2 u_3} (Ideal.{max u_2 u_3} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.instCommRing.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)))) (Submodule.instPartialOrder.{max u_2 u_3, max u_2 u_3} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' 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(CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.instCommRing.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)))))) (Semiring.toModule.{max u_2 u_3} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R 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(NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (CommRing.toCommSemiring.{max u_2 u_3} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.instCommRing.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18))))))) (fun (x : Ideal.{max u_2 u_3} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (CommSemiring.toSemiring.{max u_2 u_3} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (CommRing.toCommSemiring.{max u_2 u_3} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.instCommRing.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)))) => Membership.mem.{max u_2 u_3, max u_2 u_3} (Ideal.{max u_2 u_3} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (CommSemiring.toSemiring.{max u_2 u_3} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (CommRing.toCommSemiring.{max u_2 u_3} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.instCommRing.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)))) (Set.{max u_2 u_3} 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(CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (CommSemiring.toSemiring.{max u_2 u_3} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18))))) (Set.instMembership.{max u_2 u_3} (Ideal.{max u_2 u_3} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18) (Algebra.TensorProduct.leftAlgebra.{u_1, u_2, u_2, u_3} R R' R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18 (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) (Algebra.id.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' 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(Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (IsLocalRing.ResidueField.field.{u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (OreLocalization.instAlgebra.{u_2, u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) (Algebra.id.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)))) (CommRing.toCommSemiring.{max u_2 u_3} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.instCommRing.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.leftAlgebra.{u_1, u_2, u_2, u_3} R R' R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18 (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) (Algebra.id.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.fiberIsoOfBijectiveResidueField._proof_2.{u_1, u_2} R R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14)))))) (Subtype.instLE.{u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))) (Preorder.toLE.{u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))) (PartialOrder.toPreorder.{u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))) (Submodule.instPartialOrder.{u_3, u_3} S S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Semiring.toModule.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)))))) (fun (x : Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))) => Membership.mem.{u_3, u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))) (Set.{u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)))) (Set.instMembership.{u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)))) (Ideal.primesOver.{u_1, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) p S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18) x)) (PrimeSpectrum.primesOverOrderIsoFiber.{u_2, max u_2 u_3} R' (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Algebra.TensorProduct.instCommRing.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18) (Algebra.TensorProduct.leftAlgebra.{u_1, u_2, u_2, u_3} R R' R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18 (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) (Algebra.id.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.fiberIsoOfBijectiveResidueField._proof_2.{u_1, u_2} R R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OrderIso.trans.{max u_2 u_3, max u_1 u_3, u_3} (PrimeSpectrum.{max u_2 u_3} (Ideal.Fiber.{u_2, max u_3 u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28 (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Algebra.TensorProduct.instCommRing.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommRing.toCommSemiring.{u_3} 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14))) (Algebra.TensorProduct.instCommSemiring.{u_2, u_2, max u_2 u_3} R' (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)))) (IsLocalRing.ResidueField.algebra.{u_2, u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26 S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18) (Algebra.TensorProduct.instCommSemiring.{u_1, u_1, u_3} R (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (Semifield.toCommSemiring.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (Field.toSemifield.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (IsLocalRing.ResidueField.field.{u_1} 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(Ideal.ResidueField._proof_1.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26)))) (IsLocalRing.ResidueField.algebra.{u_1, u_1} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (OreLocalization.instCommRing.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) (Ideal.primeCompl.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18) (Algebra.TensorProduct.instCommSemiring.{u_1, u_1, u_3} R (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (Semifield.toCommSemiring.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (Field.toSemifield.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (IsLocalRing.ResidueField.field.{u_1} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18) (Algebra.TensorProduct.instCommSemiring.{u_1, u_1, u_3} R (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (Semifield.toCommSemiring.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (Field.toSemifield.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (IsLocalRing.ResidueField.field.{u_1} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18) (Algebra.TensorProduct.leftAlgebra.{u_1, u_2, u_2, u_3} R R' R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18 (CommRing.toCommSemiring.{u_2} R' 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(OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (OreLocalization.instAlgebra.{u_2, u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' 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(CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18) (Algebra.TensorProduct.leftAlgebra.{u_1, u_2, u_2, u_3} R R' R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18 (CommRing.toCommSemiring.{u_2} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26))) (Ideal.ResidueField._proof_1.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26))))) (IsLocalRing.ResidueField.algebra.{u_1, u_1} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (OreLocalization.instCommRing.{u_1} R 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(OreLocalization.instAlgebra.{u_1, u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26)) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R 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(NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R 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(CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} R' (Semiring.toNonAssocSemiring.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (OreLocalization.instAlgebra.{u_2, u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) R' 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(Ideal.fiberIsoOfBijectiveResidueField._proof_2.{u_1, u_2} R R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14))) (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)))))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semifield.toCommSemiring.{u_2} 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(CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 (OreLocalization.instAlgebra.{u_2, u_1} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26))) (Ideal.ResidueField._proof_1.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26)))) (Algebra.TensorProduct.instSemiring.{u_2, u_2, max u_2 u_3} R' (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} R' (Semiring.toNonAssocSemiring.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S 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(CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (OreLocalization.instAlgebra.{u_2, u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) (Algebra.id.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)))) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18) (Algebra.TensorProduct.leftAlgebra.{u_1, u_2, u_2, u_3} R R' R' S (CommRing.toCommSemiring.{u_1} R 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(Ideal.ResidueField._proof_1.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26)))) (IsLocalRing.ResidueField.instAlgebra.{u_1, u_2} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) 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(Field.toSemifield.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (IsLocalRing.ResidueField.field.{u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} 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(CommSemiring.toSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semifield.toCommSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Field.toSemifield.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (IsLocalRing.ResidueField.field.{u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))))) (IsLocalRing.ResidueField.algebra.{u_2, u_1} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 (OreLocalization.instAlgebra.{u_2, u_1} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14))) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18)) (Semifield.toCommSemiring.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (Field.toSemifield.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p 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(OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)))) (Algebra.TensorProduct.instSemiring.{u_2, 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(CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))))) (IsLocalRing.ResidueField.algebra.{u_2, u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (OreLocalization.instAlgebra.{u_2, u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' 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(Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18) (Algebra.TensorProduct.leftAlgebra.{u_1, u_2, u_2, u_3} R R' R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18 (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) (Algebra.id.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.fiberIsoOfBijectiveResidueField._proof_2.{u_1, u_2} R R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14))) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) S (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 (OreLocalization.instAlgebra.{u_2, u_1} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14)) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18) (IsLocalRing.ResidueField.instAlgebra.{u_1, u_2} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (OreLocalization.instAlgebra.{u_2, u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) (Algebra.id.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)))) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18) (Algebra.TensorProduct.leftAlgebra.{u_1, u_2, u_2, u_3} R R' R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18 (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) (Algebra.id.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.fiberIsoOfBijectiveResidueField._proof_2.{u_1, u_2} R R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14)) (Semifield.toCommSemiring.{u_2} (Ideal.ResidueField.{u_2} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semifield.toCommSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Field.toSemifield.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (IsLocalRing.ResidueField.field.{u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))))) (IsLocalRing.ResidueField.algebra.{u_2, u_1} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 (OreLocalization.instAlgebra.{u_2, u_1} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' 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(CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14)) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18 (Semifield.toCommSemiring.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (Field.toSemifield.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (IsLocalRing.ResidueField.field.{u_1} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) p 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(Ideal.fiberIsoOfBijectiveResidueField._proof_8.{u_1, u_2, u_3} R R' S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18 p q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.30) (Ideal.fiberIsoOfBijectiveResidueField._proof_9.{u_1, u_2, u_3} R R' S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 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(Semifield.toCommSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Field.toSemifield.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (IsLocalRing.ResidueField.field.{u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)))) (Semifield.toCommSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Field.toSemifield.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)))) (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11) (IsLocalRing.ResidueField.algebra.{u_2, u_1} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 (OreLocalization.instAlgebra.{u_2, u_1} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14)) (IsLocalRing.ResidueField.algebra.{u_2, u_1} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 (OreLocalization.instAlgebra.{u_2, u_1} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18 (Algebra.id.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semifield.toCommSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Field.toSemifield.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (IsLocalRing.ResidueField.field.{u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))))) (Ideal.fiberIsoOfBijectiveResidueField._proof_10.{u_1, u_2} R R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (IsLocalRing.ResidueField.algebra.{u_2, u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (OreLocalization.instAlgebra.{u_2, u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) (Algebra.id.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)))) (Ideal.fiberIsoOfBijectiveResidueField._proof_11.{u_1, u_2} R R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (IsLocalRing.ResidueField.algebra.{u_2, u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (OreLocalization.instAlgebra.{u_2, u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) (Algebra.id.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)))) (Ideal.fiberIsoOfBijectiveResidueField._proof_12.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Algebra.TensorProduct.congr.{u_1, u_1, u_2, u_3, u_1, u_3} R (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) S (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (Semifield.toCommSemiring.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (Field.toSemifield.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (IsLocalRing.ResidueField.field.{u_1} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (OreLocalization.instCommRing.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26)) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)))) (DivisionSemiring.toSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semifield.toDivisionSemiring.{u_2} (Ideal.ResidueField.{u_2} R' 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(CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))))) (IsLocalRing.ResidueField.algebra.{u_2, u_1} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' 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(CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Localization.AtPrime.instAlgebraOfLiesOver.{u_1, u_2} R R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.30) (instIsLocalHomAtPrimeRingHomAlgebraMap.{u_1, u_2} R R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.30)) (Ideal.fiberIsoOfBijectiveResidueField._proof_13.{u_1, u_2} R R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 p q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.30) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18 (DivisionSemiring.toSemiring.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (Semifield.toDivisionSemiring.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (Field.toSemifield.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (IsLocalRing.ResidueField.field.{u_1} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (OreLocalization.instCommRing.{u_1} R 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(CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (OreLocalization.instCommRing.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26))) (Ideal.ResidueField._proof_1.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 (OreLocalization.instAlgebra.{u_1, u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p 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(Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)))) (PrimeSpectrum.preimageEquivFiber._proof_2.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p) R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (OreLocalization.instAlgebra.{u_1, u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p))) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)))) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (fun (q : Set.Elem.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (Semifield.toCommSemiring.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (Field.toSemifield.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (IsLocalRing.ResidueField.field.{u_1} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.instCommRing.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)))) (PrimeSpectrum.preimageEquivFiber._proof_2.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)))) (IsLocalRing.ResidueField.algebra.{u_1, u_1} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.instCommRing.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)))) (PrimeSpectrum.preimageEquivFiber._proof_2.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p) R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (OreLocalization.instAlgebra.{u_1, u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p))) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)))) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) 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(CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)))) (PrimeSpectrum.preimageEquivFiber._proof_2.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p))))) (IsLocalRing.ResidueField.algebra.{u_1, u_1} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.instCommRing.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R 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(CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p))) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)))) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Semiring.toNonAssocSemiring.{u_2} (Ideal.ResidueField.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)) (Semifield.toDivisionSemiring.{u_2} (Ideal.ResidueField.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)) (Field.toSemifield.{u_2} (Ideal.ResidueField.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R 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(CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)))) (PrimeSpectrum.preimageEquivFiber._proof_7.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q))))))) (CommSemiring.toSemiring.{max u_1 u_2} (Ideal.Fiber.{u_1, u_2} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p) S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10) (Algebra.TensorProduct.instCommSemiring.{u_1, u_1, u_2} R (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (Semifield.toCommSemiring.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (Field.toSemifield.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (IsLocalRing.ResidueField.field.{u_1} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.instCommRing.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)))) (PrimeSpectrum.preimageEquivFiber._proof_2.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)))) (IsLocalRing.ResidueField.algebra.{u_1, u_1} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.instCommRing.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)))) (PrimeSpectrum.preimageEquivFiber._proof_2.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p) R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (OreLocalization.instAlgebra.{u_1, u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p))) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)))) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) 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(CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)) (Semifield.toDivisionSemiring.{u_2} (Ideal.ResidueField.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S 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(CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 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(CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)) (IsLocalRing.ResidueField.field.{u_2} (Localization.AtPrime.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)) (OreLocalization.instCommRing.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 (Ideal.primeCompl.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)) (OreLocalization.oreSetComm.{u_2} S (CommSemiring.toCommMonoid.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Ideal.primeCompl.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => 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(PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)))) (PrimeSpectrum.preimageEquivFiber._proof_7.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q))))) (RingHom.instFunLike.{max u_1 u_2, u_2} (TensorProduct.{u_1, u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R 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(PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (Semiring.toNonAssocSemiring.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (DivisionSemiring.toSemiring.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (Semifield.toDivisionSemiring.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (Field.toSemifield.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (IsLocalRing.ResidueField.field.{u_1} 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.instCommRing.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)))) (PrimeSpectrum.preimageEquivFiber._proof_2.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p) R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (OreLocalization.instAlgebra.{u_1, u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R 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(CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p))) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))))) (Algebra.toModule.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Algebra.TensorProduct.instSemiring.{u_1, u_1, u_2} R (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (DivisionSemiring.toSemiring.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (Semifield.toDivisionSemiring.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (Field.toSemifield.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (IsLocalRing.ResidueField.field.{u_1} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R 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(CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.instCommRing.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)))) (PrimeSpectrum.preimageEquivFiber._proof_2.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p) R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (OreLocalization.instAlgebra.{u_1, u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)) (OreLocalization.instCommRing.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 (Ideal.primeCompl.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)) (OreLocalization.oreSetComm.{u_2} S (CommSemiring.toCommMonoid.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Ideal.primeCompl.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q))))))) (PrimeSpectrum.preimageEquivFiber._proof_8.{u_1, u_2} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q) (AlgHom.toRingHom.{u_1, max u_1 u_2, u_2} R (TensorProduct.{u_1, u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)))) (PrimeSpectrum.preimageEquivFiber._proof_7.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q))))) (Algebra.TensorProduct.leftAlgebra.{u_1, u_1, u_1, u_2} R R (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (DivisionSemiring.toSemiring.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (Semifield.toDivisionSemiring.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (Field.toSemifield.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (IsLocalRing.ResidueField.field.{u_1} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.instCommRing.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)))) (PrimeSpectrum.preimageEquivFiber._proof_2.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p))))) (IsLocalRing.ResidueField.algebra.{u_1, u_1} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.instCommRing.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)))) (PrimeSpectrum.preimageEquivFiber._proof_2.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p) R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (OreLocalization.instAlgebra.{u_1, u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (IsLocalRing.ResidueField.algebra.{u_1, u_1} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.instCommRing.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)) (OreLocalization.instCommRing.{u_2} S 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)) (OreLocalization.oreSetComm.{u_2} S (CommSemiring.toCommMonoid.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Ideal.primeCompl.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)))) (PrimeSpectrum.preimageEquivFiber._proof_7.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q) R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (OreLocalization.instAlgebra.{u_2, u_1} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Ideal.primeCompl.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Algebra.TensorProduct.lift.{u_1, u_1, u_1, u_2, u_2} R R (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) S (Ideal.ResidueField.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R 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(Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (Semifield.toDivisionSemiring.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (Field.toSemifield.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (PrimeSpectrum.asIdeal.{u_1} R 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)))) (PrimeSpectrum.preimageEquivFiber._proof_2.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p))))) (IsLocalRing.ResidueField.algebra.{u_1, u_1} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.instCommRing.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)))) (PrimeSpectrum.preimageEquivFiber._proof_2.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p) R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (OreLocalization.instAlgebra.{u_1, u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p))) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)))) (IsLocalRing.ResidueField.algebra.{u_1, u_1} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.instCommRing.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)))) (PrimeSpectrum.preimageEquivFiber._proof_2.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p) R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (OreLocalization.instAlgebra.{u_1, u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)) (OreLocalization.oreSetComm.{u_2} S (CommSemiring.toCommMonoid.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Ideal.primeCompl.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)))) (PrimeSpectrum.preimageEquivFiber._proof_7.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q))))) (IsLocalRing.ResidueField.algebra.{u_2, u_1} (Localization.AtPrime.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)) (OreLocalization.instCommRing.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 (Ideal.primeCompl.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)) (OreLocalization.oreSetComm.{u_2} S (CommSemiring.toCommMonoid.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Ideal.primeCompl.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)))) (PrimeSpectrum.preimageEquivFiber._proof_7.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q) R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (OreLocalization.instAlgebra.{u_2, u_1} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Ideal.primeCompl.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)) (OreLocalization.oreSetComm.{u_2} S (CommSemiring.toCommMonoid.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Ideal.primeCompl.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q))) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (IsLocalRing.ResidueField.algebra.{u_2, u_1} (Localization.AtPrime.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)) (OreLocalization.instCommRing.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 (Ideal.primeCompl.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)) (OreLocalization.oreSetComm.{u_2} S (CommSemiring.toCommMonoid.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Ideal.primeCompl.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)))) (PrimeSpectrum.preimageEquivFiber._proof_7.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q) R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (OreLocalization.instAlgebra.{u_2, u_1} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Ideal.primeCompl.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)) (OreLocalization.oreSetComm.{u_2} S (CommSemiring.toCommMonoid.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Ideal.primeCompl.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q))) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (PrimeSpectrum.preimageEquivFiber._proof_11.{u_1, u_2} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q) (Ideal.ResidueField.mapₐ.{u_1, u_1, u_2} R R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q) (Algebra.ofId.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10) (PrimeSpectrum.preimageEquivFiber._proof_13.{u_1, u_2} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)) (IsScalarTower.toAlgHom.{u_1, u_2, u_2} R S (Ideal.ResidueField.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)) (OreLocalization.oreSetComm.{u_2} S (CommSemiring.toCommMonoid.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Ideal.primeCompl.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)))) (PrimeSpectrum.preimageEquivFiber._proof_7.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q))))) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 (IsLocalRing.ResidueField.algebra.{u_2, u_2} (Localization.AtPrime.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)) (OreLocalization.instCommRing.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 (Ideal.primeCompl.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)) (OreLocalization.oreSetComm.{u_2} S (CommSemiring.toCommMonoid.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Ideal.primeCompl.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)))) (PrimeSpectrum.preimageEquivFiber._proof_7.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q) S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 (OreLocalization.instAlgebra.{u_2, u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Ideal.primeCompl.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)) (OreLocalization.oreSetComm.{u_2} S (CommSemiring.toCommMonoid.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Ideal.primeCompl.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)) (OreLocalization.oreSetComm.{u_2} S (CommSemiring.toCommMonoid.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Ideal.primeCompl.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) q)) (PrimeSpectrum.preimageEquivFiber._proof_4.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)))) (PrimeSpectrum.preimageEquivFiber._proof_7.{u_2, u_1} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q) R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (OreLocalization.instAlgebra.{u_2, u_1} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Ideal.primeCompl.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.asIdeal.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} 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(CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (Subtype.val.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)) (PrimeSpectrum.preimageEquivFiber._proof_15.{u_1, u_2} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)))) (PrimeSpectrum.preimageEquivFiber._proof_16.{u_1, u_2} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q)) (fun (q : PrimeSpectrum.{max u_1 u_2} (Ideal.Fiber.{u_1, u_2} R 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(CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (Semifield.toCommSemiring.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (Field.toSemifield.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (IsLocalRing.ResidueField.field.{u_1} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.instCommRing.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)))) (PrimeSpectrum.preimageEquivFiber._proof_2.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)))) (IsLocalRing.ResidueField.algebra.{u_1, u_1} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.instCommRing.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)))) (PrimeSpectrum.preimageEquivFiber._proof_2.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p) R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 (OreLocalization.instAlgebra.{u_1, u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p)) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.asIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) p) (PrimeSpectrum.preimageEquivFiber._proof_1.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 p))) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)))) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) => Subtype.mk.{succ u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (fun (x : PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) => Membership.mem.{u_2, u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (Set.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.instMembership.{u_2} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7))) (Set.preimage.{u_2, u_1} (PrimeSpectrum.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (PrimeSpectrum.comap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7) (algebraMap.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7)) inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) (Singleton.singleton.{u_1, u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4)) (Set.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) (Set.instSingletonSet.{u_1} (PrimeSpectrum.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4))) p)) x) (PrimeSpectrum.comap.{u_2, max u_1 u_2} 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inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10)) => PrimeSpectrum.preimageEquivFiber._proof_6.{u_1, u_2} R S inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.7 inst._@.Mathlib.RingTheory.LocalRing.ResidueField.Fiber.1273674712._hygCtx._hyg.10 p q))","typeFull":"∀ (R : Type u_1) (S : Type u_2) [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : PrimeSpectrum R),\n  PrimeSpectrum.preimageEquivFiber R S p =\n    {\n      toFun := fun q =>\n        {\n          asIdeal :=\n            RingHom.ker\n              (Algebra.TensorProduct.lift (Ideal.ResidueField.mapₐ p.asIdeal (↑q).asIdeal (Algebra.ofId R S) ⋯)\n                  (IsScalarTower.toAlgHom R S (↑q).asIdeal.ResidueField) ⋯).toRingHom,\n          isPrime := ⋯ },\n      invFun := fun q => ⟨PrimeSpectrum.comap Algebra.TensorProduct.includeRight.toRingHom q, ⋯⟩, left_inv := ⋯,\n      right_inv := ⋯ }","typeReadable":"∀ (R : Type u_1) (S : Type u_2) [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : PrimeSpectrum R),\n  PrimeSpectrum.preimageEquivFiber R S p =\n    {\n      toFun := fun q =>\n        {\n          asIdeal :=\n            RingHom.ker\n              (Algebra.TensorProduct.lift (Ideal.ResidueField.mapₐ p.asIdeal (↑q).asIdeal (Algebra.ofId R S) ⋯)\n                  (IsScalarTower.toAlgHom R S (↑q).asIdeal.ResidueField) ⋯).toRingHom,\n          isPrime := ⋯ },\n      invFun := fun q => ⟨PrimeSpectrum.comap Algebra.TensorProduct.includeRight.toRingHom q, ⋯⟩, left_inv := ⋯,\n      right_inv := ⋯ }","typeReferences":[["RingHom"],["Equiv","mk"],["Singleton","singleton"],["Membership","mem"],["Ideal","ResidueField"],["PrimeSpectrum","preimageEquivFiber","_proof_4"],["IsScalarTower","toAlgHom"],["Algebra","id"],["Subtype","val"],["Equiv"],["Set","Elem"],["Ideal","ResidueField","mapₐ"],["PrimeSpectrum","isPrime"],["Semiring","toNonAssocSemiring"],["PrimeSpectrum","preimageEquivFiber","_proof_5"],["OreLocalization","instAlgebra"],["Algebra","TensorProduct","instCommSemiring"],["Semifield","toDivisionSemiring"],["Localization","AtPrime"],["PrimeSpectrum","preimageEquivFiber","_proof_14"],["TensorProduct"],["Algebra","TensorProduct","instAlgebra"],["PrimeSpectrum","mk"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DivisionSemiring","toSemiring"],["PrimeSpectrum","preimageEquivFiber","_proof_6"],["CommSemiring","toCommMonoid"],["Algebra"],["Set","instMembership"],["AlgHom","toRingHom"],["algebraMap"],["Semifield","toCommSemiring"],["PrimeSpectrum","preimageEquivFiber","_proof_1"],["PrimeSpectrum","preimageEquivFiber","_proof_7"],["IsLocalRing","ResidueField","algebra"],["OreLocalization","oreSetComm"],["RingHom","instFunLike"],["Algebra","TensorProduct","instSemiring"],["RingHom","ker"],["PrimeSpectrum","preimageEquivFiber","_proof_15"],["OreLocalization","instCommRing"],["PrimeSpectrum","preimageEquivFiber","_proof_2"],["Algebra","toModule"],["Ideal","primeCompl"],["PrimeSpectrum","asIdeal"],["Eq"],["PrimeSpectrum","comap"],["Algebra","TensorProduct","lift"],["CommRing","toCommSemiring"],["Set","preimage"],["PrimeSpectrum","preimageEquivFiber","_proof_8"],["PrimeSpectrum","preimageEquivFiber","_proof_9"],["Set"],["PrimeSpectrum","preimageEquivFiber","_proof_13"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Set","instSingletonSet"],["CommRing"],["PrimeSpectrum","preimageEquivFiber","_proof_16"],["Algebra","TensorProduct","leftAlgebra"],["Ideal","Fiber"],["PrimeSpectrum"],["Algebra","ofId"],["Ideal","ResidueField","_proof_1"],["Algebra","TensorProduct","includeRight"],["PrimeSpectrum","preimageEquivFiber","_proof_10"],["Field","toSemifield"],["Subtype","mk"],["PrimeSpectrum","preimageEquivFiber","_proof_11"],["IsLocalRing","ResidueField","field"],["PrimeSpectrum","preimageEquivFiber"],["PrimeSpectrum","preimageEquivFiber","_proof_3"]],"valueReferences":[["Semifield","toCommSemiring"],["IsLocalRing","ResidueField","algebra"],["OreLocalization","oreSetComm"],["Singleton","singleton"],["Ideal","ResidueField"],["Algebra","id"],["Set","Elem"],["Equiv"],["PrimeSpectrum","isPrime"],["OreLocalization","instCommRing"],["OreLocalization","instAlgebra"],["Algebra","TensorProduct","instCommSemiring"],["Ideal","primeCompl"],["PrimeSpectrum","asIdeal"],["PrimeSpectrum","comap"],["Localization","AtPrime"],["Set","preimage"],["CommRing","toCommSemiring"],["Set"],["CommSemiring","toSemiring"],["Set","instSingletonSet"],["CommSemiring","toCommMonoid"],["Ideal","Fiber"],["PrimeSpectrum"],["Ideal","ResidueField","_proof_1"],["Eq","refl"],["Field","toSemifield"],["IsLocalRing","ResidueField","field"],["PrimeSpectrum","preimageEquivFiber"],["algebraMap"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","RingTheory","Etale","QuasiFinite",0,"Algebra","exists_etale_isIdempotentElem_forall_liesOver_eq_aux₂","_simp_1_3"],"typeFallback":"forall (R : Type.{u}) (S : Type.{v}) (A : Type.{w}) [inst._@.Mathlib.Algebra.Algebra.Tower.1031967428._hygCtx._hyg.7 : CommSemiring.{u} R] [inst._@.Mathlib.Algebra.Algebra.Tower.1031967428._hygCtx._hyg.10 : CommSemiring.{v} S] [inst._@.Mathlib.Algebra.Algebra.Tower.1031967428._hygCtx._hyg.13 : Semiring.{w} A] [inst._@.Mathlib.Algebra.Algebra.Tower.1031967428._hygCtx._hyg.19 : Algebra.{u, v} R S inst._@.Mathlib.Algebra.Algebra.Tower.1031967428._hygCtx._hyg.7 (CommSemiring.toSemiring.{v} S inst._@.Mathlib.Algebra.Algebra.Tower.1031967428._hygCtx._hyg.10)] [inst._@.Mathlib.Algebra.Algebra.Tower.1031967428._hygCtx._hyg.23 : Algebra.{v, w} S A inst._@.Mathlib.Algebra.Algebra.Tower.1031967428._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Algebra.Tower.1031967428._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Algebra.Tower.1031967428._hygCtx._hyg.31 : Algebra.{u, w} R A inst._@.Mathlib.Algebra.Algebra.Tower.1031967428._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Algebra.Tower.1031967428._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Algebra.Tower.1031967428._hygCtx._hyg.39 : IsScalarTower.{u, v, w} R S A (Algebra.toSMul.{u, v} R S inst._@.Mathlib.Algebra.Algebra.Tower.1031967428._hygCtx._hyg.7 (CommSemiring.toSemiring.{v} S inst._@.Mathlib.Algebra.Algebra.Tower.1031967428._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Algebra.Tower.1031967428._hygCtx._hyg.19) (Algebra.toSMul.{v, w} S A inst._@.Mathlib.Algebra.Algebra.Tower.1031967428._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Algebra.Tower.1031967428._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Algebra.Tower.1031967428._hygCtx._hyg.23) (Algebra.toSMul.{u, w} R A inst._@.Mathlib.Algebra.Algebra.Tower.1031967428._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Algebra.Tower.1031967428._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Algebra.Tower.1031967428._hygCtx._hyg.31)], Eq.{max (succ u) (succ w)} (RingHom.{u, w} R A (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Algebra.Tower.1031967428._hygCtx._hyg.7)) 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inst._@.Mathlib.Algebra.Algebra.Tower.1031967428._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Algebra.Tower.1031967428._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Algebra.Tower.1031967428._hygCtx._hyg.31)","typeFull":"∀ (R : Type u) (S : Type v) (A : Type w) [inst : CommSemiring R] [inst_1 : CommSemiring S] [inst_2 : Semiring A]\n  [inst_3 : Algebra R S] [inst_4 : Algebra S A] [inst_5 : Algebra R A] [IsScalarTower R S A],\n  (algebraMap S A).comp (algebraMap R S) = algebraMap R A","typeReadable":"∀ (R : Type u) (S : Type v) (A : Type w) [inst : CommSemiring R] [inst_1 : CommSemiring S] [inst_2 : Semiring A]\n  [inst_3 : Algebra R S] [inst_4 : Algebra S A] [inst_5 : Algebra R A] [IsScalarTower R S A],\n  (algebraMap S A).comp (algebraMap R S) = algebraMap R A","typeReferences":[["RingHom"],["Semiring","toNonAssocSemiring"],["IsScalarTower"],["Algebra","toSMul"],["CommSemiring","toSemiring"],["CommSemiring"],["RingHom","comp"],["Eq"],["algebraMap"],["Algebra"],["Semiring"]],"valueReferences":[["RingHom"],["Semiring","toNonAssocSemiring"],["CommSemiring","toSemiring"],["Eq","symm"],["IsScalarTower","algebraMap_eq"],["RingHom","comp"],["algebraMap"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","RingTheory","Etale","QuasiFinite",0,"Algebra","exists_notMem_and_isIntegral_forall_mem_of_ne_of_liesOver","_proof_1_5"],"typeFallback":"forall {R : Type.{u_2}} {S : Type.{u_1}} [inst : CommRing.{u_2} R] [inst_1 : CommRing.{u_1} S] [inst_2 : Algebra.{u_2, u_1} R S (CommRing.toCommSemiring.{u_2} R inst) (CommSemiring.toSemiring.{u_1} S (CommRing.toCommSemiring.{u_1} S inst_1))] (s₂ : Subtype.{succ u_1} S (fun (x : S) => Membership.mem.{u_1, u_1} S (Subalgebra.{u_2, u_1} R S (CommRing.toCommSemiring.{u_2} R inst) 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(Submonoid.powers s₂) (Localization (Submonoid.powers s₂))","typeReferences":[["Subalgebra"],["integralClosure"],["Localization"],["CommRing","toCommSemiring"],["Subtype"],["SetLike","instMembership"],["OreLocalization","oreSetComm"],["CommSemiring","toSemiring"],["Membership","mem"],["Subalgebra","instSetLike"],["Semiring","toMonoidWithZero"],["CommRing"],["Algebra","id"],["CommSemiring","toCommMonoid"],["Algebra"],["IsLocalization"],["Subalgebra","toCommSemiring"],["OreLocalization","instAlgebra"],["Submonoid","powers"],["MonoidWithZero","toMonoid"],["OreLocalization","instCommSemiring"]],"valueReferences":[["integralClosure"],["Subalgebra"],["CommRing","toCommSemiring"],["SetLike","instMembership"],["Subtype"],["Subalgebra","instSetLike"],["CommSemiring","toSemiring"],["Membership","mem"],["Localization","awayLift","_proof_1"],["Subalgebra","toCommSemiring"]]},{"isProp":true,"kind":"theorem","name":["Ideal","fiberIsoOfBijectiveResidueField","_proof_5"],"typeFallback":"forall {R : Type.{u_1}} {R' : Type.{u_2}} [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 : CommRing.{u_1} R] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 : CommRing.{u_2} R'] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 : Algebra.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))] {p : Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5))} {q : Ideal.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))} [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26 : Ideal.IsPrime.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5)) p] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28 : Ideal.IsPrime.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.30 : Ideal.LiesOver.{u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 q p], SMulCommClass.{u_1, u_1, u_2} R (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semifield.toDivisionSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Field.toSemifield.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (IsLocalRing.ResidueField.field.{u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} 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(AddMonoid.toAddZeroClass.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (AddCommMonoid.toAddMonoid.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semiring.toNonAssocSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)))))))) (Algebra.toModule.{u_1, u_2} R (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (DivisionSemiring.toSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semifield.toDivisionSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Field.toSemifield.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))))) (IsLocalRing.ResidueField.algebra.{u_2, u_1} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 (OreLocalization.instAlgebra.{u_2, u_1} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q 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(Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (MonoidWithZero.toMonoid.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (Semiring.toMonoidWithZero.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) (Semifield.toCommSemiring.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S 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(CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18)) (Set.{max u_2 u_3} (Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S 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(Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18))) (Set.instMembership.{max u_2 u_3} (Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18) (Algebra.TensorProduct.leftAlgebra.{u_1, u_2, u_2, u_3} R R' R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18 (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8) 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(CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18)) (Ideal.primesOver.{u_2, max u_3 u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8) q (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18) (Algebra.TensorProduct.leftAlgebra.{u_1, u_2, u_2, u_3} R R' R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18 (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8) (Algebra.id.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) (Algebra.to_smulCommClass.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14)))) => Set.Elem.{u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11))) (Ideal.primesOver.{u_1, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) p S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18)) (instFunLikeOrderIso.{max u_2 u_3, u_3} (Set.Elem.{max u_2 u_3} (Ideal.{max u_3 u_2} (TensorProduct.{u_1, 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(CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S 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(NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18)) (Submodule.instPartialOrder.{max u_2 u_3, max u_2 u_3} 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(NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R 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(Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18)) (Set.{max u_2 u_3} (Ideal.{max u_3 u_2} 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(CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18))) (Ideal.primesOver.{u_2, max u_3 u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8) q (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' 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(CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18) x))) (Ideal.fiberIsoOfBijectiveResidueField.{u_1, u_2, u_3} R R' S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18 p q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.26 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.28 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.30 H) Q)) (Ideal.comap.{u_3, max u_2 u_3, max u_3 u_2} S (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18)) (AlgHom.{u_1, u_3, max u_3 u_2} R S (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} R' (Semiring.toNonAssocSemiring.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18)) (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18 (Algebra.TensorProduct.instAlgebra.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R 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(CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18)) (Set.{max u_2 u_3} (Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) R' S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18))) (Set.instMembership.{max u_2 u_3} (Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18))) (Ideal.primesOver.{u_2, max u_3 u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8) q (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18) (Algebra.TensorProduct.leftAlgebra.{u_1, u_2, u_2, u_3} R R' R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.18 (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8) (Algebra.id.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) (Algebra.to_smulCommClass.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.1875429475._hygCtx._hyg.14))) x) Q))","typeFull":"∀ {R : Type u_1} {R' : Type u_2} {S : Type u_3} [inst : CommRing R] [inst_1 : CommRing R'] [inst_2 : CommRing S]\n  [inst_3 : Algebra R R'] [inst_4 : Algebra R S] {p : Ideal R} {q : Ideal R'} [inst_5 : p.IsPrime] [inst_6 : q.IsPrime]\n  [inst_7 : q.LiesOver p] (H : Function.Bijective ⇑(Ideal.ResidueField.mapₐ p q (Algebra.ofId R R') ⋯))\n  (Q : ↑(q.primesOver (TensorProduct R R' S))),\n  ↑((Ideal.fiberIsoOfBijectiveResidueField H) Q) = Ideal.comap Algebra.TensorProduct.includeRight ↑Q","typeReadable":"∀ {R : Type u_1} {R' : Type u_2} {S : Type u_3} [inst : CommRing R] [inst_1 : CommRing R'] [inst_2 : CommRing S]\n  [inst_3 : Algebra R R'] [inst_4 : Algebra R S] {p : Ideal R} {q : Ideal R'} [inst_5 : p.IsPrime] [inst_6 : q.IsPrime]\n  [inst_7 : q.LiesOver p] (H : Function.Bijective ⇑(Ideal.ResidueField.mapₐ p q (Algebra.ofId R R') ⋯))\n  (Q : ↑(q.primesOver (TensorProduct R R' S))),\n  ↑((Ideal.fiberIsoOfBijectiveResidueField H) Q) = Ideal.comap Algebra.TensorProduct.includeRight 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(SetLike.instMembership.{u_1, u_1} (Subalgebra.{u_2, u_1} R S (CommRing.toCommSemiring.{u_2} R inst) (CommSemiring.toSemiring.{u_1} S (CommRing.toCommSemiring.{u_1} S inst_1)) inst_2) S (Subalgebra.instSetLike.{u_2, u_1} R S (CommRing.toCommSemiring.{u_2} R inst) (CommSemiring.toSemiring.{u_1} S (CommRing.toCommSemiring.{u_1} S inst_1)) inst_2)) (integralClosure.{u_2, u_1} R S inst inst_1 inst_2) x)) S (Semiring.toNonAssocSemiring.{u_1} (Subtype.{succ u_1} S (fun (x : S) => Membership.mem.{u_1, u_1} S (Subalgebra.{u_2, u_1} R S (CommRing.toCommSemiring.{u_2} R inst) (CommSemiring.toSemiring.{u_1} S (CommRing.toCommSemiring.{u_1} S inst_1)) inst_2) (SetLike.instMembership.{u_1, u_1} (Subalgebra.{u_2, u_1} R S (CommRing.toCommSemiring.{u_2} R inst) (CommSemiring.toSemiring.{u_1} S (CommRing.toCommSemiring.{u_1} S inst_1)) inst_2) S (Subalgebra.instSetLike.{u_2, u_1} R S (CommRing.toCommSemiring.{u_2} R inst) (CommSemiring.toSemiring.{u_1} S (CommRing.toCommSemiring.{u_1} S inst_1)) inst_2)) (integralClosure.{u_2, u_1} R S inst inst_1 inst_2) x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} S (fun (x : S) => Membership.mem.{u_1, u_1} S (Subalgebra.{u_2, u_1} R S (CommRing.toCommSemiring.{u_2} R inst) (CommSemiring.toSemiring.{u_1} S (CommRing.toCommSemiring.{u_1} S inst_1)) inst_2) (SetLike.instMembership.{u_1, u_1} (Subalgebra.{u_2, u_1} R S (CommRing.toCommSemiring.{u_2} R inst) (CommSemiring.toSemiring.{u_1} S (CommRing.toCommSemiring.{u_1} S inst_1)) inst_2) S (Subalgebra.instSetLike.{u_2, u_1} R S (CommRing.toCommSemiring.{u_2} R inst) (CommSemiring.toSemiring.{u_1} S (CommRing.toCommSemiring.{u_1} S inst_1)) inst_2)) (integralClosure.{u_2, u_1} R S inst inst_1 inst_2) x)) (Subalgebra.toCommSemiring.{u_2, u_1} R S (CommRing.toCommSemiring.{u_2} R inst) (CommRing.toCommSemiring.{u_1} S inst_1) inst_2 (integralClosure.{u_2, u_1} R S inst inst_1 inst_2)))) (Semiring.toNonAssocSemiring.{u_1} S (CommSemiring.toSemiring.{u_1} S (CommRing.toCommSemiring.{u_1} S inst_1))))","typeFull":"∀ {R : Type u_2} {S : Type u_1} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S],\n  RingHomClass (↥(integralClosure R S) →+* S) (↥(integralClosure R S)) S","typeReadable":"∀ {R : Type u_2} {S : Type u_1} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S],\n  RingHomClass (↥(integralClosure R S) →+* S) (↥(integralClosure R S)) S","typeReferences":[["RingHom"],["Subalgebra"],["integralClosure"],["CommRing","toCommSemiring"],["Subtype"],["SetLike","instMembership"],["CommSemiring","toSemiring"],["Membership","mem"],["Subalgebra","instSetLike"],["RingHom","instFunLike"],["CommRing"],["Subalgebra","toCommSemiring"],["Algebra"],["Semiring","toNonAssocSemiring"],["RingHomClass"]],"valueReferences":[["integralClosure"],["Subalgebra"],["CommRing","toCommSemiring"],["SetLike","instMembership"],["Subtype"],["Subalgebra","instSetLike"],["CommSemiring","toSemiring"],["Membership","mem"],["Ideal","under","_proof_1"],["Subalgebra","toCommSemiring"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","RingTheory","Etale","QuasiFinite",0,"Algebra","exists_etale_isIdempotentElem_forall_liesOver_eq_aux₂","_simp_1_2"],"typeFallback":"forall {R : Type.{uR}} {A₁ : Type.{uA₁}} {A₂ : Type.{uA₂}} [inst._@.Mathlib.Algebra.Algebra.Equiv.3191996021._hygCtx._hyg.9 : CommSemiring.{uR} R] [inst._@.Mathlib.Algebra.Algebra.Equiv.3191996021._hygCtx._hyg.12 : Semiring.{uA₁} A₁] [inst._@.Mathlib.Algebra.Algebra.Equiv.3191996021._hygCtx._hyg.15 : Semiring.{uA₂} A₂] [inst._@.Mathlib.Algebra.Algebra.Equiv.3191996021._hygCtx._hyg.30 : Algebra.{uR, uA₁} R A₁ inst._@.Mathlib.Algebra.Algebra.Equiv.3191996021._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Algebra.Equiv.3191996021._hygCtx._hyg.12] [inst._@.Mathlib.Algebra.Algebra.Equiv.3191996021._hygCtx._hyg.34 : Algebra.{uR, uA₂} R A₂ inst._@.Mathlib.Algebra.Algebra.Equiv.3191996021._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Algebra.Equiv.3191996021._hygCtx._hyg.15] (e : AlgEquiv.{uR, uA₁, uA₂} R A₁ A₂ inst._@.Mathlib.Algebra.Algebra.Equiv.3191996021._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Algebra.Equiv.3191996021._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Algebra.Equiv.3191996021._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Algebra.Equiv.3191996021._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Algebra.Equiv.3191996021._hygCtx._hyg.34), Eq.{max (succ uA₁) 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inst._@.Mathlib.Algebra.Algebra.Equiv.3191996021._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Algebra.Equiv.3191996021._hygCtx._hyg.34)) e))","typeFull":"∀ {R : Type uR} {A₁ : Type uA₁} {A₂ : Type uA₂} [inst : CommSemiring R] [inst_1 : Semiring A₁] [inst_2 : Semiring A₂]\n  [inst_3 : Algebra R A₁] [inst_4 : Algebra R A₂] (e : A₁ ≃ₐ[R] A₂), ↑e = ↑↑e","typeReadable":"∀ {R : Type uR} {A₁ : Type uA₁} {A₂ : Type uA₂} [inst : CommSemiring R] [inst_1 : Semiring A₁] [inst_2 : Semiring A₂]\n  [inst_3 : Algebra R A₁] [inst_4 : Algebra R A₂] (e : A₁ ≃ₐ[R] A₂), ↑e = ↑↑e","typeReferences":[["AlgHom"],["RingHom"],["AlgEquivClass","toAlgHomClass"],["CommSemiring"],["AlgEquiv","instFunLike"],["AlgHomClass","toAlgHom"],["AlgEquiv","instEquivLike"],["Algebra"],["Semiring","toNonAssocSemiring"],["AlgEquiv"],["AlgHom","funLike"],["AlgHomClass","toRingHomClass"],["AlgHom","algHomClass"],["Eq"],["AlgEquiv","instAlgEquivClass"],["RingHomClass","toRingHom"],["Semiring"]],"valueReferences":[["RingHom"],["AlgHom"],["AlgEquivClass","toAlgHomClass"],["AlgEquiv","instFunLike"],["AlgHomClass","toAlgHom"],["AlgEquiv","instEquivLike"],["Semiring","toNonAssocSemiring"],["AlgEquiv"],["Eq","symm"],["AlgHom","funLike"],["AlgEquiv","toAlgHom_toRingHom"],["AlgHomClass","toRingHomClass"],["AlgHom","algHomClass"],["AlgEquiv","instAlgEquivClass"],["RingHomClass","toRingHom"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","RingTheory","Etale","QuasiFinite",0,"Algebra","exists_etale_isIdempotentElem_forall_liesOver_eq_aux","_simp_1_7"],"typeFallback":"forall {A 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P)","typeReferences":[["Ideal","LiesOver"],["Ideal"],["CommSemiring","toSemiring"],["CommSemiring"],["Ideal","under"],["Eq"],["Algebra"],["Semiring"]],"valueReferences":[["Ideal","LiesOver"],["Ideal"],["Ideal","liesOver_iff"],["CommSemiring","toSemiring"],["Ideal","under"],["Eq"],["propext"]]},{"isProp":true,"kind":"theorem","name":["Ideal","comap_fiberIsoOfBijectiveResidueField_symm"],"typeFallback":"forall {R : Type.{u_1}} {R' : Type.{u_2}} {S : Type.{u_3}} [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5 : CommRing.{u_1} R] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8 : CommRing.{u_2} R'] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11 : CommRing.{u_3} S] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14 : Algebra.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' 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(CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.26) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.26)) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5)))) (IsLocalRing.ResidueField.algebra.{u_2, u_1} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.28) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5 (OreLocalization.instAlgebra.{u_2, u_1} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.28)) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14))) (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.26) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.26) => Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.28) (AlgHom.funLike.{u_1, u_1, u_2} R (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.26) (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.28) (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (DivisionSemiring.toSemiring.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.26) (Semifield.toDivisionSemiring.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.26) (Field.toSemifield.{u_1} (Ideal.ResidueField.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.26) (IsLocalRing.ResidueField.field.{u_1} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.26) (OreLocalization.instCommRing.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5 (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.26) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.26))) (Ideal.ResidueField._proof_1.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.26))))) (DivisionSemiring.toSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.28) (Semifield.toDivisionSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.28) (Field.toSemifield.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.28) (IsLocalRing.ResidueField.field.{u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.28))))) (IsLocalRing.ResidueField.algebra.{u_1, u_1} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.26) (OreLocalization.instCommRing.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5 (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.26) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.26))) (Ideal.ResidueField._proof_1.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.26) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5 (OreLocalization.instAlgebra.{u_1, u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.26) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R 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(CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.28) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5 (OreLocalization.instAlgebra.{u_2, u_1} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8 (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.26 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.28 (Algebra.ofId.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14) (Ideal.over_def.{u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14 q p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.30)))) (Q : Set.Elem.{u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11))) (Ideal.primesOver.{u_1, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) p S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)), Eq.{succ u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)) (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11))) (Semiring.toNonAssocSemiring.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} R' (Semiring.toNonAssocSemiring.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S 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(CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18))) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18) (RingHom.instFunLike.{u_3, max u_3 u_2} S (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} R' (Semiring.toNonAssocSemiring.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)) (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11))) (Semiring.toNonAssocSemiring.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18))) (RingHomClass.toRingHom.{max u_3 u_2, u_3, max u_3 u_2} (AlgHom.{u_1, u_3, max u_3 u_2} R S (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} R' (Semiring.toNonAssocSemiring.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R 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(CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11))) (Set.{u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)))) (Set.instMembership.{u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)))) (Ideal.primesOver.{u_1, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) p S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18) x)) (Subtype.instLE.{max u_2 u_3} (Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)) (Preorder.toLE.{max u_2 u_3} (Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S 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(CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)) (PartialOrder.toPreorder.{max u_2 u_3} (Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)) (Submodule.instPartialOrder.{max u_2 u_3, max u_2 u_3} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)) (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S 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(CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18) (NonUnitalNonAssocSemiring.toAddCommMonoid.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)) (Semiring.toNonAssocSemiring.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18) (Algebra.TensorProduct.leftAlgebra.{u_1, u_2, u_2, u_3} R R' R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18 (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8) (Algebra.id.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) (Algebra.to_smulCommClass.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14))) x))) (Set.Elem.{u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11))) (Ideal.primesOver.{u_1, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) p S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Set.Elem.{u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11))) (Ideal.primesOver.{u_1, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) p S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)) => Set.Elem.{max u_2 u_3} (Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)) (Ideal.primesOver.{u_2, max u_3 u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8) q (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18) (Algebra.TensorProduct.leftAlgebra.{u_1, u_2, u_2, u_3} R R' R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18 (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8) (Algebra.id.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) (Algebra.to_smulCommClass.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14)))) (instFunLikeOrderIso.{u_3, max u_2 u_3} (Set.Elem.{u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11))) (Ideal.primesOver.{u_1, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) p S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)) (Set.Elem.{max u_2 u_3} (Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)) (Ideal.primesOver.{u_2, max u_3 u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8) q (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18) (Algebra.TensorProduct.leftAlgebra.{u_1, u_2, u_2, u_3} R R' R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18 (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8) (Algebra.id.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) (Algebra.to_smulCommClass.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14)))) (Subtype.instLE.{u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11))) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)))))) (fun (x : Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11))) => Membership.mem.{u_3, u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11))) (Set.{u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)))) (Set.instMembership.{u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)))) (Ideal.primesOver.{u_1, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) p S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18) x)) (Subtype.instLE.{max u_2 u_3} (Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S 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(CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18) (Algebra.TensorProduct.leftAlgebra.{u_1, u_2, u_2, u_3} R R' R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)) (Subtype.instLE.{max u_2 u_3} (Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S 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(CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)) (Submodule.instPartialOrder.{max u_2 u_3, max u_2 u_3} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)) (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18))))) (fun (x : Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)) (Set.{max u_2 u_3} (Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' 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(Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2115617021._hygCtx._hyg.18))) (Set.instMembership.{max u_2 u_3} (Ideal.{max u_3 u_2} 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[inst_2 : CommRing S]\n  [inst_3 : Algebra R R'] [inst_4 : Algebra R S] {p : Ideal R} {q : Ideal R'} [inst_5 : p.IsPrime] [inst_6 : q.IsPrime]\n  [inst_7 : q.LiesOver p] (H : Function.Bijective ⇑(Ideal.ResidueField.mapₐ p q (Algebra.ofId R R') ⋯))\n  (Q : ↑(p.primesOver S)),\n  Ideal.comap ↑Algebra.TensorProduct.includeRight ↑((Ideal.fiberIsoOfBijectiveResidueField H).symm Q) = ↑Q","typeReadable":"∀ {R : Type u_1} {R' : Type u_2} {S : Type u_3} [inst : CommRing R] [inst_1 : CommRing R'] [inst_2 : CommRing S]\n  [inst_3 : Algebra R R'] [inst_4 : Algebra R S] {p : Ideal R} {q : Ideal R'} [inst_5 : p.IsPrime] [inst_6 : q.IsPrime]\n  [inst_7 : q.LiesOver p] (H : Function.Bijective ⇑(Ideal.ResidueField.mapₐ p q (Algebra.ofId R R') ⋯))\n  (Q : ↑(p.primesOver S)),\n  Ideal.comap ↑Algebra.TensorProduct.includeRight ↑((Ideal.fiberIsoOfBijectiveResidueField H).symm Q) = 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nAssocRing"],["DistribMulAction","toMulAction"],["PrimeSpectrum","primesOverOrderIsoFiber","_proof_1"],["Semifield","toDivisionSemiring"],["Semiring","toModule"],["NonAssocSemiring","toMulZeroOneClass"],["RingHomClass","toRingHom"],["PrimeSpectrum","preimageEquivFiber","_proof_14"],["TensorProduct"],["Ideal","fiberIsoOfBijectiveResidueField","_proof_2"],["SetLike","instMembership"],["NonUnitalNonAssocSemiring","toDistrib"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["PrimeSpectrum","mk"],["Algebra","TensorProduct","instCommRing"],["Ideal","fiberIsoOfBijectiveResidueField","_proof_3"],["TensorProduct","leftModule"],["RingHom","mem_ker","_simp_1"],["Ideal","fiberIsoOfBijectiveResidueField","_proof_13"],["HSMul","hSMul"],["id"],["instHMul"],["Ideal","primesOver"],["AlgHomClass","toRingHomClass"],["AlgEquiv","instAlgEquivClass"],["PrimeSpectrum","preimageEquivFiber","_proof_7"],["IsLocalRing","ResidueField","algebra"],["IsLocalRing","ResidueField","instAlgebra"],["Ideal","fiberIsoOfBijectiveResidueField","_proof_8"],["Algebra","TensorProduct","instSemiring"],["RingHom","instFunLike"],["CommRing","toNonUnitalCommRing"],["DFunLike","coe"],["AlgEquiv","instEquivLike"],["congrArg"],["AlgEquiv","restrictScalars"],["OreLocalization","instCommRing"],["Ideal"],["PrimeSpectrum","preimageEquivFiber","_proof_2"],["Algebra","toModule"],["RingHomClass","toMonoidWithZeroHomClass"],["Ideal","mem_comap","_simp_1"],["MonoidWithZero","toMonoid"],["Algebra","toSMul"],["OrderIso","symm"],["instHSMul"],["Ideal","primeCompl"],["Zero","toOfNat0"],["congrFun'"],["PrimeSpectrum","asIdeal"],["map_one"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Set","preimage"],["CommRing","toCommSemiring"],["PrimeSpectrum","preimageEquivFiber","_proof_8"],["Algebra","TensorProduct","congr"],["True"],["PrimeSpectrum","preimageEquivFiber","_proof_13"],["Distrib","toMul"],["Submodule","instPartialOrder"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["Set","instSingletonSet"],["Algebra","TensorProduct","leftAlgebra"],["instFunLikeOrderIso"],["Ideal","Fiber"],["Algebra","ofId"],["Algebra","TensorProduct","includeRight"],["One","toOfNat1"],["of_eq_true"],["Ideal","fiberIsoOfBijectiveResidueField","_proof_12"],["Submodule","setLike"],["AlgEquiv"],["Field","toSemifield"],["Subtype","mk"],["AlgHom","algHomClass"],["IsLocalRing","ResidueField","field"]]},{"isProp":true,"kind":"theorem","name":["Ideal","fiberIsoOfBijectiveResidueField","_proof_2"],"typeFallback":"forall {R : Type.{u_1}} {R' : Type.{u_2}} [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 : CommRing.{u_1} R] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 : CommRing.{u_2} R'] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 : Algebra.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))], SMulCommClass.{u_1, u_2, u_2} R R' R' (Algebra.toSMul.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (SMulZeroClass.toSMul.{u_2, u_2} R' R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_1} R' (CommSemiring.toCommMonoid.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))))) (IsLocalRing.ResidueField.algebra.{u_1, u_1} (Localization.AtPrime.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_1} R' (CommSemiring.toCommMonoid.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (OreLocalization.instAlgebra.{u_1, u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_1} R' (CommSemiring.toCommMonoid.{u_1} R' (CommRing.toCommSemiring.{u_1} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (AddZeroClass.toAddZero.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (AddMonoid.toAddZeroClass.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (AddMonoidWithOne.toAddMonoid.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semiring.toNonAssocSemiring.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (DivisionSemiring.toSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} (Localization.AtPrime.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_1} R' (CommSemiring.toCommMonoid.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) (Semifield.toDivisionSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} (Localization.AtPrime.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_1} R' (CommSemiring.toCommMonoid.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) (Field.toSemifield.{u_1} (IsLocalRing.ResidueField.{u_1} (Localization.AtPrime.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_1} R' (CommSemiring.toCommMonoid.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) (IsLocalRing.ResidueField.field.{u_1} (Localization.AtPrime.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_1} R' (CommSemiring.toCommMonoid.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)))))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (AddMonoid.toAddZeroClass.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (AddMonoidWithOne.toAddMonoid.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semiring.toNonAssocSemiring.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (DivisionSemiring.toSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} (Localization.AtPrime.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_1} R' (CommSemiring.toCommMonoid.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) (Semifield.toDivisionSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} (Localization.AtPrime.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_1} R' (CommSemiring.toCommMonoid.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) (Field.toSemifield.{u_1} (IsLocalRing.ResidueField.{u_1} (Localization.AtPrime.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) (Algebra.id.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)))))","typeFull":"∀ {R' : Type u_1} [inst : CommRing R'] {q : Ideal R'} [inst_1 : q.IsPrime],\n  IsScalarTower R' q.ResidueField q.ResidueField","typeReadable":"∀ {R' : Type u_1} [inst : CommRing R'] {q : Ideal R'} [inst_1 : q.IsPrime],\n  IsScalarTower R' q.ResidueField 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Semiring.toNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Subalgebra.algebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)))) (Algebra.TensorProduct.instSemiring.{u_1, u_1, u_2} R R' (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946 (Subalgebra.toSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Subalgebra.algebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))) (Algebra.TensorProduct.leftAlgebra.{u_1, u_1, u_1, u_2} R R' R' (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946 (Subalgebra.toSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Subalgebra.algebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944) (Algebra.id.{u_1} R' (CommRing.toCommSemiring.{u_1} 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R' (Semiring.toNonAssocSemiring.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Semiring.toNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Subalgebra.toSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S 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(CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (Subalgebra.toSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Subalgebra.algebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (RingHom.{max u_1 u_2, max u_1 u_2} (TensorProduct.{u_1, u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R' (Semiring.toNonAssocSemiring.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Semiring.toNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Subalgebra.toSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))))) (Algebra.toModule.{u_1, u_1} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946) (Algebra.toModule.{u_1, u_2} R (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (Subalgebra.toSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Subalgebra.algebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)))) (TensorProduct.{u_1, u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R' (Semiring.toNonAssocSemiring.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} S (Semiring.toNonAssocSemiring.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u_1, u_1} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946) (Algebra.toModule.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Semiring.toNonAssocSemiring.{max u_1 u_2} (TensorProduct.{u_1, u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R' (Semiring.toNonAssocSemiring.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Semiring.toNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Subalgebra.toSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))))) (Algebra.toModule.{u_1, u_1} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946) (Algebra.toModule.{u_1, u_2} R (Subtype.{succ 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (Subalgebra.toSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Subalgebra.algebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)))) (Algebra.TensorProduct.instSemiring.{u_1, u_1, u_2} R R' (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946 (Subalgebra.toSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Subalgebra.algebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))) (Algebra.TensorProduct.instSemiring.{u_1, u_1, u_2} R R' S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R' (Semiring.toNonAssocSemiring.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Semiring.toNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Subalgebra.toSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (Subalgebra.toSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Subalgebra.algebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)))) (TensorProduct.{u_1, u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R' (Semiring.toNonAssocSemiring.{u_1} R' (CommSemiring.toSemiring.{u_1} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Semiring.toNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Subalgebra.toSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))))) (Algebra.toModule.{u_1, u_1} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946) (Algebra.toModule.{u_1, u_2} R (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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(Subalgebra.toSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Subalgebra.algebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)))) (Algebra.TensorProduct.instSemiring.{u_1, u_1, u_2} R R' (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946 (Subalgebra.toSemiring.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Subalgebra.algebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)))) (Semiring.toNonAssocSemiring.{max u_1 u_2} (TensorProduct.{u_1, u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R' (Semiring.toNonAssocSemiring.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} S (Semiring.toNonAssocSemiring.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u_1, 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w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946 (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))) (AlgHom.toRingHom.{u_1, max u_1 u_2, max u_1 u_2} R' (TensorProduct.{u_1, u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R' (Semiring.toNonAssocSemiring.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Semiring.toNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))))) (Algebra.toModule.{u_1, u_1} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946) (Algebra.toModule.{u_1, u_2} R (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Subalgebra.algebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))) (Algebra.TensorProduct.instSemiring.{u_1, u_1, u_2} R R' S 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(SetLike.instMembership.{u_2, u_2} (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) 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(CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Semiring.toNonAssocSemiring.{u_2} (Subtype.{succ u_2} S (fun (x : S) => Membership.mem.{u_2, u_2} S (Subalgebra.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R 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(CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946 (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (PartialOrder.toPreorder.{max u_1 u_2} (Ideal.{max u_2 u_1} (TensorProduct.{u_1, u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} R' (CommRing.toNonUnitalCommRing.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} S 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(CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u_1, u_1, u_2} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946 (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)))) (Semiring.toModule.{max u_2 u_1} (TensorProduct.{u_1, u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))))) (fun (x : Ideal.{max u_2 u_1} (TensorProduct.{u_1, u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} R' (CommRing.toNonUnitalCommRing.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} S (CommRing.toNonUnitalCommRing.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u_1, u_1} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946) (Algebra.toModule.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u_1, u_1, u_2} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) 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(CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u_1, u_1, u_2} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946 (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Ideal.primesOver.{u_1, max u_2 u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944) P (TensorProduct.{u_1, u_1, u_2} R (CommRing.toCommSemiring.{u_1} R 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(CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946 (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (Algebra.TensorProduct.leftAlgebra.{u_1, u_1, u_1, u_2} R R' R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946 (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944) (Algebra.id.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) (Algebra.to_smulCommClass.{u_1, u_1} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946)))) (Subtype.instLE.{u_2} (Ideal.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))) (Preorder.toLE.{u_2} (Ideal.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S 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(CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946 (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (PartialOrder.toPreorder.{max u_1 u_2} (Ideal.{max u_2 u_1} (TensorProduct.{u_1, u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} R' (CommRing.toNonUnitalCommRing.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} S 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(CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u_1, u_1, u_2} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946 (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)))) (Semiring.toModule.{max u_2 u_1} (TensorProduct.{u_1, u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))))) (fun (x : Ideal.{max u_2 u_1} (TensorProduct.{u_1, u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} R' (CommRing.toNonUnitalCommRing.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} S (CommRing.toNonUnitalCommRing.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u_1, u_1} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946) (Algebra.toModule.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u_1, u_1, u_2} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) 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w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946 (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (Algebra.TensorProduct.leftAlgebra.{u_1, u_1, u_1, u_2} R R' R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946 (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944) (Algebra.id.{u_1} R' 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R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946)))) (Set.Elem.{u_2} (Ideal.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))) (Ideal.primesOver.{u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) p S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Subtype.instLE.{max u_1 u_2} (Ideal.{max u_2 u_1} (TensorProduct.{u_1, u_1, u_2} R (CommRing.toCommSemiring.{u_1} R 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(CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946 (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (PartialOrder.toPreorder.{max u_1 u_2} (Ideal.{max u_2 u_1} (TensorProduct.{u_1, u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} R' (CommRing.toNonUnitalCommRing.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} S 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(NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} R' (CommRing.toNonUnitalCommRing.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} S (CommRing.toNonUnitalCommRing.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u_1, u_1} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946) (Algebra.toModule.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) 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(CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946 (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (NonUnitalNonAssocSemiring.toAddCommMonoid.{max u_2 u_1} (TensorProduct.{u_1, u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} R' (CommRing.toNonUnitalCommRing.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} S 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{max u_2 u_1} (TensorProduct.{u_1, u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} R' (CommRing.toNonUnitalCommRing.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} S (CommRing.toNonUnitalCommRing.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u_1, u_1} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946) (Algebra.toModule.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Semiring.toNonAssocSemiring.{max u_2 u_1} (TensorProduct.{u_1, u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} R' (CommRing.toNonUnitalCommRing.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} S (CommRing.toNonUnitalCommRing.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u_1, u_1} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946) (Algebra.toModule.{u_1, u_2} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u_1, u_1, u_2} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944)) w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946 (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)))) (Semiring.toModule.{max u_2 u_1} (TensorProduct.{u_1, u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S 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w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.946 (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) => Membership.mem.{max u_1 u_2, max u_1 u_2} (Ideal.{max u_2 u_1} (TensorProduct.{u_1, u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} R' (CommRing.toNonUnitalCommRing.{u_1} R' w._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.944))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} S 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[inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal R)\n  [inst_3 : p.IsPrime] (q : Ideal S) [inst_4 : q.IsPrime] [inst_5 : q.LiesOver p] (R' : Type u_1) (w : CommRing R')\n  (w_1 : Algebra R R') (P : Ideal R') (w_2 : P.IsPrime) (w_3 : P.LiesOver p)\n  (hP : Function.Bijective ⇑(Ideal.ResidueField.mapₐ p P (Algebra.ofId R R') ⋯)),\n  (Ideal.comap (Algebra.TensorProduct.map (AlgHom.id R' R') (integralClosure R S).val).toRingHom\n        ↑((Ideal.fiberIsoOfBijectiveResidueField hP).symm ⟨q, ⋯⟩)).LiesOver\n    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{R : Type.{u_1}} {R' : Type.{u_2}} [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 : CommRing.{u_1} R] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 : CommRing.{u_2} R'] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 : Algebra.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))] {q : Ideal.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))} [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28 : Ideal.IsPrime.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q], IsScalarTower.{u_1, u_2, u_2} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) (Semifield.toDivisionSemiring.{u_2} (IsLocalRing.ResidueField.{u_2} 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))))) (IsLocalRing.ResidueField.algebra.{u_2, u_1} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 (OreLocalization.instAlgebra.{u_2, u_1} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14))) (SMulZeroClass.toSMul.{u_2, u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (AddZero.toZero.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (AddZeroClass.toAddZero.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (AddMonoid.toAddZeroClass.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q 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(Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) (Semifield.toDivisionSemiring.{u_2} (IsLocalRing.ResidueField.{u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)))))))))))) (DistribSMul.toSMulZeroClass.{u_2, u_2} (Ideal.ResidueField.{u_2} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) (Semifield.toDivisionSemiring.{u_2} (IsLocalRing.ResidueField.{u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' 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{R : Type.{u}} {A : Type.{v}} {B : Type.{w}} [inst._@.Mathlib.Algebra.Algebra.Hom.1886971770._hygCtx._hyg.7 : CommSemiring.{u} R] [inst._@.Mathlib.Algebra.Algebra.Hom.1886971770._hygCtx._hyg.10 : Semiring.{v} A] [inst._@.Mathlib.Algebra.Algebra.Hom.1886971770._hygCtx._hyg.13 : Semiring.{w} B] [inst._@.Mathlib.Algebra.Algebra.Hom.1886971770._hygCtx._hyg.22 : Algebra.{u, v} R A inst._@.Mathlib.Algebra.Algebra.Hom.1886971770._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Algebra.Hom.1886971770._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Algebra.Hom.1886971770._hygCtx._hyg.26 : Algebra.{u, w} R B inst._@.Mathlib.Algebra.Algebra.Hom.1886971770._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Algebra.Hom.1886971770._hygCtx._hyg.13] (f : AlgHom.{u, v, w} R A B inst._@.Mathlib.Algebra.Algebra.Hom.1886971770._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Algebra.Hom.1886971770._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Algebra.Hom.1886971770._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Algebra.Hom.1886971770._hygCtx._hyg.22 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(Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (CommSemiring.toSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semifield.toCommSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Field.toSemifield.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (IsLocalRing.ResidueField.field.{u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Field.toSemifield.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (IsLocalRing.ResidueField.field.{u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26))) (Ideal.ResidueField._proof_1.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26))))))) (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semiring.toNonAssocSemiring.{u_2} (Ideal.ResidueField.{u_2} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.18) (Ideal.fiberIsoOfBijectiveResidueField._proof_5.{u_1, u_2} R R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 p q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.30))","typeFull":"∀ {R : Type u_1} {R' : Type u_2} {S : Type u_3} [inst : CommRing R] [inst_1 : CommRing R'] [inst_2 : CommRing S]\n  [inst_3 : Algebra R R'] [inst_4 : Algebra R S] {p : Ideal R} {q : Ideal R'} [inst_5 : p.IsPrime] [inst_6 : q.IsPrime]\n  [inst_7 : q.LiesOver p], IsScalarTower p.ResidueField q.ResidueField (TensorProduct R q.ResidueField S)","typeReadable":"∀ {R : Type u_1} {R' : Type u_2} {S : Type u_3} [inst : CommRing R] [inst_1 : CommRing R'] [inst_2 : CommRing S]\n  [inst_3 : Algebra R R'] [inst_4 : Algebra R S] {p : Ideal R} {q : Ideal R'} [inst_5 : p.IsPrime] [inst_6 : q.IsPrime]\n  [inst_7 : q.LiesOver p], IsScalarTower p.ResidueField q.ResidueField (TensorProduct R q.ResidueField S)","typeReferences":[["Semifield","toCommSemiring"],["instIsLocalHomAtPrimeRingHomAlgebraMap"],["IsLocalRing","ResidueField","algebra"],["IsLocalRing","ResidueField","instAlgebra"],["IsScalarTower"],["Ideal","IsPrime"],["OreLocalization","oreSetComm"],["Localization","AtPrime","instAlgebraOfLiesOver"],["Ideal","ResidueField"],["Algebra","id"],["Ideal","LiesOver"],["Semiring","toNonAssocSemiring"],["OreLocalization","instCommRing"],["Ideal"],["OreLocalization","instAlgebra"],["Algebra","toModule"],["MonoidWithZero","toMonoid"],["Algebra","toSMul"],["Ideal","primeCompl"],["Semifield","toDivisionSemiring"],["Localization","AtPrime"],["TensorProduct"],["CommRing","toCommSemiring"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Ideal","fiberIsoOfBijectiveResidueField","_proof_7"],["Semiring","toMonoidWithZero"],["TensorProduct","leftHasSMul"],["DivisionSemiring","toSemiring"],["CommRing"],["CommSemiring","toCommMonoid"],["Algebra"],["Module","toDistribMulAction"],["Ideal","ResidueField","_proof_1"],["Field","toSemifield"],["IsLocalRing","ResidueField","field"],["Ideal","fiberIsoOfBijectiveResidueField","_proof_5"]],"valueReferences":[["Semifield","toCommSemiring"],["instIsLocalHomAtPrimeRingHomAlgebraMap"],["IsLocalRing","ResidueField","algebra"],["IsLocalRing","ResidueField","instAlgebra"],["OreLocalization","oreSetComm"],["Localization","AtPrime","instIsScalarTower"],["Localization","AtPrime","instAlgebraOfLiesOver"],["CommRing","toNonUnitalCommRing"],["AddCommMonoid","toAddMonoid"],["Ideal","ResidueField"],["Algebra","id"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["OreLocalization","instCommRing"],["Semiring","toNonAssocSemiring"],["OreLocalization","instAlgebra"],["TensorProduct","isScalarTower_left"],["Algebra","toModule"],["Algebra","toSMul"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Ideal","primeCompl"],["IsScalarTower","right"],["Semifield","toDivisionSemiring"],["Localization","AtPrime"],["CommRing","toCommSemiring"],["IsScalarTower","left"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["IsLocalRing","ResidueField"],["IsScalarTower","to_smulCommClass"],["Semiring","toMonoidWithZero"],["DivisionSemiring","toSemiring"],["CommSemiring","toCommMonoid"],["IsLocalRing","ResidueField","instIsScalarTower"],["Ideal","ResidueField","_proof_1"],["Module","toDistribMulAction"],["Field","toSemifield"],["IsLocalRing","ResidueField","field"]]},{"isProp":true,"kind":"theorem","name":["Localization","exists_finite_awayMapₐ_of_surjective_awayMapₐ"],"typeFallback":"forall {R : Type.{u}} {S : Type.{v}} {T : Type.{u_1}} [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.185859606._hygCtx._hyg.5 : CommRing.{u} R] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.185859606._hygCtx._hyg.8 : CommRing.{v} S] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.185859606._hygCtx._hyg.11 : CommRing.{u_1} T] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.185859606._hygCtx._hyg.14 : Algebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.185859606._hygCtx._hyg.5) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.185859606._hygCtx._hyg.8))] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.185859606._hygCtx._hyg.18 : Algebra.{u, u_1} R T (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.185859606._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_1} T (CommRing.toCommSemiring.{u_1} T inst._@.Mathlib.RingTheory.Etale.QuasiFinite.185859606._hygCtx._hyg.11))] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.185859606._hygCtx._hyg.22 : Algebra.FiniteType.{u, u_1} R T (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.185859606._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_1} T (CommRing.toCommSemiring.{u_1} T inst._@.Mathlib.RingTheory.Etale.QuasiFinite.185859606._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.185859606._hygCtx._hyg.18] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.185859606._hygCtx._hyg.26 : Algebra.IsIntegral.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.185859606._hygCtx._hyg.5 (CommRing.toRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.185859606._hygCtx._hyg.8) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.185859606._hygCtx._hyg.14] (f : AlgHom.{u, v, u_1} R S T (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.185859606._hygCtx._hyg.5) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14) (Algebra.toSMul.{u_2, u_2} R' (IsLocalRing.ResidueField.{u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) (DivisionSemiring.toSemiring.{u_2} (IsLocalRing.ResidueField.{u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) (Semifield.toDivisionSemiring.{u_2} (IsLocalRing.ResidueField.{u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) (Field.toSemifield.{u_2} (IsLocalRing.ResidueField.{u_2} (Localization.AtPrime.{u_2} 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) (IsLocalRing.ResidueField.field.{u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))))) (IsLocalRing.ResidueField.algebra.{u_2, u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (OreLocalization.instAlgebra.{u_2, u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) (Algebra.id.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))) (Algebra.toSMul.{u_1, u_2} R (IsLocalRing.ResidueField.{u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (DivisionSemiring.toSemiring.{u_2} (IsLocalRing.ResidueField.{u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) (Field.toSemifield.{u_2} (IsLocalRing.ResidueField.{u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) (IsLocalRing.ResidueField.field.{u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))))))))) (Module.toDistribMulAction.{u_1, u_2} R (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R 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(CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)))))))) (Algebra.toModule.{u_1, u_2} R (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} (Ideal.ResidueField.{u_2} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 (OreLocalization.instAlgebra.{u_2, u_1} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14))))))) (SMulZeroClass.toSMul.{u_2, u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (AddZero.toZero.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (AddZeroClass.toAddZero.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (AddMonoid.toAddZeroClass.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (AddCommMonoid.toAddMonoid.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semiring.toNonAssocSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (CommSemiring.toSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semifield.toCommSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Field.toSemifield.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (IsLocalRing.ResidueField.field.{u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))))))))))","typeFull":"∀ {R : Type u_1} {R' : Type u_2} [inst : CommRing R] [inst_1 : CommRing R'] [inst_2 : Algebra R R'] {q : Ideal R'}\n  [inst_3 : q.IsPrime], SMulCommClass R q.ResidueField q.ResidueField","typeReadable":"∀ {R : Type u_1} {R' : Type u_2} [inst : CommRing R] [inst_1 : CommRing R'] [inst_2 : Algebra R R'] {q : Ideal R'}\n  [inst_3 : q.IsPrime], SMulCommClass R q.ResidueField q.ResidueField","typeReferences":[["Semifield","toCommSemiring"],["IsLocalRing","ResidueField","algebra"],["Ideal","IsPrime"],["OreLocalization","oreSetComm"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Ideal","ResidueField"],["Algebra","id"],["Semiring","toNonAssocSemiring"],["OreLocalization","instCommRing"],["OreLocalization","instAlgebra"],["Ideal"],["Algebra","toModule"],["MonoidWithZero","toMonoid"],["Ideal","primeCompl"],["DistribSMul","toSMulZeroClass"],["Localization","AtPrime"],["CommRing","toCommSemiring"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["CommRing"],["CommSemiring","toCommMonoid"],["Algebra"],["SMulCommClass"],["Module","toDistribMulAction"],["Ideal","ResidueField","_proof_1"],["Field","toSemifield"],["AddZero","toZero"],["IsLocalRing","ResidueField","field"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Semifield","toCommSemiring"],["IsLocalRing","ResidueField","algebra"],["CommRing","toCommSemiring"],["OreLocalization","oreSetComm"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["IsLocalRing","ResidueField"],["IsScalarTower","to_smulCommClass"],["DivisionSemiring","toSemiring"],["Ideal","ResidueField"],["Algebra","id"],["CommSemiring","toCommMonoid"],["Ideal","ResidueField","_proof_1"],["Semiring","toNonAssocSemiring"],["OreLocalization","instCommRing"],["OreLocalization","instAlgebra"],["Algebra","toModule"],["Field","toSemifield"],["IsScalarTower","right"],["Ideal","primeCompl"],["Semifield","toDivisionSemiring"],["IsLocalRing","ResidueField","field"],["Localization","AtPrime"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","RingTheory","Etale","QuasiFinite",0,"Algebra","exists_notMem_and_isIntegral_forall_mem_of_ne_of_liesOver","_proof_1_7"],"typeFallback":"forall 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{R : Type.{u_1}} [inst._@.Mathlib.RingTheory.Localization.Defs.3823209908._hygCtx._hyg.3 : CommSemiring.{u_1} R] {M : Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.RingTheory.Localization.Defs.3823209908._hygCtx._hyg.3))))} {S : Type.{u_2}} [inst._@.Mathlib.RingTheory.Localization.Defs.3823209908._hygCtx._hyg.9 : CommSemiring.{u_2} S] [inst._@.Mathlib.RingTheory.Localization.Defs.3823209908._hygCtx._hyg.12 : Algebra.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Localization.Defs.3823209908._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} S inst._@.Mathlib.RingTheory.Localization.Defs.3823209908._hygCtx._hyg.9)] [inst._@.Mathlib.RingTheory.Localization.Defs.3823209908._hygCtx._hyg.20 : IsLocalization.{u_1, u_2} R inst._@.Mathlib.RingTheory.Localization.Defs.3823209908._hygCtx._hyg.3 M S 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(Submonoid.instSetLike.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.RingTheory.Localization.Defs.3823209908._hygCtx._hyg.3)))))) M x) y₁) x₂)))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {M : Submonoid R} {S : Type u_2} [inst_1 : CommSemiring S]\n  [inst_2 : Algebra R S] [inst_3 : IsLocalization M S] {x₁ x₂ : R} {y₁ y₂ : ↥M},\n  (IsLocalization.mk' S x₁ y₁ = IsLocalization.mk' S x₂ y₂) =\n    ((algebraMap R S) (↑y₂ * x₁) = (algebraMap R S) (↑y₁ * x₂))","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {M : Submonoid R} {S : Type u_2} [inst_1 : CommSemiring S]\n  [inst_2 : Algebra R S] [inst_3 : IsLocalization M S] {x₁ x₂ : R} {y₁ y₂ : ↥M},\n  (IsLocalization.mk' S x₁ y₁ = IsLocalization.mk' S x₂ y₂) =\n    ((algebraMap R S) (↑y₂ * x₁) = (algebraMap R S) (↑y₁ * x₂))","typeReferences":[["RingHom"],["Subtype"],["SetLike","instMembership"],["NonUnitalNonAssocSemiring","toDistrib"],["Submonoid","instSetLike"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Distrib","toMul"],["CommSemiring"],["CommSemiring","toSemiring"],["Membership","mem"],["IsLocalization","mk'"],["HMul","hMul"],["RingHom","instFunLike"],["MulZeroOneClass","toMulOneClass"],["Subtype","val"],["DFunLike","coe"],["Algebra"],["IsLocalization"],["Submonoid"],["Semiring","toNonAssocSemiring"],["instHMul"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["algebraMap"]],"valueReferences":[["RingHom"],["SetLike","instMembership"],["NonUnitalNonAssocSemiring","toDistrib"],["Submonoid","instSetLike"],["IsLocalization","mk'_eq_iff_eq"],["Membership","mem"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["CommSemiring","toSemiring"],["Distrib","toMul"],["IsLocalization","mk'"],["RingHom","instFunLike"],["HMul","hMul"],["MulZeroOneClass","toMulOneClass"],["Subtype","val"],["DFunLike","coe"],["Submonoid"],["Semiring","toNonAssocSemiring"],["instHMul"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["propext"],["algebraMap"]]},{"isProp":true,"kind":"theorem","name":["Ideal","fiberIsoOfBijectiveResidueField","_proof_6"],"typeFallback":"forall {R' : Type.{u_1}} [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 : CommRing.{u_1} R'] {q : Ideal.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))} [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28 : Ideal.IsPrime.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q], SMulCommClass.{u_1, u_1, u_1} R' (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (SMulZeroClass.toSMul.{u_1, u_1} R' (Ideal.ResidueField.{u_1} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semiring.toNonAssocSemiring.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (CommSemiring.toSemiring.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q 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(CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)))) (AddCommMonoid.toAddMonoid.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semiring.toNonAssocSemiring.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (CommSemiring.toSemiring.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semifield.toCommSemiring.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Field.toSemifield.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (IsLocalRing.ResidueField.field.{u_1} (Localization.AtPrime.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_1} R' (CommSemiring.toCommMonoid.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))))))))) (Module.toDistribMulAction.{u_1, u_1} R' (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semiring.toNonAssocSemiring.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (CommSemiring.toSemiring.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semifield.toCommSemiring.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Field.toSemifield.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (IsLocalRing.ResidueField.field.{u_1} (Localization.AtPrime.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_1} R' (CommSemiring.toCommMonoid.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)))))))) (Algebra.toModule.{u_1, u_1} R' (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (CommRing.toCommSemiring.{u_1} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_1} R' (CommSemiring.toCommMonoid.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))))) (IsLocalRing.ResidueField.algebra.{u_1, u_1} (Localization.AtPrime.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_1} R' (CommSemiring.toCommMonoid.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (OreLocalization.instAlgebra.{u_1, u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_1} R' (CommSemiring.toCommMonoid.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) (Algebra.id.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8))))))))) (SMulZeroClass.toSMul.{u_1, u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (AddZero.toZero.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (AddZeroClass.toAddZero.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (AddMonoid.toAddZeroClass.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (AddCommMonoid.toAddMonoid.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semiring.toNonAssocSemiring.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (CommSemiring.toSemiring.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semifield.toCommSemiring.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Field.toSemifield.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (IsLocalRing.ResidueField.field.{u_1} (Localization.AtPrime.{u_1} R' (CommRing.toCommSemiring.{u_1} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)))))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (AddMonoid.toAddZeroClass.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (AddCommMonoid.toAddMonoid.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} 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(Ideal.ResidueField._proof_1.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))))))) (AddCommMonoid.toAddMonoid.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semiring.toNonAssocSemiring.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semiring.toNonAssocSemiring.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (CommSemiring.toSemiring.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semifield.toCommSemiring.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Field.toSemifield.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (IsLocalRing.ResidueField.field.{u_1} (Localization.AtPrime.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_1} R' (CommSemiring.toCommMonoid.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)))))))) (Algebra.toModule.{u_1, u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semifield.toCommSemiring.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Field.toSemifield.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (IsLocalRing.ResidueField.field.{u_1} (Localization.AtPrime.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_1} R' (CommSemiring.toCommMonoid.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)))) (CommSemiring.toSemiring.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semifield.toCommSemiring.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Field.toSemifield.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (IsLocalRing.ResidueField.field.{u_1} (Localization.AtPrime.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_1} R' (CommSemiring.toCommMonoid.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))))) (Algebra.id.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Semifield.toCommSemiring.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (Field.toSemifield.{u_1} (Ideal.ResidueField.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (IsLocalRing.ResidueField.field.{u_1} (Localization.AtPrime.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_1} R' (CommSemiring.toCommMonoid.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_1} R' (CommSemiring.toSemiring.{u_1} R' (CommRing.toCommSemiring.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_1} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))))))))))","typeFull":"∀ {R' : Type u_1} [inst : CommRing R'] {q : Ideal R'} [inst_1 : q.IsPrime],\n  SMulCommClass R' q.ResidueField q.ResidueField","typeReadable":"∀ {R' : Type u_1} [inst : CommRing R'] {q : Ideal R'} [inst_1 : q.IsPrime],\n  SMulCommClass R' q.ResidueField q.ResidueField","typeReferences":[["Semifield","toCommSemiring"],["IsLocalRing","ResidueField","algebra"],["Ideal","IsPrime"],["OreLocalization","oreSetComm"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Ideal","ResidueField"],["Algebra","id"],["Semiring","toNonAssocSemiring"],["OreLocalization","instCommRing"],["OreLocalization","instAlgebra"],["Ideal"],["Algebra","toModule"],["MonoidWithZero","toMonoid"],["Ideal","primeCompl"],["Localization","AtPrime"],["DistribSMul","toSMulZeroClass"],["CommRing","toCommSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["CommRing"],["CommSemiring","toCommMonoid"],["SMulCommClass"],["Module","toDistribMulAction"],["Ideal","ResidueField","_proof_1"],["Field","toSemifield"],["AddZero","toZero"],["IsLocalRing","ResidueField","field"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Semifield","toCommSemiring"],["IsLocalRing","ResidueField","algebra"],["CommRing","toCommSemiring"],["OreLocalization","oreSetComm"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["IsLocalRing","ResidueField"],["IsScalarTower","to_smulCommClass"],["DivisionSemiring","toSemiring"],["Ideal","ResidueField"],["Algebra","id"],["CommSemiring","toCommMonoid"],["Ideal","ResidueField","_proof_1"],["Semiring","toNonAssocSemiring"],["OreLocalization","instCommRing"],["OreLocalization","instAlgebra"],["Algebra","toModule"],["Field","toSemifield"],["IsScalarTower","right"],["Ideal","primeCompl"],["Semifield","toDivisionSemiring"],["IsLocalRing","ResidueField","field"],["Localization","AtPrime"]]},{"isProp":true,"kind":"theorem","name":["Algebra","exists_etale_isIdempotentElem_forall_liesOver_eq_aux"],"typeFallback":"forall {R : Type.{u}} {S : Type.{v}} [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 : CommRing.{u} R] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 : CommRing.{v} S] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 : Algebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.34 : Algebra.FiniteType.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30] (p : Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24))) [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.40 : Ideal.IsPrime.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24)) p] (q : Ideal.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))) [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.44 : Ideal.IsPrime.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) q] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.46 : Ideal.LiesOver.{u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 q p] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.49 : Algebra.QuasiFiniteAt.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.44], Exists.{succ (succ u)} Type.{u} (fun (R' : Type.{u}) => Exists.{succ u} (CommRing.{u} R') (fun (x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66 : CommRing.{u} R') => Exists.{succ u} (Algebra.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' 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(NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (fun (e : TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) => Exists.{0} (IsIdempotentElem.{max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Algebra.TensorProduct.instMul.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.to_smulCommClass.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (IsScalarTower.right.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)))) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (Algebra.to_smulCommClass.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (IsScalarTower.right.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) e) (fun 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Algebra.TensorProduct.instMul.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.to_smulCommClass.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (IsScalarTower.right.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)))) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (Algebra.to_smulCommClass.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (IsScalarTower.right.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) e) => Exists.{max (succ u) (succ v)} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S 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(CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Subalgebra.instModuleSubtypeMem.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))) (fun (e₀ : TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S 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(SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) 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(CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Subalgebra.instModuleSubtypeMem.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))) => Exists.{0} (IsIdempotentElem.{max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} 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(CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toNonUnitalCommRing.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)))))) (Subalgebra.instModuleSubtypeMem.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Algebra.to_smulCommClass.{u, v} R (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (Subalgebra.toSemiring.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Subalgebra.algebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))) (IsScalarTower.right.{u, v} R (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (Subalgebra.toSemiring.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Subalgebra.algebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)))) e₀) (fun (x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.146 : IsIdempotentElem.{max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Subalgebra.toSemiring.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (Subalgebra.toSemiring.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Subalgebra.algebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S 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(SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R' (Semiring.toNonAssocSemiring.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Semiring.toNonAssocSemiring.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Subalgebra.toSemiring.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (Subalgebra.toSemiring.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Subalgebra.algebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S 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(integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (Subalgebra.toSemiring.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Subalgebra.algebra.{u, v} R S (CommRing.toCommSemiring.{u} R 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x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) (Algebra.id.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (IsScalarTower.right.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74))) (Algebra.TensorProduct.leftAlgebra.{u, u, u, v} R R' R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66) (Algebra.id.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) (IsScalarTower.to_smulCommClass.{u, u, u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 R' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R' (Semiring.toNonAssocSemiring.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (Algebra.toModule.{u, u} R' R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) (Algebra.id.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (IsScalarTower.right.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74)))) (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R' (Semiring.toNonAssocSemiring.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R 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(integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Semiring.toNonAssocSemiring.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Subalgebra.toSemiring.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (Subalgebra.toSemiring.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Subalgebra.algebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)))) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Semiring.toNonAssocSemiring.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Subalgebra.toSemiring.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (Subalgebra.toSemiring.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Subalgebra.algebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S 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(Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) 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(CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (Algebra.toModule.{u, u} R' R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) (Algebra.id.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (IsScalarTower.right.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74)))) (Algebra.TensorProduct.map.{u, u, u, v, u, v} R R' R' (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (Algebra.id.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) (IsScalarTower.right.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Subalgebra.toSemiring.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Subalgebra.algebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (Algebra.id.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) (IsScalarTower.right.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (AlgHom.id.{u, u} R' R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) (Algebra.id.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))) (Subalgebra.val.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))) e₀) e) => Exists.{max (succ u) (succ v)} (Ideal.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (fun (P' : Ideal.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) => Exists.{0} (Ideal.IsPrime.{max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) P') (fun (x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.192 : Ideal.IsPrime.{max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) P') => Exists.{0} (Ideal.LiesOver.{u, max u v} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66) (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (Algebra.TensorProduct.leftAlgebra.{u, u, u, v} R R' R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66) (Algebra.id.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) (Algebra.to_smulCommClass.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74)) P' P) (fun (x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.199 : Ideal.LiesOver.{u, max u v} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66) (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (Algebra.TensorProduct.leftAlgebra.{u, u, u, v} R R' R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66) (Algebra.id.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) (Algebra.to_smulCommClass.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74)) P' P) => And (Eq.{succ v} (Ideal.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))) (Ideal.comap.{v, max u v, max u v} S (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (RingHom.{v, max u v} S (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R' (Semiring.toNonAssocSemiring.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{v} S (Semiring.toNonAssocSemiring.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Semiring.toNonAssocSemiring.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))) (Semiring.toNonAssocSemiring.{max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R' (Semiring.toNonAssocSemiring.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{v} S (Semiring.toNonAssocSemiring.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (RingHom.instFunLike.{v, max u v} S (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R' (Semiring.toNonAssocSemiring.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{v} S (Semiring.toNonAssocSemiring.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Semiring.toNonAssocSemiring.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))) (Semiring.toNonAssocSemiring.{max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R' (Semiring.toNonAssocSemiring.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{v} S (Semiring.toNonAssocSemiring.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))) (AlgHom.toRingHom.{u, v, max u v} R S (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R' (Semiring.toNonAssocSemiring.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{v} S (Semiring.toNonAssocSemiring.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (Algebra.TensorProduct.instAlgebra.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (Algebra.TensorProduct.includeRight.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (RingHom.instRingHomClass.{v, max u v} S (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R' (Semiring.toNonAssocSemiring.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{v} S (Semiring.toNonAssocSemiring.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Semiring.toNonAssocSemiring.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))) (Semiring.toNonAssocSemiring.{max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R' (Semiring.toNonAssocSemiring.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{v} S (Semiring.toNonAssocSemiring.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))) P') q) (And (Not (Membership.mem.{max u v, max u v} (TensorProduct.{u, u, v} R 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x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Ideal.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (SetLike.instMembership.{max u v, max u v} (Ideal.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' 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(CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' 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(NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (NonUnitalNonAssocSemiring.toAddCommMonoid.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Semiring.toNonAssocSemiring.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)))) (Semiring.toModule.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)))) P' e)) (And (Function.Bijective.{succ u, succ u} (Ideal.ResidueField.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.40) (Ideal.ResidueField.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99) (DFunLike.coe.{succ u, succ u, succ u} (AlgHom.{u, u, u} R (Ideal.ResidueField.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.40) (Ideal.ResidueField.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99) (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (DivisionSemiring.toSemiring.{u} (Ideal.ResidueField.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.40) (Semifield.toDivisionSemiring.{u} (Ideal.ResidueField.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.40) (Field.toSemifield.{u} (Ideal.ResidueField.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.40) (IsLocalRing.ResidueField.field.{u} (Localization.AtPrime.{u} R (CommRing.toCommSemiring.{u} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.40))))) (DivisionSemiring.toSemiring.{u} (Ideal.ResidueField.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99) (Semifield.toDivisionSemiring.{u} (Ideal.ResidueField.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99) (Field.toSemifield.{u} (Ideal.ResidueField.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99) (IsLocalRing.ResidueField.field.{u} (Localization.AtPrime.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99) (OreLocalization.instCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66 (Ideal.primeCompl.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99) (OreLocalization.oreSetComm.{u} R' (CommSemiring.toCommMonoid.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) (Ideal.primeCompl.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99))) (Ideal.ResidueField._proof_1.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99))))) (IsLocalRing.ResidueField.algebra.{u, u} (Localization.AtPrime.{u} R (CommRing.toCommSemiring.{u} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.40) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 (OreLocalization.instAlgebra.{u, u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24)) (Ideal.primeCompl.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.40) (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24)) (Ideal.primeCompl.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.40)) R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (Algebra.id.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24)))) (IsLocalRing.ResidueField.algebra.{u, u} (Localization.AtPrime.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99) (OreLocalization.instCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66 (Ideal.primeCompl.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99) (OreLocalization.oreSetComm.{u} R' (CommSemiring.toCommMonoid.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) (Ideal.primeCompl.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99))) (Ideal.ResidueField._proof_1.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 (OreLocalization.instAlgebra.{u, u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) (Ideal.primeCompl.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99) (OreLocalization.oreSetComm.{u} R' (CommSemiring.toCommMonoid.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) (Ideal.primeCompl.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99)) R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74))) (Ideal.ResidueField.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.40) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Ideal.ResidueField.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.40) => Ideal.ResidueField.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99) (AlgHom.funLike.{u, u, u} R (Ideal.ResidueField.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.40) (Ideal.ResidueField.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99) (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (DivisionSemiring.toSemiring.{u} (Ideal.ResidueField.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.40) (Semifield.toDivisionSemiring.{u} (Ideal.ResidueField.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.40) (Field.toSemifield.{u} (Ideal.ResidueField.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.40) (IsLocalRing.ResidueField.field.{u} (Localization.AtPrime.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.40) (OreLocalization.instCommRing.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 (Ideal.primeCompl.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.40) (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24)) (Ideal.primeCompl.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.40))) (Ideal.ResidueField._proof_1.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.40))))) (DivisionSemiring.toSemiring.{u} (Ideal.ResidueField.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99) (Semifield.toDivisionSemiring.{u} (Ideal.ResidueField.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99) (Field.toSemifield.{u} (Ideal.ResidueField.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99) (IsLocalRing.ResidueField.field.{u} (Localization.AtPrime.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99) (OreLocalization.instCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66 (Ideal.primeCompl.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99) (OreLocalization.oreSetComm.{u} R' (CommSemiring.toCommMonoid.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) (Ideal.primeCompl.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99))) (Ideal.ResidueField._proof_1.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99))))) (IsLocalRing.ResidueField.algebra.{u, u} (Localization.AtPrime.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.40) (OreLocalization.instCommRing.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 (Ideal.primeCompl.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.40) (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24)) (Ideal.primeCompl.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.40))) (Ideal.ResidueField._proof_1.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.40) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 (OreLocalization.instAlgebra.{u, u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24)) (Ideal.primeCompl.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.40) (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24)) (Ideal.primeCompl.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.40)) R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (Algebra.id.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24)))) (IsLocalRing.ResidueField.algebra.{u, u} (Localization.AtPrime.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99) (OreLocalization.instCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66 (Ideal.primeCompl.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99) (OreLocalization.oreSetComm.{u} R' (CommSemiring.toCommMonoid.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) (Ideal.primeCompl.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99))) (Ideal.ResidueField._proof_1.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 (OreLocalization.instAlgebra.{u, u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) (Ideal.primeCompl.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99) (OreLocalization.oreSetComm.{u} R' (CommSemiring.toCommMonoid.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) (Ideal.primeCompl.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99)) R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74))) (Ideal.ResidueField.mapₐ.{u, u, u} R R R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66 (Algebra.id.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.40 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.99 (Algebra.ofId.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Ideal.over_def.{u, u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 P p x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.106)))) (And (forall (P'' : Ideal.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toNonUnitalCommRing.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Subalgebra.instModuleSubtypeMem.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (Subalgebra.toSemiring.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Subalgebra.algebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))) P'') -> (Ideal.LiesOver.{u, max u v} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66) (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toNonUnitalCommRing.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Subalgebra.instModuleSubtypeMem.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (Subalgebra.toSemiring.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Subalgebra.algebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))) (Algebra.TensorProduct.leftAlgebra.{u, u, u, v} R R' R' (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (Subalgebra.toSemiring.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Subalgebra.algebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66) (Algebra.id.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) (Algebra.to_smulCommClass.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74)) P'' P) -> (Not (Membership.mem.{max u v, max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toNonUnitalCommRing.{v} 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(Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toNonUnitalCommRing.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Subalgebra.instModuleSubtypeMem.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))) (Submodule.setLike.{max u v, max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toNonUnitalCommRing.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))) (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Subalgebra.instModuleSubtypeMem.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))) (Semiring.toNonAssocSemiring.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) 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(Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (Subalgebra.toSemiring.{u, v} 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)))))) P'' e₀)) -> (Eq.{max (succ u) (succ v)} (Ideal.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S 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(CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Subalgebra.toCommRing.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Subalgebra.instModuleSubtypeMem.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R 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(integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (Subalgebra.toSemiring.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Subalgebra.algebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)))) P'' (Ideal.comap.{max u v, max u v, max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R' (Semiring.toNonAssocSemiring.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Subalgebra.toSemiring.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R 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(CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{v} (Subtype.{succ v} S (fun (x : S) => 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Semiring.toNonAssocSemiring.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S 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(CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (Subalgebra.toSemiring.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Subalgebra.algebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (Subalgebra.toSemiring.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (Subalgebra.toSemiring.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S 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(CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R' (Semiring.toNonAssocSemiring.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{v} S (Semiring.toNonAssocSemiring.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) 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(Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{v} (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (Subalgebra.toSemiring.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Subalgebra.algebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)))) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' (Subtype.{succ v} S (fun (x : S) => Membership.mem.{v, v} S (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (SetLike.instMembership.{v, v} (Subalgebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) S (Subalgebra.instSetLike.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) x)) (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (Subalgebra.toSemiring.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Subalgebra.algebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (integralClosure.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)))) (Semiring.toNonAssocSemiring.{max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R' (Semiring.toNonAssocSemiring.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{v} S (Semiring.toNonAssocSemiring.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30))) P'))) (forall (P'' : Ideal.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)), (Ideal.IsPrime.{max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) P'') -> (Ideal.LiesOver.{u, max u v} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66) (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30) (Algebra.TensorProduct.leftAlgebra.{u, u, u, v} R R' R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30 (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66) (Algebra.id.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) (Algebra.to_smulCommClass.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74)) P'' P) -> (Not (Membership.mem.{max u v, max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.30)) (Ideal.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.269928434._hygCtx._hyg.66))))) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.30) (instIsLocalHomAtPrimeRingHomAlgebraMap.{u_1, u_2} R R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.26 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.30))) (Algebra.toSMul.{u_1, u_2} R (IsLocalRing.ResidueField.{u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) (Field.toSemifield.{u_2} (IsLocalRing.ResidueField.{u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))))) (IsLocalRing.ResidueField.algebra.{u_2, u_1} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5 (OreLocalization.instAlgebra.{u_2, u_1} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.28)) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.5) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.271995787._hygCtx._hyg.14)))","typeFull":"∀ {R : Type u_1} {R' : Type u_2} [inst : CommRing R] [inst_1 : CommRing R'] [inst_2 : Algebra R R'] {p : Ideal R}\n  {q : Ideal R'} [inst_3 : p.IsPrime] [inst_4 : q.IsPrime] [inst_5 : q.LiesOver p],\n  IsScalarTower R (IsLocalRing.ResidueField (Localization.AtPrime p))\n    (IsLocalRing.ResidueField (Localization.AtPrime q))","typeReadable":"∀ {R : Type u_1} {R' : Type u_2} [inst : CommRing R] [inst_1 : CommRing R'] [inst_2 : Algebra R R'] {p : Ideal R}\n  {q : Ideal R'} [inst_3 : p.IsPrime] [inst_4 : q.IsPrime] [inst_5 : q.LiesOver p],\n  IsScalarTower R (IsLocalRing.ResidueField (Localization.AtPrime p))\n    (IsLocalRing.ResidueField (Localization.AtPrime q))","typeReferences":[["Semifield","toCommSemiring"],["instIsLocalHomAtPrimeRingHomAlgebraMap"],["IsLocalRing","ResidueField","algebra"],["IsLocalRing","ResidueField","instAlgebra"],["OreLocalization","oreSetComm"],["IsScalarTower"],["Ideal","IsPrime"],["Localization","AtPrime","instAlgebraOfLiesOver"],["Algebra","id"],["Ideal","LiesOver"],["OreLocalization","instCommRing"],["Ideal"],["OreLocalization","instAlgebra"],["Algebra","toSMul"],["Ideal","primeCompl"],["Semifield","toDivisionSemiring"],["Localization","AtPrime"],["CommRing","toCommSemiring"],["CommSemiring","toSemiring"],["IsLocalRing","ResidueField"],["DivisionSemiring","toSemiring"],["CommRing"],["CommSemiring","toCommMonoid"],["Algebra"],["Ideal","ResidueField","_proof_1"],["Field","toSemifield"],["IsLocalRing","ResidueField","field"]],"valueReferences":[["instIsLocalHomAtPrimeRingHomAlgebraMap"],["OreLocalization","oreSetComm"],["Localization","AtPrime","instIsScalarTower"],["Localization","AtPrime","instAlgebraOfLiesOver"],["AddCommMonoid","toAddMonoid"],["CommRing","toNonUnitalCommRing"],["Algebra","id"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["OreLocalization","instCommRing"],["OreLocalization","instAlgebra"],["Algebra","toModule"],["MonoidWithZero","toMonoid"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["DistribMulAction","toMulAction"],["Ideal","primeCompl"],["Localization","AtPrime"],["CommRing","toCommSemiring"],["IsScalarTower","left"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Semiring","toMonoidWithZero"],["CommSemiring","toCommMonoid"],["IsLocalRing","ResidueField","instIsScalarTower"],["Ideal","ResidueField","_proof_1"],["Module","toDistribMulAction"]]},{"isProp":true,"kind":"theorem","name":["Algebra","exists_etale_isIdempotentElem_forall_liesOver_eq_aux₂"],"typeFallback":"forall {R : Type.{u_2}} {S : Type.{u_3}} {R' : Type.{u_4}} {R'' : Type.{u_5}} [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26 : CommRing.{u_2} R] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29 : CommRing.{u_3} S] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32 : Algebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.36 : Algebra.FiniteType.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40 : CommRing.{u_4} R'] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 : Algebra.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47 : CommRing.{u_5} R''] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50 : Algebra.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47))] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.54 : Algebra.{u_5, u_3} R'' S (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.58 : Algebra.IsIntegral.{u_2, u_5} R R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26 (CommRing.toRing.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.62 : IsScalarTower.{u_2, u_5, u_3} R R'' S (Algebra.toSMul.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50) (Algebra.toSMul.{u_5, u_3} R'' S (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.54) (Algebra.toSMul.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)] (q : Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))) (P : Ideal.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))) [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.71 : Ideal.IsPrime.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) P] (e : TensorProduct.{u_2, u_4, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_4} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_4} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_4} R' (CommRing.toNonUnitalCommRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) (e₀ : TensorProduct.{u_2, u_4, u_5} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' R'' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_4} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_4} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_4} R' (CommRing.toNonUnitalCommRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} R'' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_5} R'' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_5} R'' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_5} R'' (CommRing.toNonUnitalCommRing.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)), (IsIdempotentElem.{max u_4 u_5} (TensorProduct.{u_2, u_4, u_5} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' R'' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_4} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_4} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_4} R' (CommRing.toNonUnitalCommRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} R'' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_5} R'' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_5} R'' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_5} R'' (CommRing.toNonUnitalCommRing.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) (Algebra.TensorProduct.instMul.{u_2, u_4, u_5} R R' R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_4} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_4} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_4} R' (CommRing.toNonUnitalCommRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.to_smulCommClass.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (IsScalarTower.right.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_5} R'' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_5} R'' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_5} R'' (CommRing.toNonUnitalCommRing.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)))) (Algebra.toModule.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50) (Algebra.to_smulCommClass.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50) (IsScalarTower.right.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R 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(CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) => TensorProduct.{u_2, u_4, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) (AlgHom.funLike.{u_4, max u_4 u_5, max u_3 u_4} R' (TensorProduct.{u_2, u_4, u_5} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' R'' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} R'' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_5} R'' (Semiring.toNonAssocSemiring.{u_5} R'' (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) (TensorProduct.{u_2, u_4, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Algebra.TensorProduct.instSemiring.{u_2, u_4, u_5} R R' R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50) (Algebra.TensorProduct.instSemiring.{u_2, u_4, u_3} R R' S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32) (Algebra.TensorProduct.leftAlgebra.{u_2, u_4, u_4, u_5} R R' R' R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50 (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Algebra.id.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (IsScalarTower.to_smulCommClass.{u_2, u_4, u_4} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 R' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (Algebra.toModule.{u_4, u_4} R' R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Algebra.id.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (IsScalarTower.right.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43))) (Algebra.TensorProduct.leftAlgebra.{u_2, u_4, u_4, u_3} R R' R' S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32 (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Algebra.id.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (IsScalarTower.to_smulCommClass.{u_2, u_4, u_4} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 R' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (Algebra.toModule.{u_4, u_4} R' R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Algebra.id.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (IsScalarTower.right.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43)))) (Algebra.TensorProduct.map.{u_2, u_4, u_4, u_5, u_4, u_3} R R' R' R'' R' S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (Algebra.id.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (IsScalarTower.right.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50 (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (Algebra.id.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (IsScalarTower.right.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32 (AlgHom.id.{u_4, u_4} R' R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Algebra.id.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))) (IsScalarTower.toAlgHom.{u_2, u_5, u_3} R R'' S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.54 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.62)) e₀) e) -> (forall (P' : Ideal.{max u_3 u_4} (TensorProduct.{u_2, u_4, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_4} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_4} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_4} R' (CommRing.toNonUnitalCommRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) (Algebra.TensorProduct.instSemiring.{u_2, u_4, u_3} R R' S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)), (Eq.{succ u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))) (Ideal.comap.{u_3, max u_3 u_4, max u_4 u_3} S (TensorProduct.{u_2, u_4, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_4} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_4} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_4} R' (CommRing.toNonUnitalCommRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) (RingHom.{u_3, max u_4 u_3} S (TensorProduct.{u_2, u_4, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))) (Semiring.toNonAssocSemiring.{max u_4 u_3} (TensorProduct.{u_2, u_4, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) (Algebra.TensorProduct.instSemiring.{u_2, u_4, u_3} R R' S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32))) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) (Algebra.TensorProduct.instSemiring.{u_2, u_4, u_3} R R' S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32) (RingHom.instFunLike.{u_3, max u_4 u_3} S (TensorProduct.{u_2, u_4, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))) (Semiring.toNonAssocSemiring.{max u_4 u_3} (TensorProduct.{u_2, u_4, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) (Algebra.TensorProduct.instSemiring.{u_2, u_4, u_3} R R' S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32))) (AlgHom.toRingHom.{u_2, u_3, max u_4 u_3} R S (TensorProduct.{u_2, u_4, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) (Algebra.TensorProduct.instSemiring.{u_2, u_4, u_3} R R' S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32 (Algebra.TensorProduct.instAlgebra.{u_2, u_4, u_3} R R' S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32) (Algebra.TensorProduct.includeRight.{u_2, u_4, u_3} R R' S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) (RingHom.instRingHomClass.{u_3, max u_3 u_4} S (TensorProduct.{u_2, u_4, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))) (Semiring.toNonAssocSemiring.{max u_4 u_3} (TensorProduct.{u_2, u_4, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) (Algebra.TensorProduct.instSemiring.{u_2, u_4, u_3} R R' S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32))) P') q) -> (forall (P'' : Ideal.{max u_5 u_4} (TensorProduct.{u_2, u_4, u_5} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' R'' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_4} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_4} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_4} R' (CommRing.toNonUnitalCommRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} R'' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_5} R'' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_5} R'' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_5} R'' (CommRing.toNonUnitalCommRing.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) (Algebra.TensorProduct.instSemiring.{u_2, u_4, u_5} R R' R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)), (Ideal.IsPrime.{max u_4 u_5} (TensorProduct.{u_2, u_4, u_5} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' R'' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_4} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_4} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_4} R' (CommRing.toNonUnitalCommRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} R'' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_5} R'' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_5} R'' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_5} R'' (CommRing.toNonUnitalCommRing.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) (Algebra.TensorProduct.instSemiring.{u_2, u_4, u_5} R R' R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50) P'') -> (Ideal.LiesOver.{u_4, max u_4 u_5} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (TensorProduct.{u_2, u_4, u_5} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' R'' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_4} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_4} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_4} R' (CommRing.toNonUnitalCommRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} R'' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_5} R'' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_5} R'' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_5} R'' (CommRing.toNonUnitalCommRing.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) (Algebra.TensorProduct.instSemiring.{u_2, u_4, u_5} R R' R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50) (Algebra.TensorProduct.leftAlgebra.{u_2, u_4, u_4, u_5} R R' R' R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50 (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Algebra.id.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Algebra.to_smulCommClass.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43)) P'' P) -> (Not (Membership.mem.{max u_4 u_5, max u_4 u_5} (TensorProduct.{u_2, u_4, u_5} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' R'' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_4} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_4} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_4} R' (CommRing.toNonUnitalCommRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} R'' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_5} R'' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_5} R'' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_5} R'' (CommRing.toNonUnitalCommRing.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) (Ideal.{max u_5 u_4} (TensorProduct.{u_2, u_4, u_5} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' R'' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_4} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_4} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_4} R' (CommRing.toNonUnitalCommRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} R'' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_5} R'' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_5} R'' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_5} R'' (CommRing.toNonUnitalCommRing.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) (Algebra.TensorProduct.instSemiring.{u_2, u_4, u_5} R R' R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) (SetLike.instMembership.{max u_4 u_5, max u_4 u_5} (Ideal.{max u_5 u_4} (TensorProduct.{u_2, u_4, u_5} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' R'' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_4} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_4} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_4} R' (CommRing.toNonUnitalCommRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} R'' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_5} R'' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_5} R'' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_5} R'' (CommRing.toNonUnitalCommRing.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) (Algebra.TensorProduct.instSemiring.{u_2, u_4, u_5} R R' R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) (TensorProduct.{u_2, u_4, u_5} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' R'' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_4} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_4} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_4} R' (CommRing.toNonUnitalCommRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} R'' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_5} R'' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_5} R'' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_5} R'' (CommRing.toNonUnitalCommRing.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) (Submodule.setLike.{max u_4 u_5, max u_4 u_5} (TensorProduct.{u_2, u_4, u_5} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' R'' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_4} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_4} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_4} R' (CommRing.toNonUnitalCommRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} R'' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_5} R'' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_5} R'' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_5} R'' (CommRing.toNonUnitalCommRing.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) (TensorProduct.{u_2, u_4, u_5} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' R'' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_4} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_4} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_4} R' (CommRing.toNonUnitalCommRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} R'' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_5} R'' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_5} R'' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_5} R'' (CommRing.toNonUnitalCommRing.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) (Algebra.TensorProduct.instSemiring.{u_2, u_4, u_5} R R' R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50) (NonUnitalNonAssocSemiring.toAddCommMonoid.{max u_5 u_4} (TensorProduct.{u_2, u_4, u_5} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' R'' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_4} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_4} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_4} R' (CommRing.toNonUnitalCommRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} R'' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_5} R'' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_5} R'' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_5} R'' (CommRing.toNonUnitalCommRing.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{max u_5 u_4} (TensorProduct.{u_2, u_4, u_5} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' R'' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_4} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_4} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_4} R' (CommRing.toNonUnitalCommRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} R'' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_5} R'' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_5} R'' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_5} R'' (CommRing.toNonUnitalCommRing.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) (Semiring.toNonAssocSemiring.{max u_5 u_4} (TensorProduct.{u_2, u_4, u_5} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' R'' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_4} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_4} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_4} R' (CommRing.toNonUnitalCommRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} R'' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_5} R'' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_5} R'' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_5} R'' (CommRing.toNonUnitalCommRing.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) (Algebra.TensorProduct.instSemiring.{u_2, u_4, u_5} R R' R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)))) (Semiring.toModule.{max u_5 u_4} (TensorProduct.{u_2, u_4, u_5} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' R'' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_4} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_4} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_4} R' (CommRing.toNonUnitalCommRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} R'' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_5} R'' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_5} R'' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_5} R'' (CommRing.toNonUnitalCommRing.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) (Algebra.TensorProduct.instSemiring.{u_2, u_4, u_5} R R' R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)))) P'' e₀)) -> (Eq.{max (succ u_4) (succ u_5)} (Ideal.{max u_5 u_4} (TensorProduct.{u_2, u_4, u_5} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' R'' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_4} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_4} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_4} R' (CommRing.toNonUnitalCommRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} R'' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_5} R'' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_5} R'' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_5} R'' (CommRing.toNonUnitalCommRing.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) (Algebra.TensorProduct.instSemiring.{u_2, u_4, u_5} R R' R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) P'' (Ideal.comap.{max u_4 u_5, max u_3 u_4, max (max u_3 u_4) u_5} (TensorProduct.{u_2, u_4, u_5} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' R'' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} R'' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_5} R'' (Semiring.toNonAssocSemiring.{u_5} R'' (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) (TensorProduct.{u_2, u_4, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_4} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_4} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_4} R' (CommRing.toNonUnitalCommRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) (RingHom.{max u_4 u_5, max u_3 u_4} (TensorProduct.{u_2, u_4, u_5} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' R'' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} R'' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_5} R'' (Semiring.toNonAssocSemiring.{u_5} R'' (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) (TensorProduct.{u_2, u_4, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) (Semiring.toNonAssocSemiring.{max u_4 u_5} (TensorProduct.{u_2, u_4, u_5} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' R'' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} R'' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_5} R'' (Semiring.toNonAssocSemiring.{u_5} R'' (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) (Algebra.TensorProduct.instSemiring.{u_2, u_4, u_5} R R' R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) (Semiring.toNonAssocSemiring.{max u_3 u_4} (TensorProduct.{u_2, u_4, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) (Algebra.TensorProduct.instSemiring.{u_2, u_4, u_3} R R' S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32))) (Algebra.TensorProduct.instSemiring.{u_2, u_4, u_5} R R' R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50) (Algebra.TensorProduct.instSemiring.{u_2, u_4, u_3} R R' S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32) (RingHom.instFunLike.{max u_4 u_5, max u_3 u_4} (TensorProduct.{u_2, u_4, u_5} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' R'' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} R'' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_5} R'' (Semiring.toNonAssocSemiring.{u_5} R'' (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) (TensorProduct.{u_2, u_4, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} 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(Algebra.id.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (IsScalarTower.to_smulCommClass.{u_2, u_4, u_4} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 R' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (Algebra.toModule.{u_4, u_4} R' R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Algebra.id.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (IsScalarTower.right.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43))) (Algebra.TensorProduct.map.{u_2, u_4, u_4, u_5, u_4, u_3} R R' R' R'' R' S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (Algebra.id.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (IsScalarTower.right.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50 (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (Algebra.id.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (IsScalarTower.right.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32 (AlgHom.id.{u_4, u_4} R' R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Algebra.id.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))) (IsScalarTower.toAlgHom.{u_2, u_5, u_3} R R'' S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.54 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.62))) (RingHom.instRingHomClass.{max u_4 u_5, max u_3 u_4} (TensorProduct.{u_2, u_4, u_5} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' R'' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} R'' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_5} R'' (Semiring.toNonAssocSemiring.{u_5} R'' (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) (TensorProduct.{u_2, u_4, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) (Semiring.toNonAssocSemiring.{max u_4 u_5} (TensorProduct.{u_2, u_4, u_5} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' R'' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} R'' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_5} R'' (Semiring.toNonAssocSemiring.{u_5} R'' (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_5} R R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) (Algebra.TensorProduct.instSemiring.{u_2, u_4, u_5} R R' R'' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.50)) (Semiring.toNonAssocSemiring.{max u_3 u_4} (TensorProduct.{u_2, u_4, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) (Algebra.TensorProduct.instSemiring.{u_2, u_4, u_3} R R' S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32))) P'))) -> (forall (g : R''), (Not (Membership.mem.{u_3, u_3} S (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))) (SetLike.instMembership.{u_3, u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))) S (Submodule.setLike.{u_3, u_3} S S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))))) (Semiring.toModule.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))))) q (DFunLike.coe.{max (succ u_3) (succ u_5), succ u_5, succ u_3} (RingHom.{u_5, u_3} R'' S (Semiring.toNonAssocSemiring.{u_5} R'' (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47))) (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)))) R'' (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R'') => S) (RingHom.instFunLike.{u_5, u_3} R'' S (Semiring.toNonAssocSemiring.{u_5} R'' (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47))) (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)))) (algebraMap.{u_5, u_3} R'' S (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.54) g))) -> (Function.Surjective.{succ u_5, succ u_3} (Localization.Away.{u_5} R'' (CommSemiring.toCommMonoid.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47)) g) (Localization.Away.{u_3} S (CommSemiring.toCommMonoid.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) (DFunLike.coe.{max (succ u_5) (succ u_3), succ u_5, succ u_3} (RingHom.{u_5, u_3} R'' S (Semiring.toNonAssocSemiring.{u_5} R'' (CommSemiring.toSemiring.{u_5} R'' (CommRing.toCommSemiring.{u_5} R'' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.47))) (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)))) R'' (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R'') => 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(Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43))))) (IsScalarTower.right.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (IsScalarTower.right.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) 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(CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))))) (SemigroupAction.toSMul.{u_4, u_4} R' R' (Monoid.toSemigroup.{u_4} R' (MonoidWithZero.toMonoid.{u_4} R' (Semiring.toMonoidWithZero.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (MulAction.toSemigroupAction.{u_4, u_4} R' R' (MonoidWithZero.toMonoid.{u_4} R' (Semiring.toMonoidWithZero.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)))) (DistribMulAction.toMulAction.{u_4, u_4} R' R' (MonoidWithZero.toMonoid.{u_4} R' (Semiring.toMonoidWithZero.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)))) (AddCommMonoid.toAddMonoid.{u_4} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Algebra.id.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))))) (IsScalarTower.right.{u_4, u_4} R' R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Algebra.id.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))) (IsScalarTower.right.{u_4, u_4} R' R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Algebra.id.{u_4} R' (CommRing.toCommSemiring.{u_4} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : TensorProduct.{u_2, u_4, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 R' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (Algebra.toModule.{u_4, u_4} R' R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Algebra.id.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (IsScalarTower.right.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43))) (Algebra.TensorProduct.leftAlgebra.{u_2, u_4, u_4, u_3} R R' (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (OreLocalization.instSemiring.{u_4} R' (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' 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(Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (Algebra.id.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (IsScalarTower.right.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' 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(CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (IsScalarTower.right.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43))) (Algebra.TensorProduct.leftAlgebra.{u_2, u_4, u_4, u_3} R R' (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (OreLocalization.instSemiring.{u_4} R' (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : TensorProduct.{u_2, u_4, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S 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(CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) (Semiring.toNonAssocSemiring.{u_4} (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) (OreLocalization.instSemiring.{u_4} R' (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Submonoid.powers.{u_4} R' 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : TensorProduct.{u_2, u_4, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) => TensorProduct.{u_2, u_4, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) (Semiring.toNonAssocSemiring.{u_4} (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) (OreLocalization.instSemiring.{u_4} R' (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f) (OreLocalization.oreSetComm.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))))) (Algebra.toModule.{u_2, u_4} R (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (OreLocalization.instSemiring.{u_4} R' (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f) (OreLocalization.oreSetComm.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f))) (OreLocalization.instAlgebra.{u_4, u_2} R' (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f) (OreLocalization.oreSetComm.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f)) R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43)) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) (AlgHom.funLike.{u_4, max u_3 u_4, max u_3 u_4} R' (TensorProduct.{u_2, u_4, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) (TensorProduct.{u_2, u_4, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) (Semiring.toNonAssocSemiring.{u_4} (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) (OreLocalization.instSemiring.{u_4} R' (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f) (OreLocalization.oreSetComm.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))))) (Algebra.toModule.{u_2, u_4} R (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (OreLocalization.instSemiring.{u_4} R' (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f) (OreLocalization.oreSetComm.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f))) (OreLocalization.instAlgebra.{u_4, u_2} R' (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f) (OreLocalization.oreSetComm.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f)) R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43)) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Algebra.TensorProduct.instSemiring.{u_2, u_4, u_3} R R' S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32) (Algebra.TensorProduct.instSemiring.{u_2, u_4, u_3} R (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (OreLocalization.instSemiring.{u_4} R' (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f) (OreLocalization.oreSetComm.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f))) (OreLocalization.instAlgebra.{u_4, u_2} R' (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f) (OreLocalization.oreSetComm.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f)) R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32) (Algebra.TensorProduct.leftAlgebra.{u_2, u_4, u_4, u_3} R R' R' S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32 (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Algebra.id.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (IsScalarTower.to_smulCommClass.{u_2, u_4, u_4} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 R' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (Algebra.toModule.{u_4, u_4} R' R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Algebra.id.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (IsScalarTower.right.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43))) (Algebra.TensorProduct.leftAlgebra.{u_2, u_4, u_4, u_3} R R' (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (OreLocalization.instSemiring.{u_4} R' (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' 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(MonoidWithZero.toMonoid.{u_4} R' (Semiring.toMonoidWithZero.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)))) (AddCommMonoid.toAddMonoid.{u_4} R' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)))))) (Module.toDistribMulAction.{u_4, u_4} R' R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) 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(Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (Algebra.id.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (IsScalarTower.right.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32 (OreLocalization.instSemiring.{u_4} R' (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f) (OreLocalization.oreSetComm.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f))) (OreLocalization.instAlgebra.{u_4, u_2} R' (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f) (OreLocalization.oreSetComm.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f)) R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (OreLocalization.instAlgebra.{u_4, u_4} R' (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (SemigroupAction.toSMul.{u_4, u_4} R' R' (Monoid.toSemigroup.{u_4} R' (MonoidWithZero.toMonoid.{u_4} R' (Semiring.toMonoidWithZero.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (MulAction.toSemigroupAction.{u_4, u_4} R' R' (MonoidWithZero.toMonoid.{u_4} R' (Semiring.toMonoidWithZero.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)))) (DistribMulAction.toMulAction.{u_4, u_4} R' R' (MonoidWithZero.toMonoid.{u_4} R' (Semiring.toMonoidWithZero.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)))) (AddCommMonoid.toAddMonoid.{u_4} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : TensorProduct.{u_2, u_4, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S 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(Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (Algebra.id.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (IsScalarTower.right.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' 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(Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43))))) (IsScalarTower.right.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (IsScalarTower.right.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) 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(CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))))) (SemigroupAction.toSMul.{u_4, u_4} R' R' (Monoid.toSemigroup.{u_4} R' (MonoidWithZero.toMonoid.{u_4} R' (Semiring.toMonoidWithZero.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (MulAction.toSemigroupAction.{u_4, u_4} R' R' (MonoidWithZero.toMonoid.{u_4} R' (Semiring.toMonoidWithZero.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)))) (DistribMulAction.toMulAction.{u_4, u_4} R' R' (MonoidWithZero.toMonoid.{u_4} R' (Semiring.toMonoidWithZero.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)))) (AddCommMonoid.toAddMonoid.{u_4} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Algebra.id.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))))) (IsScalarTower.right.{u_4, u_4} R' R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Algebra.id.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))) (IsScalarTower.right.{u_4, u_4} R' R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Algebra.id.{u_4} R' (CommRing.toCommSemiring.{u_4} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : TensorProduct.{u_2, u_4, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 R' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R' (Semiring.toNonAssocSemiring.{u_4} R' (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))))) (Algebra.toModule.{u_4, u_4} R' R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Algebra.id.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40))) (Algebra.toModule.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (IsScalarTower.right.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43))) (Algebra.TensorProduct.leftAlgebra.{u_2, u_4, u_4, u_3} R R' (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (OreLocalization.instSemiring.{u_4} R' (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' 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(Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43 (Algebra.id.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (IsScalarTower.right.{u_2, u_4} R R' (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' 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(CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f)) R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43)))) (IsScalarTower.right.{u_4, max u_3 u_4} (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) (TensorProduct.{u_2, u_4, u_3} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) (Semiring.toNonAssocSemiring.{u_4} (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) (OreLocalization.instSemiring.{u_4} R' (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f) (OreLocalization.oreSetComm.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29))))) (Algebra.toModule.{u_2, u_4} R (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (OreLocalization.instSemiring.{u_4} R' (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f) (OreLocalization.oreSetComm.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f))) (OreLocalization.instAlgebra.{u_4, u_2} R' (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f) (OreLocalization.oreSetComm.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f)) R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43)) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32)) (OreLocalization.instCommSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f) (OreLocalization.oreSetComm.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f))) (OreLocalization.instAlgebra.{u_4, u_2} R' (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f) (OreLocalization.oreSetComm.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f)) R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32) (Algebra.TensorProduct.leftAlgebra.{u_2, u_4, u_4, u_3} R (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) S (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (OreLocalization.instSemiring.{u_4} R' (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f) (OreLocalization.oreSetComm.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f))) (OreLocalization.instAlgebra.{u_4, u_2} R' (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f) (OreLocalization.oreSetComm.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f)) R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.29)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.32 (OreLocalization.instCommSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f) (OreLocalization.oreSetComm.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f))) (Algebra.id.{u_4} (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) (OreLocalization.instCommSemiring.{u_4} R' (CommRing.toCommSemiring.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f) (OreLocalization.oreSetComm.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f)))) (Algebra.to_smulCommClass.{u_2, u_4} R (Localization.Away.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) f) (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) (OreLocalization.instSemiring.{u_4} R' (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f) (OreLocalization.oreSetComm.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f))) (OreLocalization.instAlgebra.{u_4, u_2} R' (Ring.toSemiring.{u_4} R' (CommRing.toRing.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f) (OreLocalization.oreSetComm.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40) (Submonoid.powers.{u_4} R' (CommMonoid.toMonoid.{u_4} R' (CommRing.toCommMonoid.{u_4} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.40)) f)) R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.26) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.3994315237._hygCtx._hyg.43))))))))))","typeFull":"∀ {R : Type u_2} {S : Type u_3} {R' : Type u_4} {R'' : Type u_5} [inst : CommRing R] [inst_1 : CommRing S]\n  [inst_2 : Algebra R S] [Algebra.FiniteType R S] [inst_4 : CommRing R'] [inst_5 : Algebra R R'] [inst_6 : CommRing R'']\n  [inst_7 : Algebra R R''] [inst_8 : Algebra R'' S] [Algebra.IsIntegral R R''] [inst_10 : IsScalarTower R R'' S]\n  (q : Ideal S) (P : Ideal R') [P.IsPrime] (e : TensorProduct R R' S) (e₀ : TensorProduct R R' R''),\n  IsIdempotentElem e₀ →\n    (Algebra.TensorProduct.map (AlgHom.id R' R') (IsScalarTower.toAlgHom R R'' S)) e₀ = e →\n      ∀ (P' : Ideal (TensorProduct R R' S)),\n        Ideal.comap Algebra.TensorProduct.includeRight.toRingHom P' = q →\n          (∀ (P'' : Ideal (TensorProduct R R' R'')),\n              P''.IsPrime →\n                P''.LiesOver P →\n                  e₀ ∉ P'' →\n                    P'' =\n                      Ideal.comap\n                        (Algebra.TensorProduct.map (AlgHom.id R' R') (IsScalarTower.toAlgHom R R'' S)).toRingHom P') →\n            ∀ (g : R''),\n              (algebraMap R'' S) g ∉ q →\n                Function.Surjective ⇑(Localization.awayMap (algebraMap R'' S) g) →\n                  ∃ f ∉ P,\n                    Module.Finite (Localization.Away f)\n                      (Localization.Away\n                        ((Algebra.TensorProduct.map (Algebra.ofId R' (Localization.Away f)) (AlgHom.id R S)) e))","typeReadable":"∀ {R : Type u_2} {S : Type u_3} {R' : Type u_4} {R'' : Type u_5} [inst : CommRing R] [inst_1 : CommRing S]\n  [inst_2 : Algebra R S] [Algebra.FiniteType R S] [inst_4 : CommRing R'] [inst_5 : Algebra R R'] [inst_6 : CommRing R'']\n  [inst_7 : Algebra R R''] [inst_8 : Algebra R'' S] [Algebra.IsIntegral R R''] [inst_10 : IsScalarTower R R'' S]\n  (q : Ideal S) (P : Ideal R') [P.IsPrime] (e : TensorProduct R R' S) (e₀ : TensorProduct R R' R''),\n  IsIdempotentElem e₀ →\n    (Algebra.TensorProduct.map (AlgHom.id R' R') (IsScalarTower.toAlgHom R R'' S)) e₀ = e →\n      ∀ (P' : Ideal (TensorProduct R R' S)),\n        Ideal.comap Algebra.TensorProduct.includeRight.toRingHom P' = q →\n          (∀ (P'' : Ideal (TensorProduct R R' R'')),\n              P''.IsPrime →\n                P''.LiesOver P →\n                  e₀ ∉ P'' →\n                    P'' =\n                      Ideal.comap\n                        (Algebra.TensorProduct.map (AlgHom.id R' R') (IsScalarTower.toAlgHom R R'' S)).toRingHom P') →\n            ∀ (g : R''),\n              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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.28) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5 (OreLocalization.instAlgebra.{u_2, u_1} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) q 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(Ideal.ResidueField._proof_1.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.26))))) (DivisionSemiring.toSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.28) (Semifield.toDivisionSemiring.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.28) (Field.toSemifield.{u_2} (Ideal.ResidueField.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.28) (IsLocalRing.ResidueField.field.{u_2} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.28))))) (IsLocalRing.ResidueField.algebra.{u_1, u_1} (Localization.AtPrime.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.26) (OreLocalization.instCommRing.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5 (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.26) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.26))) (Ideal.ResidueField._proof_1.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.26) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5 (OreLocalization.instAlgebra.{u_1, u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.26) (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5)) (Ideal.primeCompl.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.26)) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5)))) (IsLocalRing.ResidueField.algebra.{u_2, u_1} (Localization.AtPrime.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.28) (OreLocalization.instCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8 (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.28))) (Ideal.ResidueField._proof_1.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.28) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5 (OreLocalization.instAlgebra.{u_2, u_1} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.28) (OreLocalization.oreSetComm.{u_2} R' (CommSemiring.toCommMonoid.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) (Ideal.primeCompl.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.28)) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14))) (Ideal.ResidueField.mapₐ.{u_1, u_1, u_2} R R R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8 (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.26 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.28 (Algebra.ofId.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14) (Ideal.over_def.{u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14 q p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.30)))) -> (forall (P₁ : Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18)) (P₂ : Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18)) [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.67 : Ideal.IsPrime.{max u_2 u_3} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18) P₁] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.69 : Ideal.IsPrime.{max u_2 u_3} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18) P₂] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.71 : Ideal.LiesOver.{u_2, max u_2 u_3} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8) (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' 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(NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18)) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18)) (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18 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(CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18)) (RingHom.instRingHomClass.{u_3, max u_2 u_3} S (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} R' (Semiring.toNonAssocSemiring.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18)) (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11))) (Semiring.toNonAssocSemiring.{max u_2 u_3} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} R' (Semiring.toNonAssocSemiring.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18))) P₁) (Ideal.comap.{u_3, max u_2 u_3, max u_2 u_3} S (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18)) (RingHom.{u_3, max u_2 u_3} S (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} R' (Semiring.toNonAssocSemiring.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18)) (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11))) (Semiring.toNonAssocSemiring.{max u_2 u_3} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} R' (Semiring.toNonAssocSemiring.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18))) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) 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(Semiring.toNonAssocSemiring.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18))) (AlgHom.toRingHom.{u_1, u_3, max u_2 u_3} R S (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} R' (Semiring.toNonAssocSemiring.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18)) (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18 (Algebra.TensorProduct.instAlgebra.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18) (Algebra.TensorProduct.includeRight.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18)) (RingHom.instRingHomClass.{u_3, max u_2 u_3} S (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} R' (Semiring.toNonAssocSemiring.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18)) (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11))) (Semiring.toNonAssocSemiring.{max u_2 u_3} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} R' (Semiring.toNonAssocSemiring.{u_2} R' (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18))) P₂)) -> (Eq.{max (succ u_2) (succ u_3)} (Ideal.{max u_3 u_2} (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} R' (CommRing.toNonUnitalCommRing.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} S (CommRing.toNonUnitalCommRing.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11))))) (Algebra.toModule.{u_1, u_2} R R' (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14) (Algebra.toModule.{u_1, u_3} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18)) (Algebra.TensorProduct.instSemiring.{u_1, u_2, u_3} R R' S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_2} R' (CommRing.toCommSemiring.{u_2} R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.8)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.14 (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.848325382._hygCtx._hyg.18)) P₁ P₂))","typeFull":"∀ {R : Type u_1} {R' : Type u_2} {S : Type u_3} [inst : CommRing R] [inst_1 : CommRing R'] [inst_2 : CommRing S]\n  [inst_3 : Algebra R R'] [inst_4 : Algebra R S] {p : Ideal R} {q : Ideal R'} [inst_5 : p.IsPrime] [inst_6 : q.IsPrime]\n  [inst_7 : q.LiesOver p],\n  Function.Bijective ⇑(Ideal.ResidueField.mapₐ p q (Algebra.ofId R R') ⋯) →\n    ∀ (P₁ P₂ : Ideal (TensorProduct R R' S)) [P₁.IsPrime] [P₂.IsPrime] [P₁.LiesOver q] [P₂.LiesOver q],\n      Ideal.comap Algebra.TensorProduct.includeRight.toRingHom P₁ =\n          Ideal.comap Algebra.TensorProduct.includeRight.toRingHom P₂ →\n        P₁ = P₂","typeReadable":"∀ {R : Type u_1} {R' : Type u_2} {S : Type u_3} [inst : CommRing R] [inst_1 : CommRing R'] [inst_2 : CommRing S]\n  [inst_3 : Algebra R R'] [inst_4 : Algebra R S] {p : Ideal R} {q : Ideal R'} [inst_5 : p.IsPrime] [inst_6 : q.IsPrime]\n  [inst_7 : q.LiesOver p],\n  Function.Bijective ⇑(Ideal.ResidueField.mapₐ p q (Algebra.ofId R R') ⋯) →\n    ∀ (P₁ P₂ : Ideal (TensorProduct R R' S)) [P₁.IsPrime] [P₂.IsPrime] [P₁.LiesOver q] [P₂.LiesOver q],\n      Ideal.comap Algebra.TensorProduct.includeRight.toRingHom P₁ =\n          Ideal.comap Algebra.TensorProduct.includeRight.toRingHom P₂ →\n        P₁ = P₂","typeReferences":[["RingHom"],["IsLocalRing","ResidueField","algebra"],["Ideal","IsPrime"],["OreLocalization","oreSetComm"],["RingHom","instFunLike"],["Algebra","TensorProduct","instSemiring"],["CommRing","toNonUnitalCommRing"],["Ideal","ResidueField"],["RingHom","instRingHomClass"],["Algebra","to_smulCommClass"],["DFunLike","coe"],["Algebra","id"],["Ideal","ResidueField","mapₐ"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Ideal","LiesOver"],["Semiring","toNonAssocSemiring"],["OreLocalization","instCommRing"],["Ideal"],["OreLocalization","instAlgebra"],["Algebra","toModule"],["Ideal","comap"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Ideal","primeCompl"],["AlgHom","funLike"],["Semifield","toDivisionSemiring"],["Eq"],["Localization","AtPrime"],["AlgHom"],["TensorProduct"],["CommRing","toCommSemiring"],["Algebra","TensorProduct","instAlgebra"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Function","Bijective"],["DivisionSemiring","toSemiring"],["CommRing"],["Algebra","TensorProduct","leftAlgebra"],["CommSemiring","toCommMonoid"],["Algebra"],["Algebra","ofId"],["Ideal","ResidueField","_proof_1"],["Algebra","TensorProduct","includeRight"],["Ideal","over_def"],["AlgHom","toRingHom"],["Field","toSemifield"],["IsLocalRing","ResidueField","field"]],"valueReferences":[["PartialOrder","toPreorder"],["Ideal","IsPrime"],["Subtype","instLE"],["Membership","mem"],["OrderIso"],["OrderIso","injective"],["Subtype","val"],["Algebra","to_smulCommClass"],["Algebra","id"],["Set","Elem"],["Ideal","LiesOver"],["And","intro"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["Ideal","comap"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Semiring","toModule"],["TensorProduct"],["Algebra","TensorProduct","instAlgebra"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["congr_arg"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Subtype","ext"],["Set","instMembership"],["Ideal","fiberIsoOfBijectiveResidueField"],["id"],["Ideal","primesOver"],["Eq","mpr"],["AlgHomClass","toRingHomClass"],["Subtype"],["Algebra","TensorProduct","instSemiring"],["CommRing","toNonUnitalCommRing"],["DFunLike","coe"],["congrArg"],["Ideal"],["Algebra","toModule"],["congr"],["AlgHom","funLike"],["Preorder","toLE"],["Eq"],["AlgHom"],["CommRing","toCommSemiring"],["Set"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Submodule","instPartialOrder"],["Algebra","TensorProduct","leftAlgebra"],["instFunLikeOrderIso"],["Algebra","TensorProduct","includeRight"],["Ideal","comap_fiberIsoOfBijectiveResidueField_apply"],["Subtype","mk"],["AlgHom","algHomClass"]]},{"isProp":true,"kind":"theorem","name":["Algebra","exists_etale_isIdempotentElem_forall_liesOver_eq"],"typeFallback":"forall {R : Type.{u}} {S : Type.{v}} [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 : CommRing.{u} R] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27 : CommRing.{v} S] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30 : Algebra.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.34 : Algebra.FiniteType.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30] (p : Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24))) [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40 : Ideal.IsPrime.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24)) p] (q : Ideal.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))) [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.44 : Ideal.IsPrime.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) q] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.46 : Ideal.LiesOver.{u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30 q p] [inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.49 : Algebra.QuasiFiniteAt.{u, v} R S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30 q inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.44], Exists.{succ (succ u)} Type.{u} (fun (R' : Type.{u}) => Exists.{succ u} (CommRing.{u} R') (fun (x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66 : CommRing.{u} R') => Exists.{succ u} (Algebra.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))) (fun (x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 : Algebra.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))) => Exists.{0} (Algebra.Etale.{u, u} R R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66 x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (fun (x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.83 : Algebra.Etale.{u, u} R R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66 x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) => Exists.{succ u} (Ideal.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))) (fun (P : Ideal.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))) => Exists.{0} (Ideal.IsPrime.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) P) (fun (x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99 : Ideal.IsPrime.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) P) => Exists.{0} (Ideal.LiesOver.{u, u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 P p) (fun (x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.106 : Ideal.LiesOver.{u, u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 P p) => Exists.{max (succ u) (succ v)} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (fun (e : TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R 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(NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Algebra.TensorProduct.instMul.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.to_smulCommClass.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (IsScalarTower.right.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)))) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30) (Algebra.to_smulCommClass.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30) (IsScalarTower.right.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) e) (fun (x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.125 : IsIdempotentElem.{max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Algebra.TensorProduct.instMul.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.to_smulCommClass.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (IsScalarTower.right.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)))) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30) (Algebra.to_smulCommClass.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30) (IsScalarTower.right.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) e) => Exists.{max (succ u) (succ v)} (Ideal.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (fun (P' : Ideal.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) => Exists.{0} (Ideal.IsPrime.{max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30) P') (fun (x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.147 : Ideal.IsPrime.{max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30) P') => Exists.{0} (Ideal.LiesOver.{u, max u v} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66) (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30) (Algebra.TensorProduct.leftAlgebra.{u, u, u, v} R R' R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30 (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66) (Algebra.id.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) (Algebra.to_smulCommClass.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74)) P' P) (fun (x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.154 : Ideal.LiesOver.{u, max u v} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66) (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30) (Algebra.TensorProduct.leftAlgebra.{u, u, u, v} R R' R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30 (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66) (Algebra.id.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) (Algebra.to_smulCommClass.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74)) P' P) => And (Eq.{succ v} (Ideal.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))) (Ideal.comap.{v, max u v, max u v} S (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (RingHom.{v, max u v} S (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R' (Semiring.toNonAssocSemiring.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{v} S (Semiring.toNonAssocSemiring.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Semiring.toNonAssocSemiring.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))) (Semiring.toNonAssocSemiring.{max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R' (Semiring.toNonAssocSemiring.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{v} S (Semiring.toNonAssocSemiring.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30))) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30) (RingHom.instFunLike.{v, max u v} S (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R' (Semiring.toNonAssocSemiring.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{v} S (Semiring.toNonAssocSemiring.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Semiring.toNonAssocSemiring.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))) (Semiring.toNonAssocSemiring.{max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R' (Semiring.toNonAssocSemiring.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{v} S (Semiring.toNonAssocSemiring.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30))) (AlgHom.toRingHom.{u, v, max u v} R S (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R' (Semiring.toNonAssocSemiring.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{v} S (Semiring.toNonAssocSemiring.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30 (Algebra.TensorProduct.instAlgebra.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30) (Algebra.TensorProduct.includeRight.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (RingHom.instRingHomClass.{v, max u v} S (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R' (Semiring.toNonAssocSemiring.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{v} S (Semiring.toNonAssocSemiring.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Semiring.toNonAssocSemiring.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))) (Semiring.toNonAssocSemiring.{max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R' (Semiring.toNonAssocSemiring.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{v} S (Semiring.toNonAssocSemiring.{v} S (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30))) P') q) (And (Not (Membership.mem.{max u v, max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Ideal.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (SetLike.instMembership.{max u v, max u v} (Ideal.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Submodule.setLike.{max u v, max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30) (NonUnitalNonAssocSemiring.toAddCommMonoid.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Semiring.toNonAssocSemiring.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)))) (Semiring.toModule.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)))) P' e)) (And (Function.Bijective.{succ u, succ u} (Ideal.ResidueField.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40) (Ideal.ResidueField.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99) (DFunLike.coe.{succ u, succ u, succ u} (AlgHom.{u, u, u} R (Ideal.ResidueField.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40) (Ideal.ResidueField.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99) (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (DivisionSemiring.toSemiring.{u} (Ideal.ResidueField.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40) (Semifield.toDivisionSemiring.{u} (Ideal.ResidueField.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40) (Field.toSemifield.{u} (Ideal.ResidueField.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40) (IsLocalRing.ResidueField.field.{u} (Localization.AtPrime.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40) (OreLocalization.instCommRing.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 (Ideal.primeCompl.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40) (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24)) (Ideal.primeCompl.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40))) (Ideal.ResidueField._proof_1.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40))))) (DivisionSemiring.toSemiring.{u} (Ideal.ResidueField.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99) (Semifield.toDivisionSemiring.{u} (Ideal.ResidueField.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99) (Field.toSemifield.{u} (Ideal.ResidueField.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99) (IsLocalRing.ResidueField.field.{u} (Localization.AtPrime.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99) (OreLocalization.instCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66 (Ideal.primeCompl.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99) (OreLocalization.oreSetComm.{u} R' (CommSemiring.toCommMonoid.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) (Ideal.primeCompl.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99))) (Ideal.ResidueField._proof_1.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99))))) (IsLocalRing.ResidueField.algebra.{u, u} (Localization.AtPrime.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40) (OreLocalization.instCommRing.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 (Ideal.primeCompl.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40) (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24)) (Ideal.primeCompl.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40))) (Ideal.ResidueField._proof_1.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 (OreLocalization.instAlgebra.{u, u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24)) (Ideal.primeCompl.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40) (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R (CommRing.toCommSemiring.{u} R 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x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99) (OreLocalization.oreSetComm.{u} R' (CommSemiring.toCommMonoid.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) (Ideal.primeCompl.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99))) (Ideal.ResidueField._proof_1.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 (OreLocalization.instAlgebra.{u, u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) (Ideal.primeCompl.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99) (OreLocalization.oreSetComm.{u} R' (CommSemiring.toCommMonoid.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) (Ideal.primeCompl.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99)) R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74))) (Ideal.ResidueField.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Ideal.ResidueField.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40) => Ideal.ResidueField.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99) (AlgHom.funLike.{u, u, u} R (Ideal.ResidueField.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40) (Ideal.ResidueField.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99) (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (DivisionSemiring.toSemiring.{u} (Ideal.ResidueField.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40) (Semifield.toDivisionSemiring.{u} (Ideal.ResidueField.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40) (Field.toSemifield.{u} (Ideal.ResidueField.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40) (IsLocalRing.ResidueField.field.{u} (Localization.AtPrime.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40) (OreLocalization.instCommRing.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 (Ideal.primeCompl.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40) (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24)) (Ideal.primeCompl.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40))) (Ideal.ResidueField._proof_1.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40))))) (DivisionSemiring.toSemiring.{u} (Ideal.ResidueField.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99) (Semifield.toDivisionSemiring.{u} (Ideal.ResidueField.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99) (Field.toSemifield.{u} (Ideal.ResidueField.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99) (IsLocalRing.ResidueField.field.{u} (Localization.AtPrime.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99) (OreLocalization.instCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66 (Ideal.primeCompl.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99) (OreLocalization.oreSetComm.{u} R' (CommSemiring.toCommMonoid.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) (Ideal.primeCompl.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99))) (Ideal.ResidueField._proof_1.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99))))) (IsLocalRing.ResidueField.algebra.{u, u} (Localization.AtPrime.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40) (OreLocalization.instCommRing.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 (Ideal.primeCompl.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40) (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24)) (Ideal.primeCompl.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40))) (Ideal.ResidueField._proof_1.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 (OreLocalization.instAlgebra.{u, u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24)) (Ideal.primeCompl.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40) (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24)) (Ideal.primeCompl.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24)) p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40)) R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (Algebra.id.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24)))) (IsLocalRing.ResidueField.algebra.{u, u} (Localization.AtPrime.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99) (OreLocalization.instCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66 (Ideal.primeCompl.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99) (OreLocalization.oreSetComm.{u} R' (CommSemiring.toCommMonoid.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) (Ideal.primeCompl.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99))) (Ideal.ResidueField._proof_1.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99) R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 (OreLocalization.instAlgebra.{u, u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) (Ideal.primeCompl.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99) (OreLocalization.oreSetComm.{u} R' (CommSemiring.toCommMonoid.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) (Ideal.primeCompl.{u} R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99)) R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74))) (Ideal.ResidueField.mapₐ.{u, u, u} R R R' inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24 x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66 (Algebra.id.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 p inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.40 P x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.99 (Algebra.ofId.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Ideal.over_def.{u, u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 P p x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.106)))) (And (Module.Finite.{u, max u v} R' (Localization.Away.{max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (CommRing.toCommMonoid.{max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, 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inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) e) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) (OreLocalization.instAddCommMonoidOreLocalization.{max u v, max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (CommMonoid.toMonoid.{max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R 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x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Module.toDistribMulAction.{max u v, max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) 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x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Algebra.TensorProduct.instCommRing.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66 x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30))) e)) (TensorProduct.{u, u, v} R 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(NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30) (Algebra.TensorProduct.leftAlgebra.{u, u, u, v} R R' R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30 (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66) (Algebra.id.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) (Algebra.to_smulCommClass.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74))) (IsScalarTower.right.{u, max u v} R' (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30) (Algebra.TensorProduct.leftAlgebra.{u, u, u, v} R R' R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30 (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66) (Algebra.id.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) (Algebra.to_smulCommClass.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74))))) (forall (P'' : Ideal.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)), (Ideal.IsPrime.{max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30) P'') -> (Ideal.LiesOver.{u, max u v} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66) (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30) (Algebra.TensorProduct.leftAlgebra.{u, u, u, v} R R' R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30 (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66) (Algebra.id.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) (Algebra.to_smulCommClass.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74)) P'' P) -> (Not (Membership.mem.{max u v, max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Ideal.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (SetLike.instMembership.{max u v, max u v} (Ideal.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Submodule.setLike.{max u v, max u v} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30) (NonUnitalNonAssocSemiring.toAddCommMonoid.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Semiring.toNonAssocSemiring.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)) (Algebra.TensorProduct.instSemiring.{u, u, v} R R' S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74 (CommSemiring.toSemiring.{v} S (CommRing.toCommSemiring.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27)) inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.30)))) (Semiring.toModule.{max v u} (TensorProduct.{u, u, v} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) R' S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R' (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R' (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R' (CommRing.toNonUnitalCommRing.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{v} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{v} S (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{v} S (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{v} S (CommRing.toNonUnitalCommRing.{v} S inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.27))))) (Algebra.toModule.{u, u} R R' (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{u} R' (CommRing.toCommSemiring.{u} R' x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.66)) x._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.74) (Algebra.toModule.{u, v} R S (CommRing.toCommSemiring.{u} R inst._@.Mathlib.RingTheory.Etale.QuasiFinite.2444616463._hygCtx._hyg.24) (CommSemiring.toSemiring.{v} S 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{R : Type.{u_1}} {A : Type.{u_2}} [inst._@.Mathlib.RingTheory.PowerSeries.Binomial.1191813712._hygCtx._hyg.4 : CommRing.{u_1} R] [inst._@.Mathlib.RingTheory.PowerSeries.Binomial.1191813712._hygCtx._hyg.7 : BinomialRing.{u_1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} R (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} R (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} R (CommRing.toNonUnitalCommRing.{u_1} R inst._@.Mathlib.RingTheory.PowerSeries.Binomial.1191813712._hygCtx._hyg.4))))) (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.PowerSeries.Binomial.1191813712._hygCtx._hyg.4)))))] [inst._@.Mathlib.RingTheory.PowerSeries.Binomial.1191813712._hygCtx._hyg.10 : Ring.{u_2} A] [inst._@.Mathlib.RingTheory.PowerSeries.Binomial.1191813712._hygCtx._hyg.13 : Algebra.{u_1, u_2} R A 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{R : Type.{u_1}} {A : Type.{u_2}} [inst._@.Mathlib.RingTheory.PowerSeries.Binomial.3695793336._hygCtx._hyg.4 : CommRing.{u_1} R] [inst._@.Mathlib.RingTheory.PowerSeries.Binomial.3695793336._hygCtx._hyg.7 : BinomialRing.{u_1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} R (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} R (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} R (CommRing.toNonUnitalCommRing.{u_1} R inst._@.Mathlib.RingTheory.PowerSeries.Binomial.3695793336._hygCtx._hyg.4))))) (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.PowerSeries.Binomial.3695793336._hygCtx._hyg.4)))))] [inst._@.Mathlib.RingTheory.PowerSeries.Binomial.3695793336._hygCtx._hyg.10 : Ring.{u_2} A] [inst._@.Mathlib.RingTheory.PowerSeries.Binomial.3695793336._hygCtx._hyg.13 : Algebra.{u_1, u_2} R A 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(instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} R (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} R (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} R (CommRing.toNonUnitalCommRing.{u_1} R inst._@.Mathlib.RingTheory.PowerSeries.Binomial.3695793336._hygCtx._hyg.4))))))) r s)) (HMul.hMul.{u_2, u_2, u_2} (PowerSeries.{u_2} A) (PowerSeries.{u_2} A) (PowerSeries.{u_2} A) (instHMul.{u_2} (PowerSeries.{u_2} A) (MvPowerSeries.instMul.{0, u_2} Unit A (Ring.toSemiring.{u_2} A inst._@.Mathlib.RingTheory.PowerSeries.Binomial.3695793336._hygCtx._hyg.10))) (PowerSeries.binomialSeries.{u_1, u_2} R inst._@.Mathlib.RingTheory.PowerSeries.Binomial.3695793336._hygCtx._hyg.4 inst._@.Mathlib.RingTheory.PowerSeries.Binomial.3695793336._hygCtx._hyg.7 A (AddMonoidWithOne.toOne.{u_2} A (AddGroupWithOne.toAddMonoidWithOne.{u_2} A (Ring.toAddGroupWithOne.{u_2} A 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inst._@.Mathlib.RingTheory.PowerSeries.Binomial.3695793336._hygCtx._hyg.10) inst._@.Mathlib.RingTheory.PowerSeries.Binomial.3695793336._hygCtx._hyg.13) s))","typeFull":"∀ {R : Type u_1} {A : Type u_2} [inst : CommRing R] [inst_1 : BinomialRing R] [inst_2 : Ring A] [inst_3 : Algebra R A]\n  (r s : R), PowerSeries.binomialSeries A (r + s) = PowerSeries.binomialSeries A r * PowerSeries.binomialSeries A s","typeReadable":"∀ {R : Type u_1} {A : Type u_2} [inst : CommRing R] [inst_1 : BinomialRing R] [inst_2 : Ring A] [inst_3 : Algebra R A]\n  (r s : R), PowerSeries.binomialSeries A (r + s) = PowerSeries.binomialSeries A r * PowerSeries.binomialSeries A 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{R : Type.{u_1}} [inst._@.Mathlib.RingTheory.PowerSeries.Binomial.382820647._hygCtx._hyg.4 : CommRing.{u_1} R] [inst._@.Mathlib.RingTheory.PowerSeries.Binomial.382820647._hygCtx._hyg.7 : BinomialRing.{u_1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} R (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} R (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} R (CommRing.toNonUnitalCommRing.{u_1} R inst._@.Mathlib.RingTheory.PowerSeries.Binomial.382820647._hygCtx._hyg.4))))) (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.PowerSeries.Binomial.382820647._hygCtx._hyg.4)))))] (A : Type.{u_3}) [inst._@.Mathlib.RingTheory.PowerSeries.Binomial.382820647._hygCtx._hyg.11 : One.{u_3} A] [inst._@.Mathlib.RingTheory.PowerSeries.Binomial.382820647._hygCtx._hyg.14 : SMul.{u_1, u_3} R A], R -> (PowerSeries.{u_3} A)","typeFull":"{R : Type u_1} → [inst : CommRing R] → [BinomialRing R] → (A : Type u_3) → [One A] → [SMul R A] → R → PowerSeries A","typeReadable":"{R : Type u_1} → [inst : CommRing R] → [BinomialRing R] → (A : Type u_3) → [One A] → [SMul R A] → R → PowerSeries 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+[{"isProp":true,"kind":"definition","name":["_private","Mathlib","RingTheory","Valuation","LocalSubring",0,"Subring","exists_le_valuationSubring_of_isIntegrallyClosedIn","match_1_1"],"typeFallback":"forall {K : Type.{u_1}} [inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11 : Field.{u_1} K] {x : K} {R : Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))} (this : Invertible.{u_1} K (MulZeroClass.toMul.{u_1} K (MulZeroOneClass.toMulZeroClass.{u_1} K (MonoidWithZero.toMulZeroOneClass.{u_1} K (GroupWithZero.toMonoidWithZero.{u_1} K (DivisionSemiring.toGroupWithZero.{u_1} K (Semifield.toDivisionSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))))) (InvOneClass.toOne.{u_1} K 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(EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) R)) (CommSemiring.toSemiring.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) R) (Invertible.invOf.{u_1} K (Distrib.toMul.{u_1} K (NonUnitalNonAssocSemiring.toDistrib.{u_1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} K (CommRing.toNonUnitalCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))))) (AddMonoidWithOne.toOne.{u_1} K (AddGroupWithOne.toAddMonoidWithOne.{u_1} K (Ring.toAddGroupWithOne.{u_1} K (CommRing.toRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) (Inv.inv.{u_1} K (InvOneClass.toInv.{u_1} K (DivInvOneMonoid.toInvOneClass.{u_1} K (DivisionMonoid.toDivInvOneMonoid.{u_1} K (DivisionCommMonoid.toDivisionMonoid.{u_1} K (CommGroupWithZero.toDivisionCommMonoid.{u_1} K (Semifield.toCommGroupWithZero.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))))) x) (invertibleInv.{u_1} K (DivisionSemiring.toGroupWithZero.{u_1} K (Semifield.toDivisionSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))) x this))) p) (OfNat.ofNat.{u_1} K 0 (Zero.toOfNat0.{u_1} K (MulZeroClass.toZero.{u_1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} K (CommRing.toNonUnitalCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))))))))))) -> Prop) (x._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.210 : Exists.{succ u_1} (Polynomial.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) (CommRing.toCommSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) R)))) (fun (p : Polynomial.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) (CommRing.toCommSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) R)))) => And (Membership.mem.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) (Ideal.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) (CommRing.toCommSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) R)))) (SetLike.instMembership.{u_1, u_1} (Ideal.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K 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(CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) R) (Invertible.invOf.{u_1} K (Distrib.toMul.{u_1} K (NonUnitalNonAssocSemiring.toDistrib.{u_1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} K (CommRing.toNonUnitalCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))))) (AddMonoidWithOne.toOne.{u_1} K (AddGroupWithOne.toAddMonoidWithOne.{u_1} K (Ring.toAddGroupWithOne.{u_1} K (CommRing.toRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) (Inv.inv.{u_1} K (InvOneClass.toInv.{u_1} K (DivInvOneMonoid.toInvOneClass.{u_1} K (DivisionMonoid.toDivInvOneMonoid.{u_1} K (DivisionCommMonoid.toDivisionMonoid.{u_1} K (CommGroupWithZero.toDivisionCommMonoid.{u_1} K (Semifield.toCommGroupWithZero.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))))) x) (invertibleInv.{u_1} K (DivisionSemiring.toGroupWithZero.{u_1} K (Semifield.toDivisionSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))) x this))) p) (OfNat.ofNat.{u_1} K 0 (Zero.toOfNat0.{u_1} K (MulZeroClass.toZero.{u_1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} K (CommRing.toNonUnitalCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))))))))), motive (Exists.intro.{succ u_1} (Polynomial.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) (CommRing.toCommSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) R)))) (fun (p : 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(NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) (CommRing.toCommSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) R)))) => And (Membership.mem.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) (Ideal.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K 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(Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) R))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K 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inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))))))))) hp hpx))) -> (motive x._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.210)","typeFull":"∀ {K : Type u_1} [inst : Field K] {x : K} {R : Subring K} (this : Invertible x)\n  (motive : (∃ p, p.leadingCoeff - 1 ∈ ⊥ ∧ (Polynomial.aeval ⅟x⁻¹) p = 0) → Prop)\n  (x_1 : ∃ p, p.leadingCoeff - 1 ∈ ⊥ ∧ (Polynomial.aeval ⅟x⁻¹) p = 0),\n  (∀ (p : Polynomial ↥R) (hp : p.leadingCoeff - 1 ∈ ⊥) (hpx : (Polynomial.aeval ⅟x⁻¹) p = 0), motive ⋯) → motive x_1","typeReadable":"∀ {K : Type u_1} [inst : Field K] {x : K} {R : Subring K} (this : Invertible x)\n  (motive : (∃ p, p.leadingCoeff - 1 ∈ ⊥ ∧ (Polynomial.aeval ⅟x⁻¹) p = 0) → Prop)\n  (x_1 : ∃ p, p.leadingCoeff - 1 ∈ ⊥ ∧ (Polynomial.aeval ⅟x⁻¹) p = 0),\n  (∀ (p : Polynomial ↥R) (hp : p.leadingCoeff - 1 ∈ ⊥) (hpx : (Polynomial.aeval ⅟x⁻¹) p = 0), motive ⋯) → motive x_1","typeReferences":[["Invertible","invOf"],["Membership","mem"],["MulZeroClass","toMul"],["Exists","intro"],["AddGroupWithOne","toAddMonoidWithOne"],["CommGroupWithZero","toDivisionCommMonoid"],["MonoidWithZero","toMulZeroOneClass"],["Algebra","id"],["And","intro"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["Ring","toAddGroupWithOne"],["SubNegMonoid","toSub"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["HSub","hSub"],["Semifield","toDivisionSemiring"],["AddGroup","toSubNegMonoid"],["Semiring","toModule"],["Invertible"],["Subring","instSetLike"],["InvOneClass","toInv"],["SetLike","instMembership"],["Exists"],["NonUnitalNonAssocSemiring","toDistrib"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["InvOneClass","toOne"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["And"],["SubringClass","toSubsemiringClass"],["DivisionSemiring","toGroupWithZero"],["Bot","bot"],["Polynomial","aeval"],["Polynomial"],["MulZeroOneClass","toMulZeroClass"],["AddMonoidWithOne","toOne"],["Submodule","instBot"],["Polynomial","leadingCoeff"],["DivisionMonoid","toDivInvOneMonoid"],["Polynomial","algebraOfAlgebra"],["GroupWithZero","toDivisionMonoid"],["Semifield","toCommSemiring"],["Subtype"],["EuclideanDomain","toCommRing"],["SubsemiringClass","toCommSemiring"],["Field"],["Subring","toCommRing"],["CommRing","toNonUnitalCommRing"],["invertibleInv"],["DFunLike","coe"],["Field","toEuclideanDomain"],["Ideal"],["GroupWithZero","toMonoidWithZero"],["AlgHom","funLike"],["Zero","toOfNat0"],["NonAssocCommRing","toNonAssocRing"],["Eq"],["DivisionCommMonoid","toDivisionMonoid"],["AlgHom"],["Inv","inv"],["CommRing","toCommSemiring"],["CommRing","toNonAssocCommRing"],["Subring"],["Distrib","toMul"],["DivisionRing","toRing"],["Algebra","ofSubring"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Field","toDivisionRing"],["Polynomial","semiring"],["Subring","instSubringClass"],["OfNat","ofNat"],["CommRing","toRing"],["DivInvOneMonoid","toInvOneClass"],["AddGroupWithOne","toAddGroup"],["One","toOfNat1"],["MulZeroClass","toZero"],["Submodule","setLike"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Field","toSemifield"],["instHSub"],["Semifield","toCommGroupWithZero"]],"valueReferences":[["Invertible","invOf"],["Membership","mem"],["Exists","intro"],["AddGroupWithOne","toAddMonoidWithOne"],["CommGroupWithZero","toDivisionCommMonoid"],["Algebra","id"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["Ring","toAddGroupWithOne"],["SubNegMonoid","toSub"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["HSub","hSub"],["Semifield","toDivisionSemiring"],["AddGroup","toSubNegMonoid"],["Semiring","toModule"],["Subring","instSetLike"],["InvOneClass","toInv"],["SetLike","instMembership"],["NonUnitalNonAssocSemiring","toDistrib"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["And"],["SubringClass","toSubsemiringClass"],["DivisionSemiring","toGroupWithZero"],["Bot","bot"],["Polynomial","aeval"],["Exists","casesOn"],["Polynomial"],["Submodule","instBot"],["Polynomial","leadingCoeff"],["AddMonoidWithOne","toOne"],["DivisionMonoid","toDivInvOneMonoid"],["Polynomial","algebraOfAlgebra"],["Semifield","toCommSemiring"],["SubsemiringClass","toCommSemiring"],["Subtype"],["EuclideanDomain","toCommRing"],["Subring","toCommRing"],["CommRing","toNonUnitalCommRing"],["invertibleInv"],["DFunLike","coe"],["Ideal"],["Field","toEuclideanDomain"],["AlgHom","funLike"],["Zero","toOfNat0"],["NonAssocCommRing","toNonAssocRing"],["Eq"],["DivisionCommMonoid","toDivisionMonoid"],["AlgHom"],["Inv","inv"],["CommRing","toCommSemiring"],["CommRing","toNonAssocCommRing"],["Subring"],["Distrib","toMul"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Algebra","ofSubring"],["DivisionRing","toRing"],["Field","toDivisionRing"],["Polynomial","semiring"],["Subring","instSubringClass"],["OfNat","ofNat"],["CommRing","toRing"],["DivInvOneMonoid","toInvOneClass"],["AddGroupWithOne","toAddGroup"],["One","toOfNat1"],["MulZeroClass","toZero"],["Submodule","setLike"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Field","toSemifield"],["instHSub"],["And","casesOn"],["Semifield","toCommGroupWithZero"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","RingTheory","Valuation","LocalSubring",0,"Subring","exists_le_valuationSubring_of_isIntegrallyClosedIn","match_1_4"],"typeFallback":"forall {K : Type.{u_1}} [inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11 : Field.{u_1} K] {x : K} {R : Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))} (xinv : Subtype.{succ u_1} K (fun (x_1 : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (Ring.toNonAssocRing.{u_1} K (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (Ring.toNonAssocRing.{u_1} K (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) K (Subring.instSetLike.{u_1} K (Ring.toNonAssocRing.{u_1} K (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (Subalgebra.toSubring.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) K (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) R) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) R) (Algebra.adjoin.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) K (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) R K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))))) (DivisionSemiring.toSemiring.{u_1} K (Semifield.toDivisionSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) R) (Singleton.singleton.{u_1, u_1} K (Set.{u_1} K) (Set.instSingletonSet.{u_1} K) (Inv.inv.{u_1} K (InvOneClass.toInv.{u_1} K (DivInvOneMonoid.toInvOneClass.{u_1} K (DivisionMonoid.toDivInvOneMonoid.{u_1} K (DivisionCommMonoid.toDivisionMonoid.{u_1} K (CommGroupWithZero.toDivisionCommMonoid.{u_1} K (Semifield.toCommGroupWithZero.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))))) x)))) x_1)), let B : Subalgebra.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) K (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) R K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))))) (DivisionSemiring.toSemiring.{u_1} K (Semifield.toDivisionSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) R) := Algebra.adjoin.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) K (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) R K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))))) (DivisionSemiring.toSemiring.{u_1} K (Semifield.toDivisionSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) R) (Singleton.singleton.{u_1, u_1} K (Set.{u_1} K) (Set.instSingletonSet.{u_1} K) (Inv.inv.{u_1} K (InvOneClass.toInv.{u_1} K (DivInvOneMonoid.toInvOneClass.{u_1} K (DivisionMonoid.toDivInvOneMonoid.{u_1} K (DivisionCommMonoid.toDivisionMonoid.{u_1} K (CommGroupWithZero.toDivisionCommMonoid.{u_1} K (Semifield.toCommGroupWithZero.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))))) x)); forall (motive : (Exists.{succ u_1} (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11) (fun (B_1 : ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11) => And (LE.le.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (Preorder.toLE.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (PartialOrder.toPreorder.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (Subring.instPartialOrder.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))))) (Subalgebra.toSubring.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) K (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) R) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) R) B) (ValuationSubring.toSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11 B_1)) (HasSubset.Subset.{u_1} (Set.{u_1} K) (Set.instHasSubset.{u_1} K) (Set.image.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) (Subalgebra.toSubring.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) K (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) R) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) R) B) x)) K (DFunLike.coe.{succ u_1, succ u_1, succ u_1} (RingHom.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) (Subalgebra.toSubring.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) K (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) R) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) R) B) x)) K (NonAssocRing.toNonAssocSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) (Subalgebra.toSubring.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) K (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) R) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) R) B) x)) (SubringClass.toNonAssocRing.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (Subalgebra.toSubring.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) K (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) R) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) 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(Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) K (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) R) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) R) B) x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (Ring.toNonAssocRing.{u_1} K (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) (Subalgebra.toSubring.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) K (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) R) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) R) B) K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (Ring.toNonAssocRing.{u_1} K (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (Ring.toNonAssocRing.{u_1} K (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (Ring.toNonAssocRing.{u_1} K (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) (Subring.instSubringClass.{u_1} K (Ring.toNonAssocRing.{u_1} K (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))))) (Singleton.singleton.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (Ring.toNonAssocRing.{u_1} K (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (Ring.toNonAssocRing.{u_1} K (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) K (Subring.instSetLike.{u_1} K (Ring.toNonAssocRing.{u_1} K (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (Subalgebra.toSubring.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) K (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) R) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) R) B) x)) (Set.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (Ring.toNonAssocRing.{u_1} K (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (Ring.toNonAssocRing.{u_1} K (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) K (Subring.instSetLike.{u_1} K (Ring.toNonAssocRing.{u_1} K (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (Subalgebra.toSubring.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) K (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) R) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) R) B) x))) (Set.instSingletonSet.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (Ring.toNonAssocRing.{u_1} K (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (Ring.toNonAssocRing.{u_1} K (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) K (Subring.instSetLike.{u_1} K (Ring.toNonAssocRing.{u_1} K (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (Subalgebra.toSubring.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) K (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) R) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) R) B) x))) xinv)))) (SetLike.coe.{u_1, u_1} (NonUnitalSubring.{u_1} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} K (CommRing.toNonUnitalCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (NonUnitalSubring.instSetLike.{u_1} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} K (CommRing.toNonUnitalCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (ValuationSubring.nonunits.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11 B_1))))) -> Prop) (x._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.277 : Exists.{succ u_1} (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11) (fun (B_1 : ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11) => And (LE.le.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (Preorder.toLE.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (PartialOrder.toPreorder.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (Subring.instPartialOrder.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K 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(Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) R) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) R) B) (ValuationSubring.toSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11 B_1)) (HasSubset.Subset.{u_1} (Set.{u_1} 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(SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))))) R x)) K (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) R) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11)))) R) B) x))) xinv)))) (SetLike.coe.{u_1, u_1} (NonUnitalSubring.{u_1} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} K (CommRing.toNonUnitalCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (NonUnitalSubring.instSetLike.{u_1} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} K (CommRing.toNonUnitalCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (ValuationSubring.nonunits.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11 B_1)))) V hV)) -> (motive x._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.277)","typeFull":"∀ {K : Type u_1} [inst : Field K] {x : K} {R : Subring K} (xinv : ↥(↥R)[x⁻¹].toSubring),\n  let B := (↥R)[x⁻¹];\n  ∀\n    (motive :\n      (∃ B_1, B.toSubring ≤ B_1.toSubring ∧ ⇑B.toSubring.subtype '' ↑(Ideal.span {xinv}) ⊆ ↑B_1.nonunits) → Prop)\n    (x : ∃ B_1, B.toSubring ≤ B_1.toSubring ∧ ⇑B.toSubring.subtype '' ↑(Ideal.span {xinv}) ⊆ ↑B_1.nonunits),\n    (∀ (V : ValuationSubring K)\n        (hV : B.toSubring ≤ V.toSubring ∧ ⇑B.toSubring.subtype '' ↑(Ideal.span {xinv}) ⊆ ↑V.nonunits), motive ⋯) →\n      motive x","typeReadable":"∀ {K : Type u_1} [inst : Field K] {x : K} {R : Subring K} (xinv : ↥(↥R)[x⁻¹].toSubring),\n  let B := (↥R)[x⁻¹];\n  ∀\n    (motive :\n      (∃ B_1, B.toSubring ≤ B_1.toSubring ∧ ⇑B.toSubring.subtype '' ↑(Ideal.span {xinv}) ⊆ ↑B_1.nonunits) → Prop)\n    (x : ∃ B_1, B.toSubring ≤ B_1.toSubring ∧ ⇑B.toSubring.subtype '' ↑(Ideal.span {xinv}) ⊆ ↑B_1.nonunits),\n    (∀ (V : ValuationSubring K)\n        (hV : B.toSubring ≤ V.toSubring ∧ ⇑B.toSubring.subtype '' ↑(Ideal.span {xinv}) ⊆ ↑V.nonunits), motive ⋯) →\n      motive x","typeReferences":[["RingHom"],["Ring","toNonAssocRing"],["ValuationSubring"],["PartialOrder","toPreorder"],["Subring","instPartialOrder"],["Singleton","singleton"],["Membership","mem"],["Exists","intro"],["CommGroupWithZero","toDivisionCommMonoid"],["NonUnitalSubring","instSetLike"],["Algebra","id"],["Semiring","toNonAssocSemiring"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Semifield","toDivisionSemiring"],["Semiring","toModule"],["Subring","instSetLike"],["SetLike","instMembership"],["InvOneClass","toInv"],["Exists"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["And"],["SubringClass","toSubsemiringClass"],["ValuationSubring","toSubring"],["DivisionSemiring","toSemiring"],["ValuationSubring","nonunits"],["SubringClass","toNonAssocRing"],["Set","image"],["SetLike","coe"],["NonAssocRing","toNonAssocSemiring"],["HasSubset","Subset"],["Subring","subtype"],["Subalgebra","toSubring"],["DivisionMonoid","toDivInvOneMonoid"],["Semifield","toCommSemiring"],["Ideal","span"],["SubsemiringClass","toCommSemiring"],["EuclideanDomain","toCommRing"],["Subtype"],["Field"],["NonUnitalSubring"],["RingHom","instFunLike"],["Algebra","adjoin"],["Subring","toCommRing"],["CommRing","toNonUnitalCommRing"],["DFunLike","coe"],["Ideal"],["Field","toEuclideanDomain"],["NonAssocCommRing","toNonAssocRing"],["Preorder","toLE"],["DivisionCommMonoid","toDivisionMonoid"],["CommRing","toCommSemiring"],["Inv","inv"],["Field","toCommRing"],["CommRing","toNonAssocCommRing"],["Set"],["Subring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Algebra","ofSubring"],["DivisionRing","toRing"],["Field","toDivisionRing"],["Subring","instSubringClass"],["Set","instSingletonSet"],["Set","instHasSubset"],["DivInvOneMonoid","toInvOneClass"],["Submodule","setLike"],["LE","le"],["Field","toSemifield"],["Semifield","toCommGroupWithZero"]],"valueReferences":[["RingHom"],["Ring","toNonAssocRing"],["ValuationSubring"],["PartialOrder","toPreorder"],["Subring","instPartialOrder"],["Singleton","singleton"],["Membership","mem"],["CommGroupWithZero","toDivisionCommMonoid"],["NonUnitalSubring","instSetLike"],["Algebra","id"],["Semiring","toNonAssocSemiring"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Semifield","toDivisionSemiring"],["Semiring","toModule"],["Subring","instSetLike"],["SetLike","instMembership"],["InvOneClass","toInv"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["And"],["SubringClass","toSubsemiringClass"],["ValuationSubring","toSubring"],["DivisionSemiring","toSemiring"],["ValuationSubring","nonunits"],["Exists","casesOn"],["Set","image"],["SubringClass","toNonAssocRing"],["SetLike","coe"],["HasSubset","Subset"],["NonAssocRing","toNonAssocSemiring"],["Subring","subtype"],["Subalgebra","toSubring"],["DivisionMonoid","toDivInvOneMonoid"],["Semifield","toCommSemiring"],["Ideal","span"],["Subtype"],["EuclideanDomain","toCommRing"],["SubsemiringClass","toCommSemiring"],["NonUnitalSubring"],["RingHom","instFunLike"],["Algebra","adjoin"],["Subring","toCommRing"],["CommRing","toNonUnitalCommRing"],["DFunLike","coe"],["Ideal"],["Field","toEuclideanDomain"],["NonAssocCommRing","toNonAssocRing"],["Preorder","toLE"],["DivisionCommMonoid","toDivisionMonoid"],["CommRing","toCommSemiring"],["Inv","inv"],["Field","toCommRing"],["Set"],["CommRing","toNonAssocCommRing"],["Subring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["DivisionRing","toRing"],["Algebra","ofSubring"],["Field","toDivisionRing"],["Subring","instSubringClass"],["Set","instSingletonSet"],["Set","instHasSubset"],["DivInvOneMonoid","toInvOneClass"],["Submodule","setLike"],["LE","le"],["Field","toSemifield"],["Semifield","toCommGroupWithZero"]]},{"isProp":true,"kind":"theorem","name":["LocalSubring","eq_iInf_of_isIntegrallyClosedIn"],"typeFallback":"forall {K : Type.{u_3}} [inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11 : Field.{u_3} K] {R : LocalSubring.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11))} [inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.16 : IsIntegrallyClosedIn.{u_3, u_3} (Subtype.{succ u_3} K (fun (x : K) => Membership.mem.{u_3, u_3} K (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11))))) (SetLike.instMembership.{u_3, u_3} (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11)) R) x)) K (Subring.toCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11)) R)) (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11)) (Algebra.ofSubring.{u_3, u_3} K K 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inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))))) M) (LE.le.{u_1} (Ideal.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))))))) (Preorder.toLE.{u_1} (Ideal.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))))))) (PartialOrder.toPreorder.{u_1} (Ideal.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))))))) (Submodule.instPartialOrder.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (Semiring.toNonAssocSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))))))))) (Semiring.toModule.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))))))))) I M))) -> Prop) (x._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.75 : Exists.{succ u_1} (Ideal.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))))))) (fun (M : Ideal.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K 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(Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))))) 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(Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K 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(CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))))))) (PartialOrder.toPreorder.{u_1} (Ideal.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K 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(Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (Semiring.toNonAssocSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K 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(SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))))))))) (Semiring.toModule.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K 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inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))))))))) I M))), (forall (M : Ideal.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))))))) (hM : Ideal.IsMaximal.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))))) M) (le : LE.le.{u_1} (Ideal.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))))))) (Preorder.toLE.{u_1} (Ideal.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))))))) (PartialOrder.toPreorder.{u_1} (Ideal.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K 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(EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} (Subtype.{succ u_1} K (fun (x : K) => 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(CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (Semiring.toNonAssocSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))))))))) (Semiring.toModule.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))))))))) I M), motive (Exists.intro.{succ u_1} (Ideal.{u_1} (Subtype.{succ u_1} K (fun (x : K) => 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(CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K 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(NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))))))) => And (Ideal.IsMaximal.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))))) M) (LE.le.{u_1} (Ideal.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))))))) (Preorder.toLE.{u_1} (Ideal.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))))))) 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(EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (Semiring.toNonAssocSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} 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(Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))))))) (PartialOrder.toPreorder.{u_1} (Ideal.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} 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(CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))))))))) I M) hM le))) -> (motive x._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.75)","typeFull":"∀ {K : Type u_1} [inst : Field K] {A : Subring K} (I : Ideal ↥A) (motive : (∃ M, M.IsMaximal ∧ I ≤ M) → Prop)\n  (x : ∃ M, M.IsMaximal ∧ I ≤ M), (∀ (M : Ideal ↥A) (hM : M.IsMaximal) (le : I ≤ M), motive ⋯) → motive x","typeReadable":"∀ {K : Type u_1} [inst : Field K] {A : Subring K} (I : Ideal ↥A) (motive : (∃ M, M.IsMaximal ∧ I ≤ M) → Prop)\n  (x : ∃ M, M.IsMaximal ∧ I ≤ M), (∀ (M : Ideal ↥A) (hM : M.IsMaximal) (le : I ≤ M), motive ⋯) → motive 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{R : Type.{u_2}} {K : Type.{u_1}} [inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2040120653._hygCtx._hyg.5 : CommRing.{u_2} R] [inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2040120653._hygCtx._hyg.11 : Field.{u_1} K] [inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2040120653._hygCtx._hyg.23 : Algebra.{u_2, u_1} R K (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2040120653._hygCtx._hyg.5) (DivisionSemiring.toSemiring.{u_1} K (Semifield.toDivisionSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2040120653._hygCtx._hyg.11)))] (x : K) (motive : (Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2040120653._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K 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(Subring.instSetLike.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (SubsemiringClass.toCommSemiring.{u_3, u_3} (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A K (Semifield.toCommSemiring.{u_3} K (Field.toSemifield.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_3, u_3} (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))))))) (Submodule.instTop.{u_3, u_3} (Subtype.{succ u_3} K (fun (x : K) => Membership.mem.{u_3, u_3} K (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_3, u_3} (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (Subtype.{succ u_3} K (fun (x : K) => Membership.mem.{u_3, u_3} K (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_3, u_3} (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_3} (Subtype.{succ u_3} K (fun (x : K) => Membership.mem.{u_3, u_3} K (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K 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(Semifield.toCommSemiring.{u_3} K (Field.toSemifield.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_3, u_3} (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} (Subtype.{succ u_3} K (fun (x : K) => Membership.mem.{u_3, u_3} K (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_3, u_3} (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} (Subtype.{succ u_3} K (fun (x : K) => Membership.mem.{u_3, u_3} K (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_3, u_3} (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (Semiring.toNonAssocSemiring.{u_3} (Subtype.{succ u_3} K (fun (x : K) => Membership.mem.{u_3, u_3} K (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_3, u_3} (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_3} (Subtype.{succ u_3} K (fun (x : K) => Membership.mem.{u_3, u_3} K (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_3, u_3} (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (SubsemiringClass.toCommSemiring.{u_3, u_3} (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A K (Semifield.toCommSemiring.{u_3} K (Field.toSemifield.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_3, u_3} (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K 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(Semifield.toCommSemiring.{u_3} K (Field.toSemifield.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_3, u_3} (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K 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(Subring.instSetLike.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (SubsemiringClass.toCommSemiring.{u_3, u_3} (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A K (Semifield.toCommSemiring.{u_3} K (Field.toSemifield.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_3, u_3} (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))))))) 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(NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_3, u_3} (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (Subtype.{succ u_3} K (fun (x : K) => Membership.mem.{u_3, u_3} K (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K 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(Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} (Subtype.{succ u_3} K (fun (x : K) => Membership.mem.{u_3, u_3} K (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_3, u_3} (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K 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(Subring.instSetLike.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (Semiring.toNonAssocSemiring.{u_3} (Subtype.{succ u_3} K (fun (x : K) => Membership.mem.{u_3, u_3} K (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_3, u_3} (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} 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(CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SetLike.instMembership.{u_3, u_3} (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) A x)) (CommSemiring.toSemiring.{u_3} (Subtype.{succ u_3} K (fun (x : K) => Membership.mem.{u_3, u_3} K (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K 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{K : Type.{u_3}} [inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2265445946._hygCtx._hyg.11 : Field.{u_3} K], Function.Injective.{succ u_3, succ u_3} (ValuationSubring.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2265445946._hygCtx._hyg.11) (LocalSubring.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2265445946._hygCtx._hyg.11))) (ValuationSubring.toLocalSubring.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2265445946._hygCtx._hyg.11)","typeFull":"∀ {K : Type u_3} [inst : Field K], Function.Injective ValuationSubring.toLocalSubring","typeReadable":"∀ {K : Type u_3} [inst : Field K], Function.Injective ValuationSubring.toLocalSubring","typeReferences":[["EuclideanDomain","toCommRing"],["ValuationSubring"],["Field","toEuclideanDomain"],["Field"],["ValuationSubring","toLocalSubring"],["LocalSubring"],["Function","Injective"]],"valueReferences":[["ValuationSubring","toSubring_injective"],["Field","toCommRing"],["EuclideanDomain","toCommRing"],["CommRing","toNonAssocCommRing"],["Subring"],["ValuationSubring","toSubring"],["LocalSubring"],["congrArg"],["LocalSubring","toSubring"],["Field","toEuclideanDomain"],["id"],["NonAssocCommRing","toNonAssocRing"],["Eq"],["ValuationSubring","toLocalSubring"]]},{"isProp":true,"kind":"theorem","name":["LocalSubring","map_maximalIdeal_eq_top_of_isMax"],"typeFallback":"forall {K : Type.{u_3}} [inst._@.Mathlib.RingTheory.Valuation.LocalSubring.4260081845._hygCtx._hyg.11 : Field.{u_3} K] {R : LocalSubring.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K 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{R : Type.{u_1}} {S : Type.{u_2}} {K : Type.{u_3}} [inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2040120653._hygCtx._hyg.5 : CommRing.{u_1} R] [inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2040120653._hygCtx._hyg.8 : CommRing.{u_2} S] [inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2040120653._hygCtx._hyg.11 : Field.{u_3} K] [inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2040120653._hygCtx._hyg.14 : IsDomain.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2040120653._hygCtx._hyg.5))] [inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2040120653._hygCtx._hyg.17 : ValuationRing.{u_1} R inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2040120653._hygCtx._hyg.5 inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2040120653._hygCtx._hyg.14] [inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2040120653._hygCtx._hyg.20 : IsLocalRing.{u_2} S (CommSemiring.toSemiring.{u_2} S (CommRing.toCommSemiring.{u_2} S inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2040120653._hygCtx._hyg.8))] [inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2040120653._hygCtx._hyg.23 : Algebra.{u_1, u_3} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2040120653._hygCtx._hyg.5) (DivisionSemiring.toSemiring.{u_3} K (Semifield.toDivisionSemiring.{u_3} K (Field.toSemifield.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2040120653._hygCtx._hyg.11)))] [inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2040120653._hygCtx._hyg.27 : IsFractionRing.{u_1, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2040120653._hygCtx._hyg.5) K (Semifield.toCommSemiring.{u_3} K (Field.toSemifield.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2040120653._hygCtx._hyg.11)) inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2040120653._hygCtx._hyg.23] (f : RingHom.{u_1, u_2} R S 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f)","typeReferences":[["Semifield","toCommSemiring"],["RingHom"],["EuclideanDomain","toCommRing"],["Subtype"],["Field"],["Membership","mem"],["RingHom","instFunLike"],["IsDomain"],["DFunLike","coe"],["Semiring","toNonAssocSemiring"],["Field","toEuclideanDomain"],["MonoidWithZero","toMonoid"],["RingHom","range"],["NonAssocCommRing","toNonAssocRing"],["Semifield","toDivisionSemiring"],["Eq"],["RingHom","comp"],["ValuationRing"],["Subring","instSetLike"],["CommRing","toCommSemiring"],["SetLike","instMembership"],["CommRing","toNonAssocCommRing"],["Subring"],["CommSemiring","toSemiring"],["IsLocalHom"],["Semiring","toMonoidWithZero"],["Function","Bijective"],["Subring","instSubringClass"],["DivisionSemiring","toSemiring"],["IsLocalRing"],["CommRing"],["RingHom","rangeRestrict"],["Algebra"],["SubringClass","toNonAssocRing"],["NonAssocRing","toNonAssocSemiring"],["Field","toSemifield"],["IsFractionRing"],["algebraMap"]],"valueReferences":[["RingHom"],["PartialOrder","toPreorder"],["Subring","instPartialOrder"],["Eq","trans"],["Membership","mem"],["Exists","intro"],["IsLocalHom","of_surjective"],["MonoidWithZeroHomClass","toMonoidHomClass"],["RingHom","instRingHomClass"],["Subtype","val"],["Subring","inclusion"],["And","intro"],["Semiring","toNonAssocSemiring"],["_private","Mathlib","RingTheory","Valuation","LocalSubring",0,"bijective_rangeRestrict_comp_of_valuationRing","match_1_1"],["eq_of_heq"],["EuclideanDomain","toNontrivial"],["Eq","symm"],["RingHom","range"],["Semifield","toDivisionSemiring"],["Subring","inclusion","_proof_1"],["Eq","rec"],["Eq","ndrec"],["ValuationSubring","toLocalSubring"],["NonAssocSemiring","toMulZeroOneClass"],["HEq","casesOn"],["Subring","instSetLike"],["rfl"],["Exists"],["SetLike","instMembership"],["IsFractionRing","injective"],["Subtype","ext"],["SubringClass","toSubsemiringClass"],["DivisionSemiring","toSemiring"],["ValuationSubring","mk"],["Exists","casesOn"],["SubringClass","toNonAssocRing"],["NonAssocRing","toNonAssocSemiring"],["Eq","refl"],["HEq"],["id"],["Eq","mpr"],["algebraMap"],["Semifield","toCommSemiring"],["SubsemiringClass","toCommSemiring"],["Subtype"],["EuclideanDomain","toCommRing"],["Eq","mp"],["ValuationSubring","isMax_toLocalSubring"],["RingHom","instFunLike"],["Subring","toCommRing"],["DFunLike","coe"],["LocalSubring","instPartialOrder","_proof_2"],["congrArg"],["Subring","instNontrivialSubtypeMem"],["Field","toEuclideanDomain"],["IsUnit","of_map"],["RingHomClass","toMonoidWithZeroHomClass"],["Subtype","property"],["MonoidWithZero","toMonoid"],["congrFun'"],["NonAssocCommRing","toNonAssocRing"],["Preorder","toLE"],["RingHom","comp"],["Eq"],["IsLocalHom","mk"],["CommRing","toCommSemiring"],["IsLocalHom","map_nonunit"],["True"],["HEq","refl"],["CommRing","toNonAssocCommRing"],["LocalSubring","copy","_proof_2"],["Function","Injective","of_comp"],["Subring"],["ValuationRing","isInteger_or_isInteger"],["IsLocalHom"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["LocalSubring","range"],["IsUnit"],["Subring","instSubringClass"],["Function","comp"],["Function","Surjective"],["RingHom","rangeRestrict"],["Eq","casesOn"],["RingHom","rangeRestrict_surjective"],["eq_self"],["IsUnit","map"],["LocalSubring","toSubring"],["of_eq_true"],["LE","le"],["Field","toSemifield"],["Subtype","mk"],["Subtype","casesOn"],["Function","Injective"]]},{"isProp":true,"kind":"theorem","name":["Subring","exists_le_valuationSubring_of_isIntegrallyClosedIn"],"typeFallback":"forall {K : Type.{u_3}} [inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11 : Field.{u_3} K] {x : K} {R : Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))}, (Not (Membership.mem.{u_3, u_3} K (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) (SetLike.instMembership.{u_3, u_3} (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600441._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_3} K 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(CommRing.toCommSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R))))) (fun (p : Polynomial.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K 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(EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (CommRing.toCommSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R))))) => And (Membership.mem.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (Ideal.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K 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(EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (CommRing.toCommSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R))))) (SetLike.instMembership.{u_1, u_1} (Ideal.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K 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(NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (CommRing.toCommSemiring.{u_1} (Subtype.{succ 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inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R)))) (Algebra.id.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (CommRing.toCommSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R))))) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R))) (Polynomial.aeval.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) K (CommRing.toCommSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R))) (CommSemiring.toSemiring.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R)) (Invertible.invOf.{u_1} K (Distrib.toMul.{u_1} K (NonUnitalNonAssocSemiring.toDistrib.{u_1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} K (CommRing.toNonUnitalCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))))) (AddMonoidWithOne.toOne.{u_1} K (AddGroupWithOne.toAddMonoidWithOne.{u_1} K (Ring.toAddGroupWithOne.{u_1} K (CommRing.toRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) x this)) p) (OfNat.ofNat.{u_1} K 0 (Zero.toOfNat0.{u_1} K (MulZeroClass.toZero.{u_1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} K (CommRing.toNonUnitalCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))))))))))) -> Prop) (x._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.263 : Exists.{succ u_1} (Polynomial.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (CommRing.toCommSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R))))) (fun (p : Polynomial.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (CommRing.toCommSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R))))) => And (Membership.mem.{u_1, u_1} 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(EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K 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(EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R))))) (SetLike.instMembership.{u_1, u_1} (Ideal.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K 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(Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K 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(NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K 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(CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R)) (Invertible.invOf.{u_1} K (Distrib.toMul.{u_1} K (NonUnitalNonAssocSemiring.toDistrib.{u_1} K 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(NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R))))) (fun (p : Polynomial.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K 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Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R))))) => And (Membership.mem.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (Ideal.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (CommRing.toCommSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R))))) (SetLike.instMembership.{u_1, u_1} (Ideal.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (CommRing.toCommSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R))))) (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (Submodule.setLike.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (CommRing.toCommSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11)) R) x)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.920508514._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K 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(EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1605835150._hygCtx._hyg.11)) (DivisionRing.toRing.{u_3} K (Field.toDivisionRing.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1605835150._hygCtx._hyg.11)) (Algebra.id.{u_3} K (CommRing.toCommSemiring.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1605835150._hygCtx._hyg.11)))) (Subring.closure.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1605835150._hygCtx._hyg.11)))) s)) (integralClosure.{u_3, u_3} (Subtype.{succ u_3} K (fun (x : K) => Membership.mem.{u_3, u_3} K (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K 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(Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1605835150._hygCtx._hyg.11)) (Subring.closure.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1605835150._hygCtx._hyg.11)))) s)) (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1605835150._hygCtx._hyg.11)) (Algebra.ofSubring.{u_3, u_3} K K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1605835150._hygCtx._hyg.11)) (DivisionRing.toRing.{u_3} K (Field.toDivisionRing.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1605835150._hygCtx._hyg.11)) (Algebra.id.{u_3} K (CommRing.toCommSemiring.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1605835150._hygCtx._hyg.11)))) (Subring.closure.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1605835150._hygCtx._hyg.11)))) s))))","typeFull":"∀ {K : Type u_3} [inst : Field K] {s : Set K},\n  ⨅ V, (↑V).toSubring = (integralClosure (↥(Subring.closure s)) K).toSubring","typeReadable":"∀ {K : Type u_3} [inst : Field K] {s : Set K},\n  ⨅ V, (↑V).toSubring = (integralClosure (↥(Subring.closure s)) K).toSubring","typeReferences":[["integralClosure"],["ValuationSubring"],["Subtype"],["EuclideanDomain","toCommRing"],["Field"],["Membership","mem"],["Subring","toCommRing"],["Algebra","id"],["Subtype","val"],["Subring","closure"],["Field","toEuclideanDomain"],["iInf"],["NonAssocCommRing","toNonAssocRing"],["Subring","instInfSet"],["Eq"],["Subring","instSetLike"],["CommRing","toCommSemiring"],["Field","toCommRing"],["SetLike","instMembership"],["CommRing","toNonAssocCommRing"],["Set"],["Subring"],["Algebra","ofSubring"],["DivisionRing","toRing"],["Field","toDivisionRing"],["ValuationSubring","toSubring"],["Set","instHasSubset"],["SetLike","coe"],["HasSubset","Subset"],["Subalgebra","toSubring"]],"valueReferences":[["integralClosure"],["Ring","toNonAssocRing"],["PartialOrder","toPreorder"],["ValuationSubring"],["Subring","instPartialOrder"],["Eq","trans"],["Membership","mem"],["Subtype","val"],["Algebra","id"],["Subring","closure"],["funext"],["eq_of_heq"],["Eq","symm"],["Eq","ndrec"],["Subring","instSetLike"],["SetLike","instMembership"],["Algebra","ofSubsemiring"],["iff_of_eq"],["SubringClass","toCommRing"],["ValuationSubring","toSubring"],["iInf_subtype"],["SetLike","coe"],["HasSubset","Subset"],["Iff","of_eq"],["Iff"],["Eq","refl"],["HEq"],["id"],["Subalgebra","toSubring"],["Eq","mpr"],["instIsIntegrallyClosedInSubtypeMemSubringToSubringIntegralClosure"],["EuclideanDomain","toCommRing"],["Subtype"],["Subring","ofClass"],["Subring","toCommRing"],["SubringClass","toRing","_proof_8"],["congrArg"],["Subring","eq_iInf_of_isIntegrallyClosedIn"],["Field","toEuclideanDomain"],["Subring","instCompleteLattice"],["congr"],["iInf"],["Subring","toCommRing","_proof_1"],["Subring","instInfSet"],["NonAssocCommRing","toNonAssocRing"],["Preorder","toLE"],["Eq"],["propext"],["Subring","integralClosure_subring_le_iff"],["CommRing","toCommSemiring"],["Field","toCommRing"],["HEq","refl"],["CommRing","toNonAssocCommRing"],["Set"],["Subring"],["CommSemiring","toSemiring"],["DivisionRing","toRing"],["Algebra","ofSubring"],["Field","toDivisionRing"],["Iff","symm"],["Eq","casesOn"],["Subring","closure_le"],["_private","Mathlib","RingTheory","Valuation","LocalSubring",0,"iInf_valuationSubring_superset","_proof_1_1"],["Set","instHasSubset"],["CompleteSemilatticeInf","toInfSet"],["CommRing","toRing"],["LE","le"],["Subtype","mk"],["CompleteLattice","toCompleteSemilatticeInf"],["iInf_congr_Prop"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","RingTheory","Valuation","LocalSubring",0,"Subring","eq_iInf_of_isIntegrallyClosedIn","match_1_1"],"typeFallback":"forall {K : Type.{u_1}} [inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11 : Field.{u_1} K] {R : Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11))))} (x._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.61 : K) (motive : (Exists.{succ u_1} (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11) (fun (V : ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11) => And (LE.le.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11))))) (Preorder.toLE.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11))))) (PartialOrder.toPreorder.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11))))) (Subring.instPartialOrder.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11))))))) R (ValuationSubring.toSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11 V)) (Not (Membership.mem.{u_1, u_1} K (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11) K (ValuationSubring.instSetLike.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11)) V x._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.61)))) -> Prop) (x._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.79 : Exists.{succ u_1} (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11) (fun (V : ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11) => And (LE.le.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11))))) (Preorder.toLE.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11))))) (PartialOrder.toPreorder.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11))))) (Subring.instPartialOrder.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11))))))) R (ValuationSubring.toSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11 V)) (Not (Membership.mem.{u_1, u_1} K (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11) K (ValuationSubring.instSetLike.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11)) V x._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.61)))), (forall (V : ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11) (hV : And (LE.le.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11))))) (Preorder.toLE.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11))))) (PartialOrder.toPreorder.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11))))) (Subring.instPartialOrder.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11))))))) R (ValuationSubring.toSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11 V)) (Not (Membership.mem.{u_1, u_1} K (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11) K (ValuationSubring.instSetLike.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11)) V x._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.61))), motive (Exists.intro.{succ u_1} (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11) (fun (V : ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11) => And (LE.le.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11))))) (Preorder.toLE.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11))))) (PartialOrder.toPreorder.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11))))) (Subring.instPartialOrder.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11))))))) R (ValuationSubring.toSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11 V)) (Not (Membership.mem.{u_1, u_1} K (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11) K (ValuationSubring.instSetLike.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.11)) V x._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.61))) V hV)) -> (motive x._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061371._hygCtx._hyg.79)","typeFull":"∀ {K : Type u_1} [inst : Field K] {R : Subring K} (x : K) (motive : (∃ V, R ≤ V.toSubring ∧ x ∉ V) → Prop)\n  (x_1 : ∃ V, R ≤ V.toSubring ∧ x ∉ V),\n  (∀ (V : ValuationSubring K) (hV : R ≤ V.toSubring ∧ x ∉ V), motive ⋯) → motive x_1","typeReadable":"∀ {K : Type u_1} [inst : Field K] {R : Subring K} (x : K) (motive : (∃ V, R ≤ V.toSubring ∧ x ∉ V) → Prop)\n  (x_1 : ∃ V, R ≤ V.toSubring ∧ x ∉ V),\n  (∀ (V : ValuationSubring K) (hV : R ≤ V.toSubring ∧ x ∉ V), motive ⋯) → motive x_1","typeReferences":[["Not"],["SetLike","instMembership"],["EuclideanDomain","toCommRing"],["Exists"],["ValuationSubring"],["PartialOrder","toPreorder"],["Subring","instPartialOrder"],["Field"],["CommRing","toNonAssocCommRing"],["Subring"],["Membership","mem"],["And"],["Exists","intro"],["ValuationSubring","toSubring"],["ValuationSubring","instSetLike"],["Field","toEuclideanDomain"],["LE","le"],["NonAssocCommRing","toNonAssocRing"],["Preorder","toLE"]],"valueReferences":[["Not"],["SetLike","instMembership"],["ValuationSubring"],["EuclideanDomain","toCommRing"],["PartialOrder","toPreorder"],["Subring","instPartialOrder"],["CommRing","toNonAssocCommRing"],["Subring"],["Membership","mem"],["And"],["ValuationSubring","toSubring"],["ValuationSubring","instSetLike"],["Exists","casesOn"],["Field","toEuclideanDomain"],["LE","le"],["NonAssocCommRing","toNonAssocRing"],["Preorder","toLE"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","RingTheory","Valuation","LocalSubring",0,"LocalSubring","eq_iInf_of_isIntegrallyClosedIn","match_1_1"],"typeFallback":"forall {K : Type.{u_1}} [inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11 : Field.{u_1} K] {R : LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11))} (x._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.61 : K) (motive : (Exists.{succ u_1} (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11) (fun (V : ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11) => And (LE.le.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11))) (Preorder.toLE.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11))) (PartialOrder.toPreorder.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11))) (LocalSubring.instPartialOrder.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11))))) R (ValuationSubring.toLocalSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11 V)) (Not (Membership.mem.{u_1, u_1} K (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11) K (ValuationSubring.instSetLike.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11)) V x._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.61)))) -> Prop) (x._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.79 : Exists.{succ u_1} (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11) (fun (V : ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11) => And (LE.le.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11))) (Preorder.toLE.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11))) (PartialOrder.toPreorder.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11))) (LocalSubring.instPartialOrder.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11))))) R (ValuationSubring.toLocalSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11 V)) (Not (Membership.mem.{u_1, u_1} K (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11) K (ValuationSubring.instSetLike.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11)) V x._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.61)))), (forall (V : ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11) (hV : And (LE.le.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11))) (Preorder.toLE.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11))) (PartialOrder.toPreorder.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11))) (LocalSubring.instPartialOrder.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11))))) R (ValuationSubring.toLocalSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11 V)) (Not (Membership.mem.{u_1, u_1} K (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11) K (ValuationSubring.instSetLike.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11)) V x._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.61))), motive (Exists.intro.{succ u_1} (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11) (fun (V : ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11) => And (LE.le.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11))) (Preorder.toLE.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11))) (PartialOrder.toPreorder.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11))) (LocalSubring.instPartialOrder.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11))))) R (ValuationSubring.toLocalSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11 V)) (Not (Membership.mem.{u_1, u_1} K (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11) K (ValuationSubring.instSetLike.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.11)) V x._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.61))) V hV)) -> (motive x._@.Mathlib.RingTheory.Valuation.LocalSubring.2887061372._hygCtx._hyg.79)","typeFull":"∀ {K : Type u_1} [inst : Field K] {R : LocalSubring K} (x : K) (motive : (∃ V, R ≤ V.toLocalSubring ∧ x ∉ V) → Prop)\n  (x_1 : ∃ V, R ≤ V.toLocalSubring ∧ x ∉ V),\n  (∀ (V : ValuationSubring K) (hV : R ≤ V.toLocalSubring ∧ x ∉ V), motive ⋯) → motive x_1","typeReadable":"∀ {K : Type u_1} [inst : Field K] {R : LocalSubring K} (x : K) (motive : (∃ V, R ≤ V.toLocalSubring ∧ x ∉ V) → Prop)\n  (x_1 : ∃ V, R ≤ V.toLocalSubring ∧ x ∉ V),\n  (∀ (V : ValuationSubring K) (hV : R ≤ V.toLocalSubring ∧ x ∉ V), motive ⋯) → motive x_1","typeReferences":[["Not"],["SetLike","instMembership"],["EuclideanDomain","toCommRing"],["Exists"],["ValuationSubring"],["PartialOrder","toPreorder"],["Field"],["Membership","mem"],["And"],["Exists","intro"],["ValuationSubring","instSetLike"],["LocalSubring"],["Field","toEuclideanDomain"],["LocalSubring","instPartialOrder"],["LE","le"],["Preorder","toLE"],["ValuationSubring","toLocalSubring"]],"valueReferences":[["Not"],["SetLike","instMembership"],["ValuationSubring"],["EuclideanDomain","toCommRing"],["PartialOrder","toPreorder"],["Membership","mem"],["And"],["ValuationSubring","instSetLike"],["LocalSubring"],["Exists","casesOn"],["Field","toEuclideanDomain"],["LocalSubring","instPartialOrder"],["LE","le"],["Preorder","toLE"],["ValuationSubring","toLocalSubring"]]},{"isProp":true,"kind":"theorem","name":["LocalSubring","isMax_iff"],"typeFallback":"forall {K : Type.{u_3}} [inst._@.Mathlib.RingTheory.Valuation.LocalSubring.717360183._hygCtx._hyg.11 : Field.{u_3} K] {A : LocalSubring.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.717360183._hygCtx._hyg.11))}, Iff (IsMax.{u_3} (LocalSubring.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.717360183._hygCtx._hyg.11))) (Preorder.toLE.{u_3} (LocalSubring.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.717360183._hygCtx._hyg.11))) (PartialOrder.toPreorder.{u_3} (LocalSubring.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.717360183._hygCtx._hyg.11))) (LocalSubring.instPartialOrder.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.717360183._hygCtx._hyg.11))))) A) (Exists.{succ u_3} (ValuationSubring.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.717360183._hygCtx._hyg.11) (fun (B : ValuationSubring.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.717360183._hygCtx._hyg.11) => Eq.{succ u_3} (LocalSubring.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.717360183._hygCtx._hyg.11))) (ValuationSubring.toLocalSubring.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.717360183._hygCtx._hyg.11 B) A))","typeFull":"∀ {K : Type u_3} [inst : Field K] {A : LocalSubring K}, IsMax A ↔ ∃ B, B.toLocalSubring = A","typeReadable":"∀ {K : Type u_3} [inst : Field K] {A : LocalSubring K}, IsMax A ↔ ∃ B, B.toLocalSubring = A","typeReferences":[["EuclideanDomain","toCommRing"],["PartialOrder","toPreorder"],["Exists"],["ValuationSubring"],["Field"],["LocalSubring"],["Field","toEuclideanDomain"],["Iff"],["LocalSubring","instPartialOrder"],["IsMax"],["Preorder","toLE"],["Eq"],["ValuationSubring","toLocalSubring"]],"valueReferences":[["EuclideanDomain","toCommRing"],["PartialOrder","toPreorder"],["Exists"],["ValuationSubring"],["ValuationSubring","isMax_toLocalSubring"],["LocalSubring","exists_valuationRing_of_isMax"],["LocalSubring"],["Iff","intro"],["Field","toEuclideanDomain"],["LocalSubring","instPartialOrder"],["IsMax"],["_private","Mathlib","RingTheory","Valuation","LocalSubring",0,"LocalSubring","isMax_iff","match_1_1"],["Eq","rec"],["Preorder","toLE"],["Eq"],["ValuationSubring","toLocalSubring"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","RingTheory","Valuation","LocalSubring",0,"iInf_valuationSubring_superset","_proof_1_1"],"typeFallback":"forall {K : Type.{u_1}} [inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1605835150._hygCtx._hyg.11 : Field.{u_1} K] (V : ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1605835150._hygCtx._hyg.11), IsIntegrallyClosedIn.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1605835150._hygCtx._hyg.11)))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1605835150._hygCtx._hyg.11)))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1605835150._hygCtx._hyg.11))))) (ValuationSubring.toSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1605835150._hygCtx._hyg.11 V) x)) K (Subring.toCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1605835150._hygCtx._hyg.11) (ValuationSubring.toSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1605835150._hygCtx._hyg.11 V)) (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1605835150._hygCtx._hyg.11)) (Algebra.ofSubring.{u_1, u_1} K K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1605835150._hygCtx._hyg.11) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1605835150._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1605835150._hygCtx._hyg.11))) (ValuationSubring.toSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1605835150._hygCtx._hyg.11 V))","typeFull":"∀ {K : Type u_1} [inst : Field K] (V : ValuationSubring K), IsIntegrallyClosedIn (↥V.toSubring) K","typeReadable":"∀ {K : Type u_1} [inst : Field K] (V : ValuationSubring K), IsIntegrallyClosedIn (↥V.toSubring) K","typeReferences":[["CommRing","toCommSemiring"],["EuclideanDomain","toCommRing"],["ValuationSubring"],["Subtype"],["Field","toCommRing"],["SetLike","instMembership"],["Field"],["CommRing","toNonAssocCommRing"],["Subring"],["DivisionRing","toRing"],["Algebra","ofSubring"],["Membership","mem"],["Field","toDivisionRing"],["ValuationSubring","toSubring"],["Subring","toCommRing"],["Algebra","id"],["IsIntegrallyClosedIn"],["Field","toEuclideanDomain"],["NonAssocCommRing","toNonAssocRing"],["Subring","instSetLike"]],"valueReferences":[["Semifield","toCommSemiring"],["EuclideanDomain","toCommRing"],["ValuationSubring"],["Subtype"],["SubsemiringClass","toCommSemiring"],["Membership","mem"],["CommRing","toNonUnitalCommRing"],["AddCommMonoid","toAddMonoid"],["ValuationSubring","instIsFractionRingSubtypeMem"],["Algebra","id"],["Module","Finite","self"],["instIsIntegrallyClosedSubtypeMemValuationSubring"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Field","toEuclideanDomain"],["Algebra","toModule"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Algebra","IsIntegral","of_finite"],["Semifield","toDivisionSemiring"],["IsIntegralClosure","of_isIntegrallyClosed"],["NonAssocCommRing","toNonAssocRing"],["ValuationSubring","instAlgebraSubtypeMem"],["CommRing","toCommSemiring"],["SetLike","instMembership"],["ValuationSubring","instCommRingSubtypeMem"],["IsScalarTower","left"],["CommRing","toNonAssocCommRing"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["SubringClass","toSubsemiringClass"],["DivisionSemiring","toSemiring"],["ValuationSubring","instSetLike"],["CommRing","toRing"],["Module","toDistribMulAction"],["Field","toSemifield"],["ValuationSubring","instSubringClass"]]},{"isProp":true,"kind":"theorem","name":["IsLocalRing","exists_factor_valuationRing"],"typeFallback":"forall 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(Field.toCommRing.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2029001547._hygCtx._hyg.11)))) K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (Field.toCommRing.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2029001547._hygCtx._hyg.11))) (Subring.instSetLike.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (Field.toCommRing.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2029001547._hygCtx._hyg.11)))) (Subring.instSubringClass.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (Field.toCommRing.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2029001547._hygCtx._hyg.11))))) f (ValuationSubring.toSubring.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2029001547._hygCtx._hyg.11 A) h)))","typeFull":"∀ {R : Type u_1} {K : Type u_3} [inst : CommRing R] [inst_1 : Field K] [IsLocalRing R] (f : R →+* K),\n  ∃ A, ∃ (h : ∀ (x : R), f x ∈ A.toSubring), IsLocalHom (f.codRestrict A.toSubring h)","typeReadable":"∀ {R : Type u_1} {K : Type u_3} [inst : CommRing R] [inst_1 : Field K] [IsLocalRing R] (f : R →+* K),\n  ∃ A, ∃ (h : ∀ (x : R), f x ∈ A.toSubring), IsLocalHom (f.codRestrict A.toSubring h)","typeReferences":[["Semifield","toCommSemiring"],["RingHom"],["EuclideanDomain","toCommRing"],["SubsemiringClass","toCommSemiring"],["ValuationSubring"],["Subtype"],["Field"],["Membership","mem"],["RingHom","instFunLike"],["DFunLike","coe"],["Semiring","toNonAssocSemiring"],["Field","toEuclideanDomain"],["MonoidWithZero","toMonoid"],["Semifield","toDivisionSemiring"],["NonAssocCommRing","toNonAssocRing"],["Subring","instSetLike"],["RingHom","codRestrict"],["CommRing","toCommSemiring"],["Exists"],["Field","toCommRing"],["SetLike","instMembership"],["CommRing","toNonAssocCommRing"],["Subring"],["CommSemiring","toSemiring"],["IsLocalHom"],["Semiring","toMonoidWithZero"],["SubringClass","toSubsemiringClass"],["ValuationSubring","toSubring"],["Subring","instSubringClass"],["DivisionSemiring","toSemiring"],["IsLocalRing"],["CommRing"],["SubsemiringClass","toNonAssocSemiring"],["Field","toSemifield"]],"valueReferences":[["RingHom"],["ValuationSubring"],["PartialOrder","toPreorder"],["Membership","mem"],["Exists","intro"],["IsLocalHom","of_surjective"],["LocalSubring"],["Subring","inclusion"],["Semiring","toNonAssocSemiring"],["LocalSubring","instPartialOrder"],["EuclideanDomain","toNontrivial"],["Semifield","toDivisionSemiring"],["LocalSubring","exists_le_valuationSubring"],["ValuationSubring","toLocalSubring"],["Subring","instSetLike"],["RingHom","codRestrict"],["rfl"],["SetLike","instMembership"],["Exists"],["ValuationSubring","toSubring"],["SubringClass","toSubsemiringClass"],["DivisionSemiring","toSemiring"],["Exists","casesOn"],["NonAssocRing","toNonAssocSemiring"],["SubsemiringClass","toNonAssocSemiring"],["Semifield","toCommSemiring"],["EuclideanDomain","toCommRing"],["Subtype"],["SubsemiringClass","toCommSemiring"],["RingHom","instFunLike"],["Subring","toCommRing"],["DFunLike","coe"],["Subring","instNontrivialSubtypeMem"],["Field","toEuclideanDomain"],["MonoidWithZero","toMonoid"],["NonAssocCommRing","toNonAssocRing"],["Preorder","toLE"],["Eq"],["CommRing","toCommSemiring"],["Field","toCommRing"],["CommRing","toNonAssocCommRing"],["Subring"],["CommSemiring","toSemiring"],["IsLocalHom"],["LocalSubring","range"],["Semiring","toMonoidWithZero"],["Subring","instSubringClass"],["RingHom","rangeRestrict"],["RingHom","rangeRestrict_surjective"],["LocalSubring","toSubring"],["LE","le"],["Field","toSemifield"],["RingHom","isLocalHom_comp"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","RingTheory","Valuation","LocalSubring",0,"Ideal","image_subset_nonunits_valuationSubring","match_1_3"],"typeFallback":"forall 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(Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))))) M) (motive : (Exists.{succ u_1} (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11) (fun (B : ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11) => LE.le.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))) (Preorder.toLE.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K 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(CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) A K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) M hM)) (ValuationSubring.toLocalSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11 B))) -> Prop) (x._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.119 : Exists.{succ u_1} (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11) (fun (B : ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11) => LE.le.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))) (Preorder.toLE.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))) (PartialOrder.toPreorder.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))) (LocalSubring.instPartialOrder.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (LocalSubring.ofPrime.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11 A M (Ideal.IsMaximal.isPrime'.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) A K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) M hM)) (ValuationSubring.toLocalSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11 B))), (forall (V : ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11) (hV : LE.le.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))) (Preorder.toLE.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))) (PartialOrder.toPreorder.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))) (LocalSubring.instPartialOrder.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (LocalSubring.ofPrime.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11 A M (Ideal.IsMaximal.isPrime'.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) A K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) M hM)) (ValuationSubring.toLocalSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11 V)), motive (Exists.intro.{succ u_1} (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11) (fun (B : ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11) => LE.le.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))) (Preorder.toLE.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))) (PartialOrder.toPreorder.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))) (LocalSubring.instPartialOrder.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) (LocalSubring.ofPrime.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11 A M (Ideal.IsMaximal.isPrime'.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11))))) A x)) (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) A K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11)))))) M hM)) (ValuationSubring.toLocalSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.11 B)) V hV)) -> (motive x._@.Mathlib.RingTheory.Valuation.LocalSubring.2882904079._hygCtx._hyg.119)","typeFull":"∀ {K : Type u_1} [inst : Field K] {A : Subring K} (M : Ideal ↥A) (hM : M.IsMaximal)\n  (motive : (∃ B, LocalSubring.ofPrime A M ≤ B.toLocalSubring) → Prop)\n  (x : ∃ B, LocalSubring.ofPrime A M ≤ B.toLocalSubring),\n  (∀ (V : ValuationSubring K) (hV : LocalSubring.ofPrime A M ≤ V.toLocalSubring), motive ⋯) → motive x","typeReadable":"∀ {K : Type u_1} [inst : Field K] {A : Subring K} (M : Ideal ↥A) (hM : M.IsMaximal)\n  (motive : (∃ B, LocalSubring.ofPrime A M ≤ B.toLocalSubring) → Prop)\n  (x : ∃ B, LocalSubring.ofPrime A M ≤ B.toLocalSubring),\n  (∀ (V : ValuationSubring K) (hV : LocalSubring.ofPrime A M ≤ V.toLocalSubring), motive ⋯) → motive x","typeReferences":[["Semifield","toCommSemiring"],["SubsemiringClass","toCommSemiring"],["Subtype"],["EuclideanDomain","toCommRing"],["ValuationSubring"],["PartialOrder","toPreorder"],["Field"],["Membership","mem"],["Exists","intro"],["LocalSubring"],["Ideal"],["Field","toEuclideanDomain"],["LocalSubring","instPartialOrder"],["NonAssocCommRing","toNonAssocRing"],["Ideal","IsMaximal","isPrime'"],["Preorder","toLE"],["ValuationSubring","toLocalSubring"],["Subring","instSetLike"],["Ideal","IsMaximal"],["Field","toCommRing"],["SetLike","instMembership"],["Exists"],["CommRing","toNonAssocCommRing"],["Subring"],["CommSemiring","toSemiring"],["SubringClass","toSubsemiringClass"],["Subring","instSubringClass"],["LE","le"],["Field","toSemifield"],["LocalSubring","ofPrime"]],"valueReferences":[["Semifield","toCommSemiring"],["Subtype"],["PartialOrder","toPreorder"],["EuclideanDomain","toCommRing"],["ValuationSubring"],["SubsemiringClass","toCommSemiring"],["Membership","mem"],["LocalSubring"],["Field","toEuclideanDomain"],["LocalSubring","instPartialOrder"],["NonAssocCommRing","toNonAssocRing"],["Ideal","IsMaximal","isPrime'"],["Preorder","toLE"],["ValuationSubring","toLocalSubring"],["Subring","instSetLike"],["Field","toCommRing"],["SetLike","instMembership"],["CommRing","toNonAssocCommRing"],["Subring"],["SubringClass","toSubsemiringClass"],["Subring","instSubringClass"],["Exists","casesOn"],["LE","le"],["Field","toSemifield"],["LocalSubring","ofPrime"]]},{"isProp":true,"kind":"theorem","name":["LocalSubring","mem_of_isMax_of_isIntegral"],"typeFallback":"forall {K : Type.{u_3}} [inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1914405336._hygCtx._hyg.11 : Field.{u_3} K] {R : LocalSubring.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1914405336._hygCtx._hyg.11))}, (IsMax.{u_3} (LocalSubring.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1914405336._hygCtx._hyg.11))) (Preorder.toLE.{u_3} (LocalSubring.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1914405336._hygCtx._hyg.11))) (PartialOrder.toPreorder.{u_3} (LocalSubring.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1914405336._hygCtx._hyg.11))) (LocalSubring.instPartialOrder.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1914405336._hygCtx._hyg.11))))) R) -> (forall {x : K}, (IsIntegral.{u_3, u_3} (Subtype.{succ u_3} K (fun (x : K) => Membership.mem.{u_3, u_3} K (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1914405336._hygCtx._hyg.11))))) (SetLike.instMembership.{u_3, u_3} (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1914405336._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1914405336._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1914405336._hygCtx._hyg.11)) R) x)) K (Subring.toCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1914405336._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1914405336._hygCtx._hyg.11)) R)) (DivisionRing.toRing.{u_3} K (Field.toDivisionRing.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1914405336._hygCtx._hyg.11)) (Algebra.ofSubring.{u_3, u_3} K K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1914405336._hygCtx._hyg.11)) (DivisionRing.toRing.{u_3} K (Field.toDivisionRing.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1914405336._hygCtx._hyg.11)) (Algebra.id.{u_3} K (CommRing.toCommSemiring.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1914405336._hygCtx._hyg.11)))) (LocalSubring.toSubring.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1914405336._hygCtx._hyg.11)) R)) x) -> (Membership.mem.{u_3, u_3} K (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1914405336._hygCtx._hyg.11))))) (SetLike.instMembership.{u_3, u_3} (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K 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Membership.mem.{u_1, u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))) (Set.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11)))) (Set.instMembership.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11)))) s x) i))) x)) -> Prop) (h._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx.157.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.170 : Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K 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inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11)) (Subtype.val.{succ u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))) (fun (x : LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))) => Membership.mem.{u_1, u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))) (Set.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11)))) (Set.instMembership.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K 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(NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))))) (Set.Elem.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))) s) (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))))) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))))) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))))) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))))) (Subring.instCompleteLattice.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))))))))) (fun (i : Set.Elem.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))) s) => LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11)) (Subtype.val.{succ u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))) (fun (x : LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))) => Membership.mem.{u_1, u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))) (Set.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11)))) (Set.instMembership.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11)))) s x) i))) a), motive (Subtype.mk.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11)))))) (iSup.{u_1, succ u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))))) (Set.Elem.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))) s) (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))))) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))))) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))))) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))))) (Subring.instCompleteLattice.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))))))))) (fun (i : Set.Elem.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))) s) => LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11)) (Subtype.val.{succ u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))) (fun (x : LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))) => Membership.mem.{u_1, u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11))) (Set.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11)))) (Set.instMembership.{u_1} (LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.11)))) s x) i))) x) a ha)) -> (motive h._@.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx.157.Mathlib.RingTheory.Valuation.LocalSubring.1439105013._hygCtx._hyg.170)","typeFull":"∀ {K : Type u_1} [inst : Field K] (s : Set (LocalSubring K)) (motive : ↥(⨆ i, (↑i).toSubring) → Prop)\n  (h : ↥(⨆ i, (↑i).toSubring)), (∀ (a : K) (ha : a ∈ ⨆ i, (↑i).toSubring), motive ⟨a, ha⟩) → motive h","typeReadable":"∀ {K : Type u_1} [inst : Field K] (s : Set (LocalSubring K)) (motive : ↥(⨆ i, (↑i).toSubring) → Prop)\n  (h : ↥(⨆ i, (↑i).toSubring)), (∀ (a : K) (ha : a ∈ ⨆ i, (↑i).toSubring), motive ⟨a, ha⟩) → motive h","typeReferences":[["EuclideanDomain","toCommRing"],["Subtype"],["SetLike","instMembership"],["Field"],["Set"],["CommRing","toNonAssocCommRing"],["Subring"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["Membership","mem"],["Subtype","val"],["LocalSubring"],["Set","Elem"],["Set","instMembership"],["LocalSubring","toSubring"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderSup"],["iSup"],["Field","toEuclideanDomain"],["ConditionallyCompletePartialOrderSup","toSupSet"],["Subring","instCompleteLattice"],["Subtype","mk"],["NonAssocCommRing","toNonAssocRing"],["CompleteLattice","toConditionallyCompleteLattice"],["Subring","instSetLike"]],"valueReferences":[["EuclideanDomain","toCommRing"],["SetLike","instMembership"],["Set"],["CommRing","toNonAssocCommRing"],["Subring"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["Membership","mem"],["Subtype","val"],["LocalSubring"],["Set","Elem"],["Set","instMembership"],["LocalSubring","toSubring"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderSup"],["iSup"],["Field","toEuclideanDomain"],["ConditionallyCompletePartialOrderSup","toSupSet"],["Subring","instCompleteLattice"],["Subtype","casesOn"],["NonAssocCommRing","toNonAssocRing"],["CompleteLattice","toConditionallyCompleteLattice"],["Subring","instSetLike"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","RingTheory","Valuation","LocalSubring",0,"LocalSubring","exists_le_valuationSubring_of_isIntegrallyClosedIn","match_1_1"],"typeFallback":"forall {K : Type.{u_1}} [inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11 : Field.{u_1} K] {x : K} {R : LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))} (this : Invertible.{u_1} K (MulZeroClass.toMul.{u_1} K (MulZeroOneClass.toMulZeroClass.{u_1} K (MonoidWithZero.toMulZeroOneClass.{u_1} K (GroupWithZero.toMonoidWithZero.{u_1} K (DivisionSemiring.toGroupWithZero.{u_1} K (Semifield.toDivisionSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))))) (InvOneClass.toOne.{u_1} K (DivInvOneMonoid.toInvOneClass.{u_1} K (DivisionMonoid.toDivInvOneMonoid.{u_1} K (GroupWithZero.toDivisionMonoid.{u_1} K (DivisionSemiring.toGroupWithZero.{u_1} K (Semifield.toDivisionSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))))) x) (motive : (Exists.{succ u_1} (Polynomial.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) (CommRing.toCommSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R))))) (fun (p : Polynomial.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) (CommRing.toCommSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R))))) => And (Membership.mem.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K 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(LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K 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(NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) (CommSemiring.toSemiring.{u_1} 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(EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) K (CommRing.toCommSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R))) (CommSemiring.toSemiring.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R)) (Invertible.invOf.{u_1} K (Distrib.toMul.{u_1} K (NonUnitalNonAssocSemiring.toDistrib.{u_1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} K (CommRing.toNonUnitalCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))))) (AddMonoidWithOne.toOne.{u_1} K (AddGroupWithOne.toAddMonoidWithOne.{u_1} K (Ring.toAddGroupWithOne.{u_1} K (CommRing.toRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (Inv.inv.{u_1} K (InvOneClass.toInv.{u_1} K (DivInvOneMonoid.toInvOneClass.{u_1} K (DivisionMonoid.toDivInvOneMonoid.{u_1} K (DivisionCommMonoid.toDivisionMonoid.{u_1} K (CommGroupWithZero.toDivisionCommMonoid.{u_1} K (Semifield.toCommGroupWithZero.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))))) x) (invertibleInv.{u_1} K (DivisionSemiring.toGroupWithZero.{u_1} K (Semifield.toDivisionSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))) x this))) p) (OfNat.ofNat.{u_1} K 0 (Zero.toOfNat0.{u_1} K (MulZeroClass.toZero.{u_1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} K (CommRing.toNonUnitalCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))))))))))) -> Prop) (x._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.217 : Exists.{succ u_1} (Polynomial.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) (CommRing.toCommSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R))))) (fun (p : Polynomial.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) (CommRing.toCommSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R))))) => And (Membership.mem.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) (Ideal.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) (CommRing.toCommSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} 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inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R))))) (fun (p : Polynomial.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) (CommSemiring.toSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K 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K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) (CommRing.toCommSemiring.{u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K 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(NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R))) (CommSemiring.toSemiring.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R)) (Invertible.invOf.{u_1} K (Distrib.toMul.{u_1} K (NonUnitalNonAssocSemiring.toDistrib.{u_1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} K (CommRing.toNonUnitalCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))))) (AddMonoidWithOne.toOne.{u_1} K (AddGroupWithOne.toAddMonoidWithOne.{u_1} K (Ring.toAddGroupWithOne.{u_1} K (CommRing.toRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (Inv.inv.{u_1} K (InvOneClass.toInv.{u_1} K (DivInvOneMonoid.toInvOneClass.{u_1} K (DivisionMonoid.toDivInvOneMonoid.{u_1} K (DivisionCommMonoid.toDivisionMonoid.{u_1} K (CommGroupWithZero.toDivisionCommMonoid.{u_1} K (Semifield.toCommGroupWithZero.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))))) x) (invertibleInv.{u_1} K (DivisionSemiring.toGroupWithZero.{u_1} K (Semifield.toDivisionSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))) x this))) p) (OfNat.ofNat.{u_1} K 0 (Zero.toOfNat0.{u_1} K (MulZeroClass.toZero.{u_1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} K (CommRing.toNonUnitalCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))))))))) hp hpx))) -> (motive x._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.217)","typeFull":"∀ {K : Type u_1} [inst : Field K] {x : K} {R : LocalSubring K} (this : Invertible x)\n  (motive : (∃ p, p.leadingCoeff - 1 ∈ IsLocalRing.maximalIdeal ↥R.toSubring ∧ (Polynomial.aeval ⅟x⁻¹) p = 0) → Prop)\n  (x_1 : ∃ p, p.leadingCoeff - 1 ∈ IsLocalRing.maximalIdeal ↥R.toSubring ∧ (Polynomial.aeval ⅟x⁻¹) p = 0),\n  (∀ (p : Polynomial ↥R.toSubring) (hp : p.leadingCoeff - 1 ∈ IsLocalRing.maximalIdeal ↥R.toSubring)\n      (hpx : (Polynomial.aeval ⅟x⁻¹) p = 0), motive ⋯) →\n    motive x_1","typeReadable":"∀ {K : Type u_1} [inst : Field K] {x : K} {R : LocalSubring K} (this : Invertible x)\n  (motive : (∃ p, p.leadingCoeff - 1 ∈ IsLocalRing.maximalIdeal ↥R.toSubring ∧ (Polynomial.aeval ⅟x⁻¹) p = 0) → Prop)\n  (x_1 : ∃ p, p.leadingCoeff - 1 ∈ IsLocalRing.maximalIdeal ↥R.toSubring ∧ (Polynomial.aeval ⅟x⁻¹) p = 0),\n  (∀ (p : Polynomial ↥R.toSubring) (hp : p.leadingCoeff - 1 ∈ IsLocalRing.maximalIdeal ↥R.toSubring)\n      (hpx : (Polynomial.aeval ⅟x⁻¹) p = 0), motive ⋯) →\n    motive x_1","typeReferences":[["LocalSubring","isLocalRing"],["Invertible","invOf"],["Membership","mem"],["MulZeroClass","toMul"],["Exists","intro"],["AddGroupWithOne","toAddMonoidWithOne"],["CommGroupWithZero","toDivisionCommMonoid"],["MonoidWithZero","toMulZeroOneClass"],["Algebra","id"],["LocalSubring"],["IsLocalRing","maximalIdeal"],["And","intro"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["Ring","toAddGroupWithOne"],["SubNegMonoid","toSub"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["HSub","hSub"],["Semifield","toDivisionSemiring"],["AddGroup","toSubNegMonoid"],["Semiring","toModule"],["Invertible"],["Subring","instSetLike"],["InvOneClass","toInv"],["SetLike","instMembership"],["Exists"],["NonUnitalNonAssocSemiring","toDistrib"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["InvOneClass","toOne"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["And"],["SubringClass","toSubsemiringClass"],["DivisionSemiring","toGroupWithZero"],["Polynomial","aeval"],["Polynomial"],["MulZeroOneClass","toMulZeroClass"],["AddMonoidWithOne","toOne"],["Polynomial","leadingCoeff"],["DivisionMonoid","toDivInvOneMonoid"],["Polynomial","algebraOfAlgebra"],["GroupWithZero","toDivisionMonoid"],["Semifield","toCommSemiring"],["Subtype"],["EuclideanDomain","toCommRing"],["SubsemiringClass","toCommSemiring"],["Field"],["Subring","toCommRing"],["CommRing","toNonUnitalCommRing"],["invertibleInv"],["DFunLike","coe"],["Field","toEuclideanDomain"],["Ideal"],["GroupWithZero","toMonoidWithZero"],["AlgHom","funLike"],["Zero","toOfNat0"],["NonAssocCommRing","toNonAssocRing"],["Eq"],["DivisionCommMonoid","toDivisionMonoid"],["AlgHom"],["Inv","inv"],["CommRing","toCommSemiring"],["CommRing","toNonAssocCommRing"],["Subring"],["Distrib","toMul"],["DivisionRing","toRing"],["Algebra","ofSubring"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Field","toDivisionRing"],["Polynomial","semiring"],["Subring","instSubringClass"],["OfNat","ofNat"],["CommRing","toRing"],["LocalSubring","toSubring"],["DivInvOneMonoid","toInvOneClass"],["AddGroupWithOne","toAddGroup"],["One","toOfNat1"],["MulZeroClass","toZero"],["Submodule","setLike"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Field","toSemifield"],["instHSub"],["Semifield","toCommGroupWithZero"]],"valueReferences":[["LocalSubring","isLocalRing"],["Invertible","invOf"],["Membership","mem"],["Exists","intro"],["AddGroupWithOne","toAddMonoidWithOne"],["CommGroupWithZero","toDivisionCommMonoid"],["Algebra","id"],["IsLocalRing","maximalIdeal"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["Ring","toAddGroupWithOne"],["SubNegMonoid","toSub"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["HSub","hSub"],["Semifield","toDivisionSemiring"],["AddGroup","toSubNegMonoid"],["Semiring","toModule"],["Subring","instSetLike"],["InvOneClass","toInv"],["SetLike","instMembership"],["NonUnitalNonAssocSemiring","toDistrib"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["And"],["SubringClass","toSubsemiringClass"],["DivisionSemiring","toGroupWithZero"],["Polynomial","aeval"],["Exists","casesOn"],["Polynomial"],["Polynomial","leadingCoeff"],["AddMonoidWithOne","toOne"],["DivisionMonoid","toDivInvOneMonoid"],["Polynomial","algebraOfAlgebra"],["Semifield","toCommSemiring"],["SubsemiringClass","toCommSemiring"],["Subtype"],["EuclideanDomain","toCommRing"],["Subring","toCommRing"],["CommRing","toNonUnitalCommRing"],["invertibleInv"],["DFunLike","coe"],["Ideal"],["Field","toEuclideanDomain"],["AlgHom","funLike"],["Zero","toOfNat0"],["NonAssocCommRing","toNonAssocRing"],["Eq"],["DivisionCommMonoid","toDivisionMonoid"],["AlgHom"],["Inv","inv"],["CommRing","toCommSemiring"],["CommRing","toNonAssocCommRing"],["Subring"],["Distrib","toMul"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Algebra","ofSubring"],["DivisionRing","toRing"],["Field","toDivisionRing"],["Polynomial","semiring"],["Subring","instSubringClass"],["OfNat","ofNat"],["CommRing","toRing"],["LocalSubring","toSubring"],["DivInvOneMonoid","toInvOneClass"],["AddGroupWithOne","toAddGroup"],["One","toOfNat1"],["MulZeroClass","toZero"],["Submodule","setLike"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Field","toSemifield"],["instHSub"],["And","casesOn"],["Semifield","toCommGroupWithZero"]]},{"isProp":false,"kind":"definition","name":["ValuationSubring","toLocalSubring"],"typeFallback":"forall {K : Type.{u_3}} [inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1754168285._hygCtx._hyg.11 : Field.{u_3} K], (ValuationSubring.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1754168285._hygCtx._hyg.11) -> (LocalSubring.{u_3} K (EuclideanDomain.toCommRing.{u_3} K (Field.toEuclideanDomain.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1754168285._hygCtx._hyg.11)))","typeFull":"{K : Type u_3} → [inst : Field K] → ValuationSubring K → LocalSubring K","typeReadable":"{K : Type u_3} → [inst : Field K] → ValuationSubring K → LocalSubring K","typeReferences":[["EuclideanDomain","toCommRing"],["ValuationSubring"],["Field","toEuclideanDomain"],["Field"],["LocalSubring"]],"valueReferences":[["EuclideanDomain","toCommRing"],["Field","toEuclideanDomain"],["LocalSubring","mk"],["ValuationSubring","toSubring"],["ValuationSubring","isLocalRing"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","RingTheory","Valuation","LocalSubring",0,"ValuationSubring","isMax_toLocalSubring","_simp_1_1"],"typeFallback":"forall {α : Sort.{u}} {p : α -> Prop} {a1 : Subtype.{u} α (fun (x : α) => p x)} {a2 : Subtype.{u} α (fun (x : α) => p x)}, Eq.{1} Prop (Eq.{max 1 u} (Subtype.{u} α (fun (x : α) => p x)) a1 a2) (Eq.{u} α (Subtype.val.{u} α (fun (x : α) => p x) a1) (Subtype.val.{u} α (fun (x : α) => p x) a2))","typeFull":"∀ {α : Sort u} {p : α → Prop} {a1 a2 : { x // p x }}, (a1 = a2) = (↑a1 = ↑a2)","typeReadable":"∀ {α : Sort u} {p : α → Prop} {a1 a2 : { x // p x }}, (a1 = a2) = (↑a1 = ↑a2)","typeReferences":[["Subtype"],["Eq"],["Subtype","val"]],"valueReferences":[["Subtype"],["Subtype","ext_iff"],["Eq"],["Subtype","val"],["propext"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","RingTheory","Valuation","LocalSubring",0,"LocalSubring","exists_le_valuationSubring_of_isIntegrallyClosedIn","match_1_3"],"typeFallback":"forall {K : Type.{u_1}} [inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11 : Field.{u_1} K] {x : K} {R : LocalSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))} (xinv : Subtype.{succ u_1} K (fun (x_1 : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (Ring.toNonAssocRing.{u_1} K (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (Ring.toNonAssocRing.{u_1} K (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))) K (Subring.instSetLike.{u_1} K (Ring.toNonAssocRing.{u_1} K (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (Subalgebra.toSubring.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) K (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R)) (Algebra.adjoin.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) K (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))))) (DivisionSemiring.toSemiring.{u_1} K (Semifield.toDivisionSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R)) (Singleton.singleton.{u_1, u_1} K (Set.{u_1} K) (Set.instSingletonSet.{u_1} K) (Inv.inv.{u_1} K (InvOneClass.toInv.{u_1} K (DivInvOneMonoid.toInvOneClass.{u_1} K (DivisionMonoid.toDivInvOneMonoid.{u_1} K (DivisionCommMonoid.toDivisionMonoid.{u_1} K (CommGroupWithZero.toDivisionCommMonoid.{u_1} K (Semifield.toCommGroupWithZero.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))))) x)))) x_1)), let B : Subalgebra.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) K (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))))) (DivisionSemiring.toSemiring.{u_1} K (Semifield.toDivisionSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R)) := Algebra.adjoin.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) K (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))))) (DivisionSemiring.toSemiring.{u_1} K (Semifield.toDivisionSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R)) (Singleton.singleton.{u_1, u_1} K (Set.{u_1} K) (Set.instSingletonSet.{u_1} K) (Inv.inv.{u_1} K (InvOneClass.toInv.{u_1} K (DivInvOneMonoid.toInvOneClass.{u_1} K (DivisionMonoid.toDivInvOneMonoid.{u_1} K (DivisionCommMonoid.toDivisionMonoid.{u_1} K (CommGroupWithZero.toDivisionCommMonoid.{u_1} K (Semifield.toCommGroupWithZero.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))))) x)); forall (motive : (Exists.{succ u_1} (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11) (fun (B_1 : ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11) => And (LE.le.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (Preorder.toLE.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (PartialOrder.toPreorder.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (Subring.instPartialOrder.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))))) (Subalgebra.toSubring.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) K (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R)) B) (ValuationSubring.toSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11 B_1)) (HasSubset.Subset.{u_1} (Set.{u_1} K) (Set.instHasSubset.{u_1} K) (Set.image.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (Subalgebra.toSubring.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) K (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R)) B) x)) K (DFunLike.coe.{succ u_1, succ u_1, succ u_1} (RingHom.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (Subalgebra.toSubring.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) K (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K 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(NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K 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inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R))) K (Subalgebra.instSetLike.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) K (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))))) (DivisionSemiring.toSemiring.{u_1} K (Semifield.toDivisionSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R)))) B x))) xinv))))) (SetLike.coe.{u_1, u_1} (NonUnitalSubring.{u_1} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} K (CommRing.toNonUnitalCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (NonUnitalSubring.instSetLike.{u_1} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} K (CommRing.toNonUnitalCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (ValuationSubring.nonunits.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11 B_1))))) -> Prop) (x._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.318 : Exists.{succ u_1} (ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11) (fun (B_1 : ValuationSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11) => And (LE.le.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (Preorder.toLE.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (PartialOrder.toPreorder.{u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (Subring.instPartialOrder.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))))) (Subalgebra.toSubring.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) K (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R)) B) (ValuationSubring.toSubring.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11 B_1)) (HasSubset.Subset.{u_1} (Set.{u_1} K) (Set.instHasSubset.{u_1} K) (Set.image.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (Subalgebra.toSubring.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) K (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R)) B) x)) K (DFunLike.coe.{succ u_1, succ u_1, succ u_1} (RingHom.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (Subalgebra.toSubring.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) K (Subring.toCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) 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inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R))) K (Subalgebra.instSetLike.{u_1, u_1} (Subtype.{succ u_1} K (fun (x : K) => Membership.mem.{u_1, u_1} K (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SetLike.instMembership.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) x)) K (SubsemiringClass.toCommSemiring.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R) K (Semifield.toCommSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (SubringClass.toSubsemiringClass.{u_1, u_1} (Subring.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))) (Subring.instSetLike.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (Subring.instSubringClass.{u_1} K (NonAssocCommRing.toNonAssocRing.{u_1} K (CommRing.toNonAssocCommRing.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))))) (DivisionSemiring.toSemiring.{u_1} K (Semifield.toDivisionSemiring.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))) (Algebra.ofSubring.{u_1, u_1} K K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) (Algebra.id.{u_1} K (CommRing.toCommSemiring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)))) (LocalSubring.toSubring.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11)) R)))) B x))) xinv))))) (SetLike.coe.{u_1, u_1} (NonUnitalSubring.{u_1} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} K (CommRing.toNonUnitalCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) K (NonUnitalSubring.instSetLike.{u_1} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} K (CommRing.toNonUnitalCommRing.{u_1} K (Field.toCommRing.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11))))) (ValuationSubring.nonunits.{u_1} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.11 B_1)))) V hV)) -> (motive x._@.Mathlib.RingTheory.Valuation.LocalSubring.1071600442._hygCtx._hyg.318)","typeFull":"∀ {K : Type u_1} [inst : Field K] {x : K} {R : LocalSubring K} (xinv : ↥(↥R.toSubring)[x⁻¹].toSubring),\n  let B := (↥R.toSubring)[x⁻¹];\n  ∀\n    (motive :\n      (∃ B_1,\n          B.toSubring ≤ B_1.toSubring ∧\n            ⇑B.toSubring.subtype ''\n                ↑(Ideal.map (algebraMap ↥R.toSubring ↥B) (IsLocalRing.maximalIdeal ↥R.toSubring) + Ideal.span {xinv}) ⊆\n              ↑B_1.nonunits) →\n        Prop)\n    (x :\n      ∃ B_1,\n        B.toSubring ≤ B_1.toSubring ∧\n          ⇑B.toSubring.subtype ''\n              ↑(Ideal.map (algebraMap ↥R.toSubring ↥B) (IsLocalRing.maximalIdeal ↥R.toSubring) + Ideal.span {xinv}) ⊆\n            ↑B_1.nonunits),\n    (∀ (V : ValuationSubring K)\n        (hV :\n          B.toSubring ≤ V.toSubring ∧\n            ⇑B.toSubring.subtype ''\n                ↑(Ideal.map (algebraMap ↥R.toSubring ↥B) (IsLocalRing.maximalIdeal ↥R.toSubring) + Ideal.span {xinv}) ⊆\n              ↑V.nonunits),\n        motive ⋯) →\n      motive x","typeReadable":"∀ {K : Type u_1} [inst : Field K] {x : K} {R : LocalSubring K} (xinv : ↥(↥R.toSubring)[x⁻¹].toSubring),\n  let B := (↥R.toSubring)[x⁻¹];\n  ∀\n    (motive :\n      (∃ B_1,\n          B.toSubring ≤ B_1.toSubring ∧\n            ⇑B.toSubring.subtype ''\n                ↑(Ideal.map (algebraMap ↥R.toSubring ↥B) (IsLocalRing.maximalIdeal ↥R.toSubring) + Ideal.span {xinv}) ⊆\n              ↑B_1.nonunits) →\n        Prop)\n    (x :\n      ∃ B_1,\n        B.toSubring ≤ B_1.toSubring ∧\n          ⇑B.toSubring.subtype ''\n              ↑(Ideal.map (algebraMap ↥R.toSubring ↥B) (IsLocalRing.maximalIdeal ↥R.toSubring) + Ideal.span {xinv}) ⊆\n            ↑B_1.nonunits),\n    (∀ (V : ValuationSubring K)\n        (hV :\n          B.toSubring ≤ V.toSubring ∧\n            ⇑B.toSubring.subtype ''\n                ↑(Ideal.map (algebraMap ↥R.toSubring ↥B) (IsLocalRing.maximalIdeal ↥R.toSubring) + Ideal.span {xinv}) ⊆\n              ↑V.nonunits),\n        motive ⋯) →\n      motive x","typeReferences":[["LocalSubring","isLocalRing"],["RingHom"],["Ring","toNonAssocRing"],["ValuationSubring"],["PartialOrder","toPreorder"],["Subring","instPartialOrder"],["Singleton","singleton"],["Membership","mem"],["Exists","intro"],["CommGroupWithZero","toDivisionCommMonoid"],["NonUnitalSubring","instSetLike"],["Algebra","id"],["LocalSubring"],["IsLocalRing","maximalIdeal"],["Semiring","toNonAssocSemiring"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Semifield","toDivisionSemiring"],["Semiring","toModule"],["Subring","instSetLike"],["Subalgebra"],["SetLike","instMembership"],["InvOneClass","toInv"],["Exists"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["And"],["SubringClass","toSubsemiringClass"],["ValuationSubring","toSubring"],["DivisionSemiring","toSemiring"],["ValuationSubring","nonunits"],["Submodule","pointwiseAdd"],["SubringClass","toNonAssocRing"],["Set","image"],["SetLike","coe"],["NonAssocRing","toNonAssocSemiring"],["HasSubset","Subset"],["Subring","subtype"],["Subalgebra","toSubring"],["DivisionMonoid","toDivInvOneMonoid"],["algebraMap"],["Semifield","toCommSemiring"],["Ideal","span"],["SubsemiringClass","toCommSemiring"],["EuclideanDomain","toCommRing"],["Subtype"],["Field"],["NonUnitalSubring"],["RingHom","instFunLike"],["Algebra","adjoin"],["Subring","toCommRing"],["CommRing","toNonUnitalCommRing"],["Subalgebra","algebra"],["DFunLike","coe"],["Ideal","map"],["Ideal"],["Field","toEuclideanDomain"],["NonAssocCommRing","toNonAssocRing"],["Preorder","toLE"],["DivisionCommMonoid","toDivisionMonoid"],["CommRing","toCommSemiring"],["Inv","inv"],["Field","toCommRing"],["instHAdd"],["CommRing","toNonAssocCommRing"],["Set"],["Subring"],["Subalgebra","instSetLike"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Algebra","ofSubring"],["DivisionRing","toRing"],["Field","toDivisionRing"],["Subring","instSubringClass"],["Set","instSingletonSet"],["Subalgebra","toSemiring"],["HAdd","hAdd"],["Set","instHasSubset"],["LocalSubring","toSubring"],["DivInvOneMonoid","toInvOneClass"],["Submodule","setLike"],["LE","le"],["Field","toSemifield"],["Semifield","toCommGroupWithZero"]],"valueReferences":[["LocalSubring","isLocalRing"],["RingHom"],["Ring","toNonAssocRing"],["ValuationSubring"],["PartialOrder","toPreorder"],["Subring","instPartialOrder"],["Singleton","singleton"],["Membership","mem"],["CommGroupWithZero","toDivisionCommMonoid"],["NonUnitalSubring","instSetLike"],["Algebra","id"],["IsLocalRing","maximalIdeal"],["Semiring","toNonAssocSemiring"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Semifield","toDivisionSemiring"],["Semiring","toModule"],["Subring","instSetLike"],["Subalgebra"],["SetLike","instMembership"],["InvOneClass","toInv"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["And"],["SubringClass","toSubsemiringClass"],["ValuationSubring","toSubring"],["DivisionSemiring","toSemiring"],["ValuationSubring","nonunits"],["Exists","casesOn"],["Submodule","pointwiseAdd"],["SubringClass","toNonAssocRing"],["Set","image"],["SetLike","coe"],["HasSubset","Subset"],["NonAssocRing","toNonAssocSemiring"],["Subring","subtype"],["Subalgebra","toSubring"],["DivisionMonoid","toDivInvOneMonoid"],["algebraMap"],["Semifield","toCommSemiring"],["Ideal","span"],["Subtype"],["EuclideanDomain","toCommRing"],["SubsemiringClass","toCommSemiring"],["NonUnitalSubring"],["RingHom","instFunLike"],["Algebra","adjoin"],["Subring","toCommRing"],["CommRing","toNonUnitalCommRing"],["Subalgebra","algebra"],["DFunLike","coe"],["Ideal","map"],["Ideal"],["Field","toEuclideanDomain"],["NonAssocCommRing","toNonAssocRing"],["Preorder","toLE"],["DivisionCommMonoid","toDivisionMonoid"],["CommRing","toCommSemiring"],["Inv","inv"],["Field","toCommRing"],["instHAdd"],["Set"],["CommRing","toNonAssocCommRing"],["Subring"],["Subalgebra","instSetLike"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["DivisionRing","toRing"],["Algebra","ofSubring"],["Field","toDivisionRing"],["Subring","instSubringClass"],["Set","instSingletonSet"],["Subalgebra","toSemiring"],["HAdd","hAdd"],["Set","instHasSubset"],["LocalSubring","toSubring"],["DivInvOneMonoid","toInvOneClass"],["Submodule","setLike"],["LE","le"],["Field","toSemifield"],["Semifield","toCommGroupWithZero"]]},{"isProp":true,"kind":"theorem","name":["instIsIntegrallyClosedSubtypeMemSubring"],"typeFallback":"forall {K : Type.{u_3}} [inst._@.Mathlib.RingTheory.Valuation.LocalSubring.396702554._hygCtx._hyg.11 : Field.{u_3} K] (V : ValuationSubring.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.396702554._hygCtx._hyg.11), IsIntegrallyClosed.{u_3} (Subtype.{succ u_3} K (fun (x : K) => Membership.mem.{u_3, u_3} K (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (Field.toCommRing.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.396702554._hygCtx._hyg.11)))) (SetLike.instMembership.{u_3, u_3} (Subring.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (Field.toCommRing.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.396702554._hygCtx._hyg.11)))) K (Subring.instSetLike.{u_3} K (NonAssocCommRing.toNonAssocRing.{u_3} K (CommRing.toNonAssocCommRing.{u_3} K (Field.toCommRing.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.396702554._hygCtx._hyg.11))))) (ValuationSubring.toSubring.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.396702554._hygCtx._hyg.11 V) x)) (Subring.toCommRing.{u_3} K (Field.toCommRing.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.396702554._hygCtx._hyg.11) (ValuationSubring.toSubring.{u_3} K inst._@.Mathlib.RingTheory.Valuation.LocalSubring.396702554._hygCtx._hyg.11 V))","typeFull":"∀ {K : Type u_3} [inst : Field K] (V : ValuationSubring K), IsIntegrallyClosed ↥V.toSubring","typeReadable":"∀ {K : Type u_3} [inst : Field K] (V : ValuationSubring K), IsIntegrallyClosed ↥V.toSubring","typeReferences":[["ValuationSubring"],["Subtype"],["Field","toCommRing"],["SetLike","instMembership"],["Field"],["CommRing","toNonAssocCommRing"],["Subring"],["Membership","mem"],["IsIntegrallyClosed"],["ValuationSubring","toSubring"],["Subring","toCommRing"],["NonAssocCommRing","toNonAssocRing"],["Subring","instSetLike"]],"valueReferences":[["Subtype"],["Ring","toNonAssocRing"],["ValuationSubring","instLinearOrderedCommGroupWithZeroValueGroup"],["ValuationSubring","valuation"],["Membership","mem"],["ValuationSubring","integer_valuation"],["IsIntegrallyClosed"],["Subring","toCommRing"],["Valuation","integer"],["ValuationSubring","ValueGroup"],["congrArg"],["Eq","symm"],["NonAssocCommRing","toNonAssocRing"],["Eq"],["Subring","instSetLike"],["Valuation","Integers","isIntegrallyClosed_integers"],["SetLike","instMembership"],["Field","toCommRing"],["CommRing","toNonAssocCommRing"],["Subring"],["DivisionRing","toRing"],["Field","toDivisionRing"],["ValuationSubring","toSubring"],["id"],["inferInstance"],["Eq","mpr"]]}]
diff --git a/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Topology.Algebra.GroupWithZero.sym.json b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Topology.Algebra.GroupWithZero.sym.json
new file mode 100644
index 0000000000000000000000000000000000000000..530bef133fd9dc947e7b848f75d4508996f14ee7
--- /dev/null
+++ b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Topology.Algebra.GroupWithZero.sym.json
@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["map_mul_right_nhds_one₀"],"typeFallback":"forall {G₀ : Type.{u_3}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2172790485._hygCtx._hyg.5 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2172790485._hygCtx._hyg.8 : GroupWithZero.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2172790485._hygCtx._hyg.11 : SeparatelyContinuousMul.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2172790485._hygCtx._hyg.5 (MulZeroClass.toMul.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2172790485._hygCtx._hyg.8))))] {a : G₀}, (Ne.{succ u_3} G₀ a (OfNat.ofNat.{u_3} G₀ 0 (Zero.toOfNat0.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2172790485._hygCtx._hyg.8))))))) -> (Eq.{succ u_3} (Filter.{u_3} G₀) (Filter.map.{u_3, u_3} G₀ G₀ (fun (x._@.Mathlib.Topology.Algebra.GroupWithZero.2172790485._hygCtx._hyg.26 : G₀) => HMul.hMul.{u_3, u_3, u_3} G₀ G₀ G₀ (instHMul.{u_3} G₀ (MulZeroClass.toMul.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2172790485._hygCtx._hyg.8))))) x._@.Mathlib.Topology.Algebra.GroupWithZero.2172790485._hygCtx._hyg.26 a) (nhds.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2172790485._hygCtx._hyg.5 (OfNat.ofNat.{u_3} G₀ 1 (One.toOfNat1.{u_3} G₀ (InvOneClass.toOne.{u_3} G₀ (DivInvOneMonoid.toInvOneClass.{u_3} G₀ (DivisionMonoid.toDivInvOneMonoid.{u_3} G₀ (GroupWithZero.toDivisionMonoid.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2172790485._hygCtx._hyg.8)))))))) (nhds.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2172790485._hygCtx._hyg.5 a))","typeFull":"∀ {G₀ : Type u_3} [inst : TopologicalSpace G₀] [inst_1 : GroupWithZero G₀] [SeparatelyContinuousMul G₀] {a : G₀},\n  a ≠ 0 → Filter.map (fun x => x * a) (nhds 1) = nhds a","typeReadable":"∀ {G₀ : Type u_3} [inst : TopologicalSpace G₀] [inst_1 : GroupWithZero G₀] [SeparatelyContinuousMul G₀] {a : G₀},\n  a ≠ 0 → Filter.map (fun x => x * a) (nhds 1) = nhds 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{α : Type.{u_1}} {G₀ : Type.{u_3}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392308._hygCtx._hyg.5 : Zero.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392308._hygCtx._hyg.8 : Inv.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392308._hygCtx._hyg.11 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392308._hygCtx._hyg.14 : ContinuousInv₀.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392308._hygCtx._hyg.5 inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392308._hygCtx._hyg.8 inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392308._hygCtx._hyg.11] {f : α -> G₀} {s : Set.{u_1} α} {a : α} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392308._hygCtx._hyg.25 : TopologicalSpace.{u_1} α], (ContinuousWithinAt.{u_1, u_3} α G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392308._hygCtx._hyg.25 inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392308._hygCtx._hyg.11 f s a) -> (Ne.{succ u_3} G₀ (f 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0","typeReferences":[["DivInvMonoid","toInv"],["Inv","inv"],["Subtype"],["GroupWithZero","toDivInvMonoid"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["Subtype","val"],["OfNat","ofNat"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["GroupWithZero","toMonoidWithZero"],["Ne"],["Zero","toOfNat0"]],"valueReferences":[["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Subtype","property"],["GroupWithZero","toMonoidWithZero"],["MonoidWithZero","toMulZeroOneClass"],["Zero","toOfNat0"],["Ne"],["Subtype","val"],["OfNat","ofNat"],["inv_ne_zero"]]},{"isProp":true,"kind":"theorem","name":["Filter","Tendsto","div"],"typeFallback":"forall {α : Type.{u_1}} {G₀ : Type.{u_3}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639154._hygCtx._hyg.5 : GroupWithZero.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639154._hygCtx._hyg.8 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639154._hygCtx._hyg.11 : ContinuousInv₀.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639154._hygCtx._hyg.5)))) (InvOneClass.toInv.{u_3} G₀ (DivInvOneMonoid.toInvOneClass.{u_3} G₀ (DivisionMonoid.toDivInvOneMonoid.{u_3} G₀ (GroupWithZero.toDivisionMonoid.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639154._hygCtx._hyg.5)))) inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639154._hygCtx._hyg.8] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639154._hygCtx._hyg.14 : ContinuousMul.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639154._hygCtx._hyg.8 (MulZeroClass.toMul.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639154._hygCtx._hyg.5))))] {f : α -> G₀} {g : α -> G₀} {l : 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{G₀ : Type.{u_3}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280310._hygCtx._hyg.5 : GroupWithZero.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280310._hygCtx._hyg.8 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280310._hygCtx._hyg.11 : ContinuousInv₀.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280310._hygCtx._hyg.5)))) (InvOneClass.toInv.{u_3} G₀ (DivInvOneMonoid.toInvOneClass.{u_3} G₀ (DivisionMonoid.toDivInvOneMonoid.{u_3} G₀ (GroupWithZero.toDivisionMonoid.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280310._hygCtx._hyg.5)))) inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280310._hygCtx._hyg.8] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280310._hygCtx._hyg.14 : ContinuousMul.{u_3} G₀ 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Type.{u_3}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106155._hygCtx._hyg.5 : DivInvMonoid.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106155._hygCtx._hyg.8 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106155._hygCtx._hyg.11 : SeparatelyContinuousMul.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106155._hygCtx._hyg.8 (MulOne.toMul.{u_3} G₀ (MulOneClass.toMulOne.{u_3} G₀ (Monoid.toMulOneClass.{u_3} G₀ (DivInvMonoid.toMonoid.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106155._hygCtx._hyg.5))))] {f : α -> G₀} {s : Set.{u_1} α} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106155._hygCtx._hyg.21 : TopologicalSpace.{u_1} α] {a : α}, (ContinuousWithinAt.{u_1, u_3} α G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106155._hygCtx._hyg.21 inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106155._hygCtx._hyg.8 f s a) -> (forall (y : G₀), ContinuousWithinAt.{u_1, u_3} α G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106155._hygCtx._hyg.21 inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106155._hygCtx._hyg.8 (fun (x : α) => HDiv.hDiv.{u_3, u_3, u_3} G₀ G₀ G₀ (instHDiv.{u_3} G₀ (DivInvMonoid.toDiv.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106155._hygCtx._hyg.5)) (f x) y) s a)","typeFull":"∀ {α : Type u_1} {G₀ : Type u_3} [inst : DivInvMonoid G₀] [inst_1 : TopologicalSpace G₀] [SeparatelyContinuousMul G₀]\n  {f : α → G₀} {s : Set α} [inst_3 : TopologicalSpace α] {a : α},\n  ContinuousWithinAt f s a → ∀ (y : G₀), ContinuousWithinAt (fun x => f x / y) s a","typeReadable":"∀ {α : Type u_1} {G₀ : Type u_3} [inst : DivInvMonoid G₀] [inst_1 : TopologicalSpace G₀] [SeparatelyContinuousMul G₀]\n  {f : α → G₀} {s : Set α} [inst_3 : TopologicalSpace α] {a : α},\n  ContinuousWithinAt f s a → ∀ (y : G₀), ContinuousWithinAt (fun x => f x / y) s 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α","typeReferences":[["TopologicalSpace"],["MulZeroOneClass","toMulZeroClass"],["SeparatelyContinuousMul"],["MulZeroClass","toZero"],["GroupWithZero","toMonoidWithZero"],["MulZeroClass","toMul"],["Zero","toOfNat0"],["Ne"],["MonoidWithZero","toMulZeroOneClass"],["Homeomorph"],["GroupWithZero"],["OfNat","ofNat"]],"valueReferences":[["Equiv","mulLeft₀"],["Homeomorph","mk"],["Homeomorph","mulLeft₀","_proof_2"],["Homeomorph","mulLeft₀","_proof_1"]]},{"isProp":false,"kind":"definition","name":["ContinuousInv₀","recOn"],"typeFallback":"forall {G₀ : Type.{u_4}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.9 : Zero.{u_4} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.12 : Inv.{u_4} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.15 : TopologicalSpace.{u_4} G₀] {motive : (ContinuousInv₀.{u_4} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.9 inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.12 inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.15) -> Sort.{u}} (t : ContinuousInv₀.{u_4} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.9 inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.12 inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.15), (forall (continuousAt_inv₀ : forall {{x : G₀}}, (Ne.{succ u_4} G₀ x (OfNat.ofNat.{u_4} G₀ 0 (Zero.toOfNat0.{u_4} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.9))) -> (ContinuousAt.{u_4, u_4} G₀ G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.15 inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.15 (Inv.inv.{u_4} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.12) x)), motive (ContinuousInv₀.mk.{u_4} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.9 inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.12 inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.15 continuousAt_inv₀)) -> (motive t)","typeFull":"{G₀ : Type u_4} →\n  [inst : Zero G₀] →\n    [inst_1 : Inv G₀] →\n      [inst_2 : TopologicalSpace G₀] →\n        {motive : ContinuousInv₀ G₀ → Sort u} →\n          (t : ContinuousInv₀ G₀) →\n            ((continuousAt_inv₀ : ∀ ⦃x : G₀⦄, x ≠ 0 → ContinuousAt Inv.inv x) → motive ⋯) → motive t","typeReadable":"{G₀ : Type u_4} →\n  [inst : Zero G₀] →\n    [inst_1 : Inv G₀] →\n      [inst_2 : TopologicalSpace G₀] →\n        {motive : ContinuousInv₀ G₀ → Sort u} →\n          (t : ContinuousInv₀ G₀) →\n            ((continuousAt_inv₀ : ∀ ⦃x : G₀⦄, x ≠ 0 → ContinuousAt Inv.inv x) → motive ⋯) → motive t","typeReferences":[["TopologicalSpace"],["Inv","inv"],["Inv"],["ContinuousInv₀"],["ContinuousAt"],["Zero","toOfNat0"],["Ne"],["Zero"],["ContinuousInv₀","mk"],["OfNat","ofNat"]],"valueReferences":[["ContinuousInv₀","rec"]]},{"isProp":true,"kind":"theorem","name":["Continuous","div_const"],"typeFallback":"forall {α : Type.{u_1}} {G₀ : Type.{u_3}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106157._hygCtx._hyg.5 : DivInvMonoid.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106157._hygCtx._hyg.8 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106157._hygCtx._hyg.11 : SeparatelyContinuousMul.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106157._hygCtx._hyg.8 (MulOne.toMul.{u_3} G₀ (MulOneClass.toMulOne.{u_3} G₀ (Monoid.toMulOneClass.{u_3} G₀ (DivInvMonoid.toMonoid.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106157._hygCtx._hyg.5))))] {f : α -> G₀} 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[SeparatelyContinuousMul G₀]\n  {f : α → G₀} [inst_3 : TopologicalSpace α], Continuous f → ∀ (y : G₀), Continuous fun x => f x / y","typeReferences":[["HDiv","hDiv"],["MulOneClass","toMulOne"],["TopologicalSpace"],["Continuous"],["DivInvMonoid"],["MulOne","toMul"],["DivInvMonoid","toMonoid"],["SeparatelyContinuousMul"],["Monoid","toMulOneClass"],["instHDiv"],["DivInvMonoid","toDiv"]],"valueReferences":[["div_eq_mul_inv"],["MulOneClass","toMulOne"],["DivInvMonoid","toInv"],["Inv","inv"],["HMul","hMul"],["instHDiv"],["Continuous","mul_const"],["DivInvMonoid","toDiv"],["congrArg"],["HDiv","hDiv"],["Continuous"],["MulOne","toMul"],["DivInvMonoid","toMonoid"],["funext"],["Monoid","toMulOneClass"],["id"],["instHMul"],["Eq","mpr"],["Eq"]]},{"isProp":true,"kind":"theorem","name":["ContinuousAt","comp_div_cases"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} {G₀ : Type.{u_3}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598047._hygCtx._hyg.5 : GroupWithZero.{u_3} G₀] 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(MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598047._hygCtx._hyg.5))))))) -> (Filter.Tendsto.{max u_1 u_3, u_2} (Prod.{u_1, u_3} α G₀) β (Function.HasUncurry.uncurry.{max (max u_1 u_2) u_3, max u_1 u_3, u_2} (α -> G₀ -> β) (Prod.{u_1, u_3} α G₀) β (Function.hasUncurryInduction.{u_1, max u_2 u_3, u_3, u_2} α (G₀ -> β) G₀ β (Function.hasUncurryBase.{u_3, u_2} G₀ β)) h) (SProd.sprod.{u_1, u_3, max u_1 u_3} (Filter.{u_1} α) (Filter.{u_3} G₀) (Filter.{max u_3 u_1} (Prod.{u_1, u_3} α G₀)) (Filter.instSProd.{u_1, u_3} α G₀) (nhds.{u_1} α inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598047._hygCtx._hyg.23 a) (Top.top.{u_3} (Filter.{u_3} G₀) (Filter.instTop.{u_3} G₀))) (nhds.{u_2} β inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598047._hygCtx._hyg.26 (h a (OfNat.ofNat.{u_3} G₀ 0 (Zero.toOfNat0.{u_3} G₀ 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{G₀ : Type.{u_3}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280309._hygCtx._hyg.5 : GroupWithZero.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280309._hygCtx._hyg.8 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280309._hygCtx._hyg.11 : ContinuousInv₀.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280309._hygCtx._hyg.5)))) (InvOneClass.toInv.{u_3} G₀ (DivInvOneMonoid.toInvOneClass.{u_3} G₀ (DivisionMonoid.toDivInvOneMonoid.{u_3} G₀ (GroupWithZero.toDivisionMonoid.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280309._hygCtx._hyg.5)))) inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280309._hygCtx._hyg.8] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280309._hygCtx._hyg.14 : ContinuousMul.{u_3} G₀ 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G₀},\n  ContinuousWithinAt f s a → ∀ (m : ℤ), f a ≠ 0 ∨ 0 ≤ m → ContinuousWithinAt (fun x => f x ^ m) s a","typeReferences":[["DivInvMonoid","toZPow"],["GroupWithZero","toDivInvMonoid"],["MulZeroClass","toMul"],["MonoidWithZero","toMulZeroOneClass"],["ContinuousWithinAt"],["Or"],["GroupWithZero","toMonoidWithZero"],["Int","instLEInt"],["Zero","toOfNat0"],["instHPow"],["InvOneClass","toInv"],["Set"],["ContinuousInv₀"],["GroupWithZero"],["HPow","hPow"],["OfNat","ofNat"],["Int"],["TopologicalSpace"],["DivInvOneMonoid","toInvOneClass"],["MulZeroOneClass","toMulZeroClass"],["instOfNat"],["MulZeroClass","toZero"],["LE","le"],["Ne"],["ContinuousMul"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["nhdsWithin"],["Filter","Tendsto","zpow₀"]]},{"isProp":true,"kind":"theorem","name":["Continuous","inv₀"],"typeFallback":"forall {α : Type.{u_1}} {G₀ : Type.{u_3}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392310._hygCtx._hyg.5 : Zero.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392310._hygCtx._hyg.8 : Inv.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392310._hygCtx._hyg.11 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392310._hygCtx._hyg.14 : ContinuousInv₀.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392310._hygCtx._hyg.5 inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392310._hygCtx._hyg.8 inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392310._hygCtx._hyg.11] {f : α -> G₀} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392310._hygCtx._hyg.25 : TopologicalSpace.{u_1} α], (Continuous.{u_1, u_3} α G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392310._hygCtx._hyg.25 inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392310._hygCtx._hyg.11 f) -> (forall (x : α), Ne.{succ u_3} G₀ (f x) (OfNat.ofNat.{u_3} G₀ 0 (Zero.toOfNat0.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392310._hygCtx._hyg.5))) -> (Continuous.{u_1, u_3} α G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392310._hygCtx._hyg.25 inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392310._hygCtx._hyg.11 (fun (x : α) => Inv.inv.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392310._hygCtx._hyg.8 (f x)))","typeFull":"∀ {α : Type u_1} {G₀ : Type u_3} [inst : Zero G₀] [inst_1 : Inv G₀] [inst_2 : TopologicalSpace G₀] [ContinuousInv₀ G₀]\n  {f : α → G₀} [inst_4 : TopologicalSpace α], Continuous f → (∀ (x : α), f x ≠ 0) → Continuous fun x => (f x)⁻¹","typeReadable":"∀ {α : Type u_1} {G₀ : Type u_3} [inst : Zero G₀] [inst_1 : Inv G₀] [inst_2 : TopologicalSpace G₀] [ContinuousInv₀ G₀]\n  {f : α → G₀} [inst_4 : TopologicalSpace α], Continuous f → (∀ (x : α), f x ≠ 0) → Continuous fun x => (f x)⁻¹","typeReferences":[["TopologicalSpace"],["Continuous"],["Inv","inv"],["Inv"],["ContinuousInv₀"],["Zero","toOfNat0"],["Ne"],["Zero"],["OfNat","ofNat"]],"valueReferences":[["Continuous"],["Continuous","tendsto"],["Inv","inv"],["continuous_iff_continuousAt"],["Iff","mpr"],["ContinuousAt"],["nhds"],["Filter","Tendsto","inv₀"]]},{"isProp":false,"kind":"definition","name":["unitsHomeomorphNeZero"],"typeFallback":"forall {G₀ : Type.{u_3}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1815514143._hygCtx._hyg.5 : GroupWithZero.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1815514143._hygCtx._hyg.8 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1815514143._hygCtx._hyg.11 : ContinuousInv₀.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.1815514143._hygCtx._hyg.5)))) 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(GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.1815514143._hygCtx._hyg.5))) (instTopologicalSpaceSubtype.{u_3} G₀ (fun (g : G₀) => Ne.{succ u_3} G₀ g (OfNat.ofNat.{u_3} G₀ 0 (Zero.toOfNat0.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.1815514143._hygCtx._hyg.5))))))) inst._@.Mathlib.Topology.Algebra.GroupWithZero.1815514143._hygCtx._hyg.8)","typeFull":"{G₀ : Type u_3} →\n  [inst : GroupWithZero G₀] → [inst_1 : TopologicalSpace G₀] → [ContinuousInv₀ G₀] → G₀ˣ ≃ₜ { g // g ≠ 0 }","typeReadable":"{G₀ : Type u_3} →\n  [inst : GroupWithZero G₀] → [inst_1 : TopologicalSpace G₀] → [ContinuousInv₀ G₀] → G₀ˣ ≃ₜ { g // g ≠ 0 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{G₀ : Type.{u_3}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280308._hygCtx._hyg.5 : GroupWithZero.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280308._hygCtx._hyg.8 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280308._hygCtx._hyg.11 : ContinuousInv₀.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280308._hygCtx._hyg.5)))) (InvOneClass.toInv.{u_3} G₀ (DivInvOneMonoid.toInvOneClass.{u_3} G₀ (DivisionMonoid.toDivInvOneMonoid.{u_3} G₀ (GroupWithZero.toDivisionMonoid.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280308._hygCtx._hyg.5)))) inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280308._hygCtx._hyg.8] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280308._hygCtx._hyg.14 : ContinuousMul.{u_3} G₀ 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ContinuousAt (fun x => f x ^ m) a","typeReferences":[["DivInvMonoid","toZPow"],["GroupWithZero","toDivInvMonoid"],["MulZeroClass","toMul"],["MonoidWithZero","toMulZeroOneClass"],["Or"],["GroupWithZero","toMonoidWithZero"],["Int","instLEInt"],["ContinuousAt"],["Zero","toOfNat0"],["instHPow"],["InvOneClass","toInv"],["ContinuousInv₀"],["GroupWithZero"],["HPow","hPow"],["OfNat","ofNat"],["Int"],["TopologicalSpace"],["DivInvOneMonoid","toInvOneClass"],["MulZeroOneClass","toMulZeroClass"],["instOfNat"],["MulZeroClass","toZero"],["LE","le"],["Ne"],["ContinuousMul"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["Filter","Tendsto","zpow₀"],["nhds"]]},{"isProp":true,"kind":"theorem","name":["Continuous","zpow₀"],"typeFallback":"forall {G₀ : Type.{u_3}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280311._hygCtx._hyg.5 : GroupWithZero.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280311._hygCtx._hyg.8 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280311._hygCtx._hyg.11 : ContinuousInv₀.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280311._hygCtx._hyg.5)))) (InvOneClass.toInv.{u_3} G₀ (DivInvOneMonoid.toInvOneClass.{u_3} G₀ (DivisionMonoid.toDivInvOneMonoid.{u_3} G₀ (GroupWithZero.toDivisionMonoid.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280311._hygCtx._hyg.5)))) inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280311._hygCtx._hyg.8] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280311._hygCtx._hyg.14 : ContinuousMul.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280311._hygCtx._hyg.8 (MulZeroClass.toMul.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280311._hygCtx._hyg.5))))] {X : Type.{u_4}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280311._hygCtx._hyg.18 : TopologicalSpace.{u_4} X] {f : X -> G₀}, (Continuous.{u_4, u_3} X G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280311._hygCtx._hyg.18 inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280311._hygCtx._hyg.8 f) -> (forall (m : Int), (forall (a : X), Or (Ne.{succ u_3} G₀ (f a) (OfNat.ofNat.{u_3} G₀ 0 (Zero.toOfNat0.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280311._hygCtx._hyg.5))))))) (LE.le.{0} Int Int.instLEInt (OfNat.ofNat.{0} Int 0 (instOfNat 0)) m)) -> (Continuous.{u_4, u_3} X G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280311._hygCtx._hyg.18 inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280311._hygCtx._hyg.8 (fun (x : X) => HPow.hPow.{u_3, 0, u_3} G₀ Int G₀ (instHPow.{u_3, 0} G₀ Int (DivInvMonoid.toZPow.{u_3} G₀ (GroupWithZero.toDivInvMonoid.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280311._hygCtx._hyg.5))) (f x) m)))","typeFull":"∀ {G₀ : Type u_3} [inst : GroupWithZero G₀] [inst_1 : TopologicalSpace G₀] [ContinuousInv₀ G₀] [ContinuousMul G₀]\n  {X : Type u_4} [inst_4 : TopologicalSpace X] {f : X → G₀},\n  Continuous f → ∀ (m : ℤ), (∀ (a : X), f a ≠ 0 ∨ 0 ≤ m) → Continuous fun x => f x ^ m","typeReadable":"∀ {G₀ : Type u_3} [inst : GroupWithZero G₀] [inst_1 : TopologicalSpace G₀] [ContinuousInv₀ G₀] [ContinuousMul G₀]\n  {X : Type u_4} [inst_4 : TopologicalSpace X] {f : X → G₀},\n  Continuous f → ∀ (m : ℤ), (∀ (a : X), f a ≠ 0 ∨ 0 ≤ m) → Continuous fun x => f x ^ m","typeReferences":[["DivInvMonoid","toZPow"],["GroupWithZero","toDivInvMonoid"],["MulZeroClass","toMul"],["MonoidWithZero","toMulZeroOneClass"],["Or"],["GroupWithZero","toMonoidWithZero"],["Int","instLEInt"],["Zero","toOfNat0"],["instHPow"],["InvOneClass","toInv"],["ContinuousInv₀"],["GroupWithZero"],["HPow","hPow"],["OfNat","ofNat"],["Int"],["TopologicalSpace"],["Continuous"],["DivInvOneMonoid","toInvOneClass"],["MulZeroOneClass","toMulZeroClass"],["instOfNat"],["MulZeroClass","toZero"],["LE","le"],["Ne"],["ContinuousMul"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["instHPow"],["Continuous"],["Continuous","tendsto"],["continuous_iff_continuousAt"],["Filter","Tendsto","zpow₀"],["DivInvMonoid","toZPow"],["GroupWithZero","toDivInvMonoid"],["Iff","mpr"],["ContinuousAt"],["HPow","hPow"],["nhds"],["Int"]]},{"isProp":true,"kind":"theorem","name":["Homeomorph","mulLeft₀","_proof_1"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2381840360._hygCtx._hyg.5 : TopologicalSpace.{u_1} α] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2381840360._hygCtx._hyg.8 : GroupWithZero.{u_1} α] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2381840360._hygCtx._hyg.11 : SeparatelyContinuousMul.{u_1} α inst._@.Mathlib.Topology.Algebra.GroupWithZero.2381840360._hygCtx._hyg.5 (MulZeroClass.toMul.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Topology.Algebra.GroupWithZero.2381840360._hygCtx._hyg.8))))] (c : α) (hc : Ne.{succ u_1} α c (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Topology.Algebra.GroupWithZero.2381840360._hygCtx._hyg.8))))))), Continuous.{u_1, u_1} α α inst._@.Mathlib.Topology.Algebra.GroupWithZero.2381840360._hygCtx._hyg.5 inst._@.Mathlib.Topology.Algebra.GroupWithZero.2381840360._hygCtx._hyg.5 (fun (x._@.Mathlib.Topology.Algebra.Monoid.Defs.793145910._hygCtx._hyg.16 : α) => HMul.hMul.{u_1, u_1, u_1} α α α (instHMul.{u_1} α (MulOne.toMul.{u_1} α (MulOneClass.toMulOne.{u_1} α (Monoid.toMulOneClass.{u_1} α (MonoidWithZero.toMonoid.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Topology.Algebra.GroupWithZero.2381840360._hygCtx._hyg.8)))))) (Units.val.{u_1} α (MonoidWithZero.toMonoid.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Topology.Algebra.GroupWithZero.2381840360._hygCtx._hyg.8)) (Units.mk0.{u_1} α inst._@.Mathlib.Topology.Algebra.GroupWithZero.2381840360._hygCtx._hyg.8 c hc)) x._@.Mathlib.Topology.Algebra.Monoid.Defs.793145910._hygCtx._hyg.16)","typeFull":"∀ {α : Type u_1} [inst : TopologicalSpace α] [inst_1 : GroupWithZero α] [SeparatelyContinuousMul α] (c : α)\n  (hc : c ≠ 0), Continuous fun x => ↑(Units.mk0 c hc) * x","typeReadable":"∀ {α : Type u_1} [inst : TopologicalSpace α] [inst_1 : GroupWithZero α] [SeparatelyContinuousMul α] (c : α)\n  (hc : c ≠ 0), Continuous fun x => ↑(Units.mk0 c hc) * x","typeReferences":[["MulOneClass","toMulOne"],["MulZeroClass","toMul"],["HMul","hMul"],["Units","mk0"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["OfNat","ofNat"],["TopologicalSpace"],["Continuous"],["MulOne","toMul"],["SeparatelyContinuousMul"],["MulZeroOneClass","toMulZeroClass"],["Units","val"],["MulZeroClass","toZero"],["MonoidWithZero","toMonoid"],["Monoid","toMulOneClass"],["GroupWithZero","toMonoidWithZero"],["instHMul"],["Ne"],["Zero","toOfNat0"]],"valueReferences":[["MulOneClass","toMulOne"],["MulOne","toMul"],["Units","val"],["continuous_const_mul"],["GroupWithZero","toMonoidWithZero"],["MonoidWithZero","toMonoid"],["Monoid","toMulOneClass"],["Units","mk0"]]},{"isProp":false,"kind":"definition","name":["Homeomorph","inv₀"],"typeFallback":"forall (G₀ : Type.{u_3}) [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392312._hygCtx._hyg.5 : GroupWithZero.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392312._hygCtx._hyg.8 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392312._hygCtx._hyg.11 : ContinuousInv₀.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392312._hygCtx._hyg.5)))) (InvOneClass.toInv.{u_3} G₀ (DivInvOneMonoid.toInvOneClass.{u_3} G₀ (DivisionMonoid.toDivInvOneMonoid.{u_3} G₀ (GroupWithZero.toDivisionMonoid.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392312._hygCtx._hyg.5)))) inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392312._hygCtx._hyg.8], Homeomorph.{u_3, u_3} (Subtype.{succ u_3} G₀ (fun (g : G₀) => Ne.{succ u_3} G₀ g (OfNat.ofNat.{u_3} G₀ 0 (Zero.toOfNat0.{u_3} G₀ 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inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392312._hygCtx._hyg.8) (instTopologicalSpaceSubtype.{u_3} G₀ (fun (g : G₀) => Ne.{succ u_3} G₀ g (OfNat.ofNat.{u_3} G₀ 0 (Zero.toOfNat0.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392312._hygCtx._hyg.5))))))) inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392312._hygCtx._hyg.8)","typeFull":"(G₀ : Type u_3) →\n  [inst : GroupWithZero G₀] → [inst_1 : TopologicalSpace G₀] → [ContinuousInv₀ G₀] → { g // g ≠ 0 } ≃ₜ { g // g ≠ 0 }","typeReadable":"(G₀ : Type u_3) →\n  [inst : GroupWithZero G₀] → [inst_1 : TopologicalSpace G₀] → [ContinuousInv₀ G₀] → { g // g ≠ 0 } ≃ₜ { g // g ≠ 0 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TopologicalSpace G₀] [inst_1 : GroupWithZero G₀] [SeparatelyContinuousMul G₀] {a : G₀},\n  a ≠ 0 → ∀ (b : G₀), Filter.map (fun x => x * a) (nhds b) = nhds (b * a)","typeReferences":[["MulZeroClass","toMul"],["Filter","map"],["HMul","hMul"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["OfNat","ofNat"],["Filter"],["TopologicalSpace"],["SeparatelyContinuousMul"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["GroupWithZero","toMonoidWithZero"],["instHMul"],["Ne"],["Zero","toOfNat0"],["Eq"],["nhds"]],"valueReferences":[["Homeomorph","map_nhds_eq"],["Homeomorph","mulRight₀"]]},{"isProp":true,"kind":"theorem","name":["Homeomorph","mulRight₀","congr_simp"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3978221141._hygCtx._hyg.5 : TopologicalSpace.{u_1} α] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3978221141._hygCtx._hyg.8 : GroupWithZero.{u_1} α] 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[inst : GroupWithZero G₀] [inst_1 : TopologicalSpace G₀] [ContinuousInv₀ G₀] [ContinuousMul G₀]\n  (x : G₀) (m : ℤ), x ≠ 0 ∨ 0 ≤ m → ContinuousAt (fun x => x ^ m) x","typeReadable":"∀ {G₀ : Type u_3} [inst : GroupWithZero G₀] [inst_1 : TopologicalSpace G₀] [ContinuousInv₀ G₀] [ContinuousMul G₀]\n  (x : G₀) (m : ℤ), x ≠ 0 ∨ 0 ≤ m → ContinuousAt (fun x => x ^ m) 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{G₀ : Type.{u_3}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3007347988._hygCtx._hyg.5 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3007347988._hygCtx._hyg.8 : GroupWithZero.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3007347988._hygCtx._hyg.11 : SeparatelyContinuousMul.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3007347988._hygCtx._hyg.5 (MulZeroClass.toMul.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3007347988._hygCtx._hyg.8))))] {a : G₀}, (Ne.{succ u_3} G₀ a (OfNat.ofNat.{u_3} G₀ 0 (Zero.toOfNat0.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3007347988._hygCtx._hyg.8))))))) -> (forall (b : G₀), Eq.{succ u_3} (Filter.{u_3} 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TopologicalSpace G₀] [inst_1 : GroupWithZero G₀] [SeparatelyContinuousMul G₀] {a : G₀},\n  a ≠ 0 → ∀ (b : G₀), Filter.map (fun x => a * x) (nhds b) = nhds (a * b)","typeReadable":"∀ {G₀ : Type u_3} [inst : TopologicalSpace G₀] [inst_1 : GroupWithZero G₀] [SeparatelyContinuousMul G₀] {a : G₀},\n  a ≠ 0 → ∀ (b : G₀), Filter.map (fun x => a * x) (nhds b) = nhds (a * b)","typeReferences":[["MulZeroClass","toMul"],["Filter","map"],["HMul","hMul"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["OfNat","ofNat"],["Filter"],["TopologicalSpace"],["SeparatelyContinuousMul"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["GroupWithZero","toMonoidWithZero"],["instHMul"],["Ne"],["Zero","toOfNat0"],["Eq"],["nhds"]],"valueReferences":[["Homeomorph","map_nhds_eq"],["Homeomorph","mulLeft₀"]]},{"isProp":true,"kind":"theorem","name":["ContinuousAt","inv₀"],"typeFallback":"forall {α : Type.{u_1}} {G₀ : Type.{u_3}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392309._hygCtx._hyg.5 : Zero.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392309._hygCtx._hyg.8 : Inv.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392309._hygCtx._hyg.11 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392309._hygCtx._hyg.14 : ContinuousInv₀.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392309._hygCtx._hyg.5 inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392309._hygCtx._hyg.8 inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392309._hygCtx._hyg.11] {f : α -> G₀} {a : α} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392309._hygCtx._hyg.25 : TopologicalSpace.{u_1} α], (ContinuousAt.{u_1, u_3} α G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392309._hygCtx._hyg.25 inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392309._hygCtx._hyg.11 f a) -> (Ne.{succ u_3} G₀ (f a) (OfNat.ofNat.{u_3} G₀ 0 (Zero.toOfNat0.{u_3} G₀ 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(Homeomorph.mulLeft₀.{u_1} α inst._@.Mathlib.Topology.Algebra.GroupWithZero.991527267._hygCtx._hyg.5 inst._@.Mathlib.Topology.Algebra.GroupWithZero.991527267._hygCtx._hyg.8 inst._@.Mathlib.Topology.Algebra.GroupWithZero.991527267._hygCtx._hyg.11 c hc)) (fun (x._@.Mathlib.Topology.Algebra.GroupWithZero.991527267._hygCtx._hyg.32 : α) => HMul.hMul.{u_1, u_1, u_1} α α α (instHMul.{u_1} α (MulZeroClass.toMul.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Topology.Algebra.GroupWithZero.991527267._hygCtx._hyg.8))))) c x._@.Mathlib.Topology.Algebra.GroupWithZero.991527267._hygCtx._hyg.32)","typeFull":"∀ {α : Type u_1} [inst : TopologicalSpace α] [inst_1 : GroupWithZero α] [inst_2 : SeparatelyContinuousMul α] (c : α)\n  (hc : c ≠ 0), ⇑(Homeomorph.mulLeft₀ c hc) = fun x => c * x","typeReadable":"∀ {α : Type u_1} [inst : TopologicalSpace α] [inst_1 : GroupWithZero α] [inst_2 : SeparatelyContinuousMul α] (c : α)\n  (hc : c ≠ 0), ⇑(Homeomorph.mulLeft₀ c hc) = fun x => c * x","typeReferences":[["Homeomorph","mulLeft₀"],["Homeomorph","instEquivLike"],["MulZeroClass","toMul"],["HMul","hMul"],["MonoidWithZero","toMulZeroOneClass"],["Homeomorph"],["GroupWithZero"],["DFunLike","coe"],["OfNat","ofNat"],["TopologicalSpace"],["SeparatelyContinuousMul"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["EquivLike","toFunLike"],["GroupWithZero","toMonoidWithZero"],["instHMul"],["Ne"],["Zero","toOfNat0"],["Eq"]],"valueReferences":[["rfl"],["Homeomorph","mulLeft₀"],["Homeomorph","instEquivLike"],["EquivLike","toFunLike"],["Homeomorph"],["DFunLike","coe"]]},{"isProp":true,"kind":"theorem","name":["ContinuousOn","div"],"typeFallback":"forall {α : Type.{u_1}} {G₀ : Type.{u_3}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639156._hygCtx._hyg.5 : GroupWithZero.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639156._hygCtx._hyg.8 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639156._hygCtx._hyg.11 : ContinuousInv₀.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639156._hygCtx._hyg.5)))) (InvOneClass.toInv.{u_3} G₀ (DivInvOneMonoid.toInvOneClass.{u_3} G₀ (DivisionMonoid.toDivInvOneMonoid.{u_3} G₀ (GroupWithZero.toDivisionMonoid.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639156._hygCtx._hyg.5)))) inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639156._hygCtx._hyg.8] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639156._hygCtx._hyg.14 : ContinuousMul.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639156._hygCtx._hyg.8 (MulZeroClass.toMul.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639156._hygCtx._hyg.5))))] {f : α -> G₀} {g : α -> G₀} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639156._hygCtx._hyg.23 : TopologicalSpace.{u_1} α] {s : Set.{u_1} α}, (ContinuousOn.{u_1, u_3} α G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639156._hygCtx._hyg.23 inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639156._hygCtx._hyg.8 f s) -> (ContinuousOn.{u_1, u_3} α G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639156._hygCtx._hyg.23 inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639156._hygCtx._hyg.8 g s) -> (forall (x : α), (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s x) -> (Ne.{succ u_3} G₀ (g x) (OfNat.ofNat.{u_3} G₀ 0 (Zero.toOfNat0.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639156._hygCtx._hyg.5)))))))) -> (ContinuousOn.{u_1, u_3} α G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639156._hygCtx._hyg.23 inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639156._hygCtx._hyg.8 (HDiv.hDiv.{max u_1 u_3, max u_1 u_3, max u_1 u_3} (α -> G₀) (α -> G₀) (α -> G₀) (instHDiv.{max u_1 u_3} (α -> G₀) (Pi.instDiv.{u_1, u_3} α (fun (a._@._internal._hyg.0 : α) => G₀) (fun (i : α) => DivInvMonoid.toDiv.{u_3} G₀ (GroupWithZero.toDivInvMonoid.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639156._hygCtx._hyg.5)))) f g) s)","typeFull":"∀ {α : Type u_1} {G₀ : Type u_3} [inst : GroupWithZero G₀] [inst_1 : TopologicalSpace G₀] [ContinuousInv₀ G₀]\n  [ContinuousMul G₀] {f g : α → G₀} [inst_4 : TopologicalSpace α] {s : Set α},\n  ContinuousOn f s → ContinuousOn g s → (∀ x ∈ s, g x ≠ 0) → ContinuousOn (f / g) s","typeReadable":"∀ {α : Type u_1} {G₀ : Type u_3} [inst : GroupWithZero G₀] [inst_1 : TopologicalSpace G₀] [ContinuousInv₀ G₀]\n  [ContinuousMul G₀] {f g : α → G₀} [inst_4 : TopologicalSpace α] {s : Set α},\n  ContinuousOn f s → ContinuousOn g s → (∀ x ∈ s, g x ≠ 0) → ContinuousOn (f / g) s","typeReferences":[["GroupWithZero","toDivInvMonoid"],["MulZeroClass","toMul"],["Membership","mem"],["ContinuousOn"],["MonoidWithZero","toMulZeroOneClass"],["instHDiv"],["HDiv","hDiv"],["GroupWithZero","toMonoidWithZero"],["Zero","toOfNat0"],["Pi","instDiv"],["InvOneClass","toInv"],["Set"],["ContinuousInv₀"],["GroupWithZero"],["OfNat","ofNat"],["DivInvMonoid","toDiv"],["Set","instMembership"],["TopologicalSpace"],["DivInvOneMonoid","toInvOneClass"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Ne"],["ContinuousMul"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["ContinuousWithinAt","div"]]},{"isProp":true,"kind":"definition","name":["ContinuousInv₀","mk","_flat_ctor"],"typeFallback":"forall {G₀ : Type.{u_4}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.9 : Zero.{u_4} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.12 : Inv.{u_4} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.15 : TopologicalSpace.{u_4} G₀], (forall {{x : G₀}}, (Ne.{succ u_4} G₀ x (OfNat.ofNat.{u_4} G₀ 0 (Zero.toOfNat0.{u_4} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.9))) -> (ContinuousAt.{u_4, u_4} G₀ G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.15 inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.15 (Inv.inv.{u_4} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.12) x)) -> (ContinuousInv₀.{u_4} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.9 inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.12 inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.15)","typeFull":"∀ {G₀ : Type u_4} [inst : Zero G₀] [inst_1 : Inv G₀] [inst_2 : TopologicalSpace G₀],\n  (∀ ⦃x : G₀⦄, x ≠ 0 → ContinuousAt Inv.inv x) → ContinuousInv₀ G₀","typeReadable":"∀ {G₀ : Type u_4} [inst : Zero G₀] [inst_1 : Inv G₀] [inst_2 : TopologicalSpace G₀],\n  (∀ ⦃x : G₀⦄, x ≠ 0 → ContinuousAt Inv.inv x) → ContinuousInv₀ G₀","typeReferences":[["TopologicalSpace"],["Inv","inv"],["Inv"],["ContinuousInv₀"],["ContinuousAt"],["Zero","toOfNat0"],["Ne"],["Zero"],["OfNat","ofNat"]],"valueReferences":[["ContinuousInv₀","mk"]]},{"isProp":true,"kind":"theorem","name":["Filter","Tendsto","div_const"],"typeFallback":"forall {α : Type.{u_1}} {G₀ : Type.{u_3}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106153._hygCtx._hyg.5 : DivInvMonoid.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106153._hygCtx._hyg.8 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106153._hygCtx._hyg.11 : SeparatelyContinuousMul.{u_3} G₀ 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G₀] [inst_1 : TopologicalSpace G₀] [SeparatelyContinuousMul G₀]\n  {f : α → G₀} {l : Filter α} {x : G₀},\n  Filter.Tendsto f l (nhds x) → ∀ (y : G₀), Filter.Tendsto (fun a => f a / y) l (nhds (x / y))","typeReadable":"∀ {α : Type u_1} {G₀ : Type u_3} [inst : DivInvMonoid G₀] [inst_1 : TopologicalSpace G₀] [SeparatelyContinuousMul G₀]\n  {f : α → G₀} {l : Filter α} {x : G₀},\n  Filter.Tendsto f l (nhds x) → ∀ (y : G₀), Filter.Tendsto (fun a => f a / y) l (nhds (x / y))","typeReferences":[["MulOneClass","toMulOne"],["DivInvMonoid"],["instHDiv"],["Filter"],["DivInvMonoid","toDiv"],["TopologicalSpace"],["HDiv","hDiv"],["MulOne","toMul"],["SeparatelyContinuousMul"],["DivInvMonoid","toMonoid"],["Monoid","toMulOneClass"],["Filter","Tendsto"],["nhds"]],"valueReferences":[["div_eq_mul_inv"],["DivInvMonoid","toInv"],["MulOneClass","toMulOne"],["Inv","inv"],["HMul","hMul"],["instHDiv"],["Filter"],["DivInvMonoid","toDiv"],["congrArg"],["HDiv","hDiv"],["MulOne","toMul"],["DivInvMonoid","toMonoid"],["congr"],["Filter","Tendsto","mul_const"],["funext"],["Monoid","toMulOneClass"],["id"],["instHMul"],["congrFun'"],["Eq","mpr"],["Eq"],["Filter","Tendsto"],["nhds"]]},{"isProp":false,"kind":"inductive","name":["ContinuousInv₀"],"typeFallback":"forall (G₀ : Type.{u_4}) [inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.9 : Zero.{u_4} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.12 : Inv.{u_4} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.15 : TopologicalSpace.{u_4} G₀], Prop","typeFull":"(G₀ : Type u_4) → [Zero G₀] → [Inv G₀] → [TopologicalSpace G₀] → Prop","typeReadable":"(G₀ : Type u_4) → [Zero G₀] → [Inv G₀] → [TopologicalSpace G₀] → Prop","typeReferences":[["TopologicalSpace"],["Inv"],["Zero"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["ContinuousWithinAt","div"],"typeFallback":"forall {α : Type.{u_1}} {G₀ : Type.{u_3}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639155._hygCtx._hyg.5 : GroupWithZero.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639155._hygCtx._hyg.8 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639155._hygCtx._hyg.11 : ContinuousInv₀.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639155._hygCtx._hyg.5)))) (InvOneClass.toInv.{u_3} G₀ (DivInvOneMonoid.toInvOneClass.{u_3} G₀ (DivisionMonoid.toDivInvOneMonoid.{u_3} G₀ (GroupWithZero.toDivisionMonoid.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639155._hygCtx._hyg.5)))) inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639155._hygCtx._hyg.8] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639155._hygCtx._hyg.14 : ContinuousMul.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639155._hygCtx._hyg.8 (MulZeroClass.toMul.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639155._hygCtx._hyg.5))))] {f : α -> G₀} {g : α -> G₀} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639155._hygCtx._hyg.23 : TopologicalSpace.{u_1} α] {s : Set.{u_1} α} {a : α}, (ContinuousWithinAt.{u_1, u_3} α G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639155._hygCtx._hyg.23 inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639155._hygCtx._hyg.8 f s a) -> (ContinuousWithinAt.{u_1, u_3} α G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639155._hygCtx._hyg.23 inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639155._hygCtx._hyg.8 g s a) -> (Ne.{succ u_3} G₀ (g a) (OfNat.ofNat.{u_3} G₀ 0 (Zero.toOfNat0.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639155._hygCtx._hyg.5))))))) -> (ContinuousWithinAt.{u_1, u_3} α G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639155._hygCtx._hyg.23 inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639155._hygCtx._hyg.8 (HDiv.hDiv.{max u_1 u_3, max u_1 u_3, max u_1 u_3} (α -> G₀) (α -> G₀) (α -> G₀) (instHDiv.{max u_1 u_3} (α -> G₀) (Pi.instDiv.{u_1, u_3} α (fun (a._@._internal._hyg.0 : α) => G₀) (fun (i : α) => DivInvMonoid.toDiv.{u_3} G₀ (GroupWithZero.toDivInvMonoid.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639155._hygCtx._hyg.5)))) f g) s a)","typeFull":"∀ {α : Type u_1} {G₀ : Type u_3} [inst : GroupWithZero G₀] [inst_1 : TopologicalSpace G₀] [ContinuousInv₀ G₀]\n  [ContinuousMul G₀] {f g : α → G₀} [inst_4 : TopologicalSpace α] {s : Set α} {a : α},\n  ContinuousWithinAt f s a → ContinuousWithinAt g s a → g a ≠ 0 → ContinuousWithinAt (f / g) s a","typeReadable":"∀ {α : Type u_1} {G₀ : Type u_3} [inst : GroupWithZero G₀] [inst_1 : TopologicalSpace G₀] [ContinuousInv₀ G₀]\n  [ContinuousMul G₀] {f g : α → G₀} [inst_4 : TopologicalSpace α] {s : Set α} {a : α},\n  ContinuousWithinAt f s a → ContinuousWithinAt g s a → g a ≠ 0 → ContinuousWithinAt (f / g) s a","typeReferences":[["InvOneClass","toInv"],["Set"],["GroupWithZero","toDivInvMonoid"],["ContinuousInv₀"],["MulZeroClass","toMul"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["instHDiv"],["DivInvMonoid","toDiv"],["OfNat","ofNat"],["ContinuousWithinAt"],["HDiv","hDiv"],["TopologicalSpace"],["DivInvOneMonoid","toInvOneClass"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["GroupWithZero","toMonoidWithZero"],["Zero","toOfNat0"],["Ne"],["Pi","instDiv"],["DivisionMonoid","toDivInvOneMonoid"],["ContinuousMul"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["nhdsWithin"],["Filter","Tendsto","div"]]},{"isProp":true,"kind":"theorem","name":["continuousOn_div"],"typeFallback":"forall {G₀ : Type.{u_3}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.4011998232._hygCtx._hyg.5 : GroupWithZero.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.4011998232._hygCtx._hyg.8 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.4011998232._hygCtx._hyg.11 : ContinuousInv₀.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.4011998232._hygCtx._hyg.5)))) (InvOneClass.toInv.{u_3} G₀ (DivInvOneMonoid.toInvOneClass.{u_3} G₀ (DivisionMonoid.toDivInvOneMonoid.{u_3} G₀ (GroupWithZero.toDivisionMonoid.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.4011998232._hygCtx._hyg.5)))) inst._@.Mathlib.Topology.Algebra.GroupWithZero.4011998232._hygCtx._hyg.8] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.4011998232._hygCtx._hyg.14 : ContinuousMul.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.4011998232._hygCtx._hyg.8 (MulZeroClass.toMul.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.4011998232._hygCtx._hyg.5))))], ContinuousOn.{u_3, u_3} (Prod.{u_3, u_3} G₀ G₀) G₀ (instTopologicalSpaceProd.{u_3, u_3} G₀ G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.4011998232._hygCtx._hyg.8 inst._@.Mathlib.Topology.Algebra.GroupWithZero.4011998232._hygCtx._hyg.8) inst._@.Mathlib.Topology.Algebra.GroupWithZero.4011998232._hygCtx._hyg.8 (fun (p : Prod.{u_3, u_3} G₀ G₀) => HDiv.hDiv.{u_3, u_3, u_3} G₀ G₀ G₀ (instHDiv.{u_3} G₀ (DivInvMonoid.toDiv.{u_3} G₀ (GroupWithZero.toDivInvMonoid.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.4011998232._hygCtx._hyg.5))) (Prod.fst.{u_3, u_3} G₀ G₀ p) (Prod.snd.{u_3, u_3} G₀ G₀ p)) (setOf.{u_3} (Prod.{u_3, u_3} G₀ G₀) (fun (p : Prod.{u_3, u_3} G₀ G₀) => Ne.{succ u_3} G₀ (Prod.snd.{u_3, u_3} G₀ G₀ p) (OfNat.ofNat.{u_3} G₀ 0 (Zero.toOfNat0.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ 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Inv G₀] →\n      [inst_2 : TopologicalSpace G₀] →\n        {motive : ContinuousInv₀ G₀ → Sort u} →\n          ((continuousAt_inv₀ : ∀ ⦃x : G₀⦄, x ≠ 0 → ContinuousAt Inv.inv x) → motive ⋯) →\n            (t : ContinuousInv₀ G₀) → motive t","typeReferences":[["TopologicalSpace"],["Inv","inv"],["Inv"],["ContinuousInv₀"],["ContinuousAt"],["Zero","toOfNat0"],["Ne"],["Zero"],["ContinuousInv₀","mk"],["OfNat","ofNat"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["Filter","Tendsto","inv₀"],"typeFallback":"forall {α : Type.{u_1}} {G₀ : Type.{u_3}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392307._hygCtx._hyg.5 : Zero.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392307._hygCtx._hyg.8 : Inv.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392307._hygCtx._hyg.11 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392307._hygCtx._hyg.14 : ContinuousInv₀.{u_3} G₀ 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[inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280307._hygCtx._hyg.8 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280307._hygCtx._hyg.11 : ContinuousInv₀.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280307._hygCtx._hyg.5)))) (InvOneClass.toInv.{u_3} G₀ (DivInvOneMonoid.toInvOneClass.{u_3} G₀ (DivisionMonoid.toDivInvOneMonoid.{u_3} G₀ (GroupWithZero.toDivisionMonoid.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280307._hygCtx._hyg.5)))) inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280307._hygCtx._hyg.8] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280307._hygCtx._hyg.14 : ContinuousMul.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3564280307._hygCtx._hyg.8 (MulZeroClass.toMul.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ 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{inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.9 : Zero.{u_4} G₀} {inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.12 : Inv.{u_4} G₀} {inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.15 : TopologicalSpace.{u_4} G₀} [self : ContinuousInv₀.{u_4} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.9 inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.12 inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.15] {{x : G₀}}, (Ne.{succ u_4} G₀ x (OfNat.ofNat.{u_4} G₀ 0 (Zero.toOfNat0.{u_4} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.9))) -> (ContinuousAt.{u_4, u_4} G₀ G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.15 inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.15 (Inv.inv.{u_4} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.12) x)","typeFull":"∀ {G₀ : Type u_4} {inst : Zero G₀} {inst_1 : Inv G₀} {inst_2 : TopologicalSpace G₀} [self : ContinuousInv₀ G₀] ⦃x : G₀⦄,\n  x ≠ 0 → ContinuousAt Inv.inv x","typeReadable":"∀ {G₀ : Type u_4} {inst : Zero G₀} {inst_1 : Inv G₀} {inst_2 : TopologicalSpace G₀} [self : ContinuousInv₀ G₀] ⦃x : G₀⦄,\n  x ≠ 0 → ContinuousAt Inv.inv x","typeReferences":[["TopologicalSpace"],["Inv","inv"],["Inv"],["ContinuousInv₀"],["ContinuousAt"],["Zero","toOfNat0"],["Ne"],["Zero"],["OfNat","ofNat"]],"valueReferences":[]},{"isProp":true,"kind":"theorem","name":["nhds_inv₀"],"typeFallback":"forall {G₀ : Type.{u_3}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3405481537._hygCtx._hyg.5 : GroupWithZero.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3405481537._hygCtx._hyg.8 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3405481537._hygCtx._hyg.11 : ContinuousInv₀.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ 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{α : Type.{u_1}} {G₀ : Type.{u_3}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392311._hygCtx._hyg.5 : Zero.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392311._hygCtx._hyg.8 : Inv.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392311._hygCtx._hyg.11 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392311._hygCtx._hyg.14 : ContinuousInv₀.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392311._hygCtx._hyg.5 inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392311._hygCtx._hyg.8 inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392311._hygCtx._hyg.11] {f : α -> G₀} {s : Set.{u_1} α} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392311._hygCtx._hyg.25 : TopologicalSpace.{u_1} α], (ContinuousOn.{u_1, u_3} α G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392311._hygCtx._hyg.25 inst._@.Mathlib.Topology.Algebra.GroupWithZero.1578392311._hygCtx._hyg.11 f s) -> (forall (x : α), 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ContinuousOn f s → (∀ x ∈ s, f x ≠ 0) → ContinuousOn (fun x => (f x)⁻¹) s","typeReferences":[["TopologicalSpace"],["Inv","inv"],["Inv"],["Set"],["Membership","mem"],["ContinuousInv₀"],["ContinuousOn"],["Zero","toOfNat0"],["Ne"],["Zero"],["OfNat","ofNat"],["Set","instMembership"]],"valueReferences":[["ContinuousWithinAt","inv₀"]]},{"isProp":true,"kind":"theorem","name":["ContinuousAt","div"],"typeFallback":"forall {α : Type.{u_1}} {G₀ : Type.{u_3}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639157._hygCtx._hyg.5 : GroupWithZero.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639157._hygCtx._hyg.8 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639157._hygCtx._hyg.11 : ContinuousInv₀.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2515639157._hygCtx._hyg.5)))) 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inst._@.Mathlib.Topology.Algebra.GroupWithZero.868821435._hygCtx._hyg.5)","typeFull":"∀ {G₀ : Type u_3} [inst : TopologicalSpace G₀] [inst_1 : GroupWithZero G₀] [SeparatelyContinuousMul G₀],\n  Filter.Tendsto Inv.inv (nhds 1) (nhds 1) → ContinuousInv₀ G₀","typeReadable":"∀ {G₀ : Type u_3} [inst : TopologicalSpace G₀] [inst_1 : GroupWithZero G₀] [SeparatelyContinuousMul G₀],\n  Filter.Tendsto Inv.inv (nhds 1) (nhds 1) → ContinuousInv₀ 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{G₀ : Type.{u_3}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3149965788._hygCtx._hyg.5 : GroupWithZero.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3149965788._hygCtx._hyg.8 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3149965788._hygCtx._hyg.11 : ContinuousInv₀.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3149965788._hygCtx._hyg.5)))) (InvOneClass.toInv.{u_3} G₀ (DivInvOneMonoid.toInvOneClass.{u_3} G₀ (DivisionMonoid.toDivInvOneMonoid.{u_3} G₀ (GroupWithZero.toDivisionMonoid.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3149965788._hygCtx._hyg.5)))) inst._@.Mathlib.Topology.Algebra.GroupWithZero.3149965788._hygCtx._hyg.8] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3149965788._hygCtx._hyg.14 : ContinuousMul.{u_3} G₀ 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(MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3149965788._hygCtx._hyg.5))))))))","typeFull":"∀ {G₀ : Type u_3} [inst : GroupWithZero G₀] [inst_1 : TopologicalSpace G₀] [ContinuousInv₀ G₀] [ContinuousMul G₀]\n  (m : ℤ), ContinuousOn (fun x => x ^ m) {0}ᶜ","typeReadable":"∀ {G₀ : Type u_3} [inst : GroupWithZero G₀] [inst_1 : TopologicalSpace G₀] [ContinuousInv₀ G₀] [ContinuousMul G₀]\n  (m : ℤ), ContinuousOn (fun x => x ^ m) {0}ᶜ","typeReferences":[["DivInvMonoid","toZPow"],["GroupWithZero","toDivInvMonoid"],["Compl","compl"],["Singleton","singleton"],["MulZeroClass","toMul"],["ContinuousOn"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero","toMonoidWithZero"],["Zero","toOfNat0"],["instHPow"],["InvOneClass","toInv"],["Set"],["ContinuousInv₀"],["GroupWithZero"],["HPow","hPow"],["Set","instSingletonSet"],["OfNat","ofNat"],["Int"],["Set","instCompl"],["TopologicalSpace"],["DivInvOneMonoid","toInvOneClass"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["ContinuousMul"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["instHPow"],["DivInvMonoid","toZPow"],["Set"],["GroupWithZero","toDivInvMonoid"],["Compl","compl"],["Singleton","singleton"],["MonoidWithZero","toMulZeroOneClass"],["HPow","hPow"],["Set","instSingletonSet"],["OfNat","ofNat"],["Int"],["Set","instCompl"],["instOfNat"],["Or","inl"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["LE","le"],["GroupWithZero","toMonoidWithZero"],["ContinuousAt","continuousWithinAt"],["Int","instLEInt"],["Ne"],["Zero","toOfNat0"],["continuousAt_zpow₀"]]},{"isProp":true,"kind":"theorem","name":["ContinuousAt","div_const"],"typeFallback":"forall {α : Type.{u_1}} {G₀ : Type.{u_3}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106154._hygCtx._hyg.5 : DivInvMonoid.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106154._hygCtx._hyg.8 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106154._hygCtx._hyg.11 : SeparatelyContinuousMul.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106154._hygCtx._hyg.8 (MulOne.toMul.{u_3} G₀ (MulOneClass.toMulOne.{u_3} G₀ (Monoid.toMulOneClass.{u_3} G₀ (DivInvMonoid.toMonoid.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106154._hygCtx._hyg.5))))] {f : α -> G₀} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106154._hygCtx._hyg.21 : TopologicalSpace.{u_1} α] {a : α}, (ContinuousAt.{u_1, u_3} α G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106154._hygCtx._hyg.21 inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106154._hygCtx._hyg.8 f a) -> (forall (y : G₀), ContinuousAt.{u_1, u_3} α G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106154._hygCtx._hyg.21 inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106154._hygCtx._hyg.8 (fun (x : α) => HDiv.hDiv.{u_3, u_3, u_3} G₀ G₀ G₀ (instHDiv.{u_3} G₀ (DivInvMonoid.toDiv.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3705106154._hygCtx._hyg.5)) (f x) y) a)","typeFull":"∀ {α : Type u_1} {G₀ : Type u_3} [inst : DivInvMonoid G₀] [inst_1 : TopologicalSpace G₀] [SeparatelyContinuousMul G₀]\n  {f : α → G₀} [inst_3 : TopologicalSpace α] {a : α}, ContinuousAt f a → ∀ (y : G₀), ContinuousAt (fun x => f x / y) a","typeReadable":"∀ {α : Type u_1} {G₀ : Type u_3} [inst : DivInvMonoid G₀] [inst_1 : TopologicalSpace G₀] [SeparatelyContinuousMul G₀]\n  {f : α → G₀} [inst_3 : TopologicalSpace α] {a : α}, ContinuousAt f a → ∀ (y : G₀), ContinuousAt (fun x => f x / y) a","typeReferences":[["HDiv","hDiv"],["MulOneClass","toMulOne"],["TopologicalSpace"],["DivInvMonoid"],["MulOne","toMul"],["DivInvMonoid","toMonoid"],["SeparatelyContinuousMul"],["Monoid","toMulOneClass"],["ContinuousAt"],["instHDiv"],["DivInvMonoid","toDiv"]],"valueReferences":[["Filter","Tendsto","div_const"],["nhds"]]},{"isProp":true,"kind":"theorem","name":["Continuous","comp_div_cases"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} {G₀ : Type.{u_3}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598048._hygCtx._hyg.5 : GroupWithZero.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598048._hygCtx._hyg.8 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598048._hygCtx._hyg.11 : ContinuousInv₀.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598048._hygCtx._hyg.5)))) (InvOneClass.toInv.{u_3} G₀ (DivInvOneMonoid.toInvOneClass.{u_3} G₀ (DivisionMonoid.toDivInvOneMonoid.{u_3} G₀ (GroupWithZero.toDivisionMonoid.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598048._hygCtx._hyg.5)))) inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598048._hygCtx._hyg.8] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598048._hygCtx._hyg.14 : ContinuousMul.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598048._hygCtx._hyg.8 (MulZeroClass.toMul.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598048._hygCtx._hyg.5))))] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598048._hygCtx._hyg.23 : TopologicalSpace.{u_1} α] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598048._hygCtx._hyg.26 : TopologicalSpace.{u_2} β] {f : α -> G₀} {g : α -> G₀} (h : α -> G₀ -> β), (Continuous.{u_1, u_3} α G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598048._hygCtx._hyg.23 inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598048._hygCtx._hyg.8 f) -> (Continuous.{u_1, u_3} α G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598048._hygCtx._hyg.23 inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598048._hygCtx._hyg.8 g) -> (forall (a : α), (Ne.{succ u_3} G₀ (g a) (OfNat.ofNat.{u_3} G₀ 0 (Zero.toOfNat0.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598048._hygCtx._hyg.5))))))) -> (ContinuousAt.{max u_1 u_3, u_2} (Prod.{u_1, u_3} α G₀) β (instTopologicalSpaceProd.{u_1, u_3} α G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598048._hygCtx._hyg.23 inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598048._hygCtx._hyg.8) inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598048._hygCtx._hyg.26 (Function.HasUncurry.uncurry.{max (max u_1 u_2) u_3, max u_1 u_3, u_2} (α -> G₀ -> β) (Prod.{u_1, u_3} α G₀) β (Function.hasUncurryInduction.{u_1, max u_2 u_3, u_3, u_2} α (G₀ -> β) G₀ β (Function.hasUncurryBase.{u_3, u_2} G₀ β)) h) (Prod.mk.{u_1, u_3} α G₀ a (HDiv.hDiv.{u_3, u_3, u_3} G₀ G₀ G₀ (instHDiv.{u_3} G₀ (DivInvMonoid.toDiv.{u_3} G₀ (GroupWithZero.toDivInvMonoid.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598048._hygCtx._hyg.5))) (f a) (g a))))) -> (forall (a : α), (Eq.{succ u_3} G₀ (g a) (OfNat.ofNat.{u_3} G₀ 0 (Zero.toOfNat0.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598048._hygCtx._hyg.5))))))) -> (Filter.Tendsto.{max u_1 u_3, u_2} (Prod.{u_1, u_3} α G₀) β (Function.HasUncurry.uncurry.{max (max u_1 u_2) u_3, max u_1 u_3, u_2} (α -> G₀ -> β) (Prod.{u_1, u_3} α G₀) β (Function.hasUncurryInduction.{u_1, max u_2 u_3, u_3, u_2} α (G₀ -> β) G₀ β (Function.hasUncurryBase.{u_3, u_2} G₀ β)) h) (SProd.sprod.{u_1, u_3, max u_1 u_3} (Filter.{u_1} α) (Filter.{u_3} G₀) (Filter.{max u_3 u_1} (Prod.{u_1, u_3} α G₀)) (Filter.instSProd.{u_1, u_3} α G₀) (nhds.{u_1} α inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598048._hygCtx._hyg.23 a) (Top.top.{u_3} (Filter.{u_3} G₀) (Filter.instTop.{u_3} G₀))) (nhds.{u_2} β inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598048._hygCtx._hyg.26 (h a (OfNat.ofNat.{u_3} G₀ 0 (Zero.toOfNat0.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598048._hygCtx._hyg.5)))))))))) -> (Continuous.{u_1, u_2} α β inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598048._hygCtx._hyg.23 inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598048._hygCtx._hyg.26 (fun (x : α) => h x (HDiv.hDiv.{u_3, u_3, u_3} G₀ G₀ G₀ (instHDiv.{u_3} G₀ (DivInvMonoid.toDiv.{u_3} G₀ (GroupWithZero.toDivInvMonoid.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3032598048._hygCtx._hyg.5))) (f x) (g x))))","typeFull":"∀ {α : Type u_1} {β : Type u_2} {G₀ : Type u_3} [inst : GroupWithZero G₀] [inst_1 : TopologicalSpace G₀]\n  [ContinuousInv₀ G₀] [ContinuousMul G₀] [inst_4 : TopologicalSpace α] [inst_5 : TopologicalSpace β] {f g : α → G₀}\n  (h : α → G₀ → β),\n  Continuous f →\n    Continuous g →\n      (∀ (a : α), g a ≠ 0 → ContinuousAt ↿h (a, f a / g a)) →\n        (∀ (a : α), g a = 0 → Filter.Tendsto (↿h) (nhds a ×ˢ ⊤) (nhds (h a 0))) → Continuous fun x => h x (f x / g x)","typeReadable":"∀ {α : Type u_1} {β : Type u_2} {G₀ : Type u_3} [inst : GroupWithZero G₀] [inst_1 : TopologicalSpace G₀]\n  [ContinuousInv₀ G₀] [ContinuousMul G₀] [inst_4 : TopologicalSpace α] [inst_5 : TopologicalSpace β] {f g : α → G₀}\n  (h : α → G₀ → β),\n  Continuous f →\n    Continuous g →\n      (∀ (a : α), g a ≠ 0 → ContinuousAt ↿h (a, f a / g a)) →\n        (∀ (a : α), g a = 0 → Filter.Tendsto (↿h) (nhds a ×ˢ ⊤) (nhds (h a 0))) → Continuous fun x => h x (f x / g x)","typeReferences":[["Prod","mk"],["GroupWithZero","toDivInvMonoid"],["MulZeroClass","toMul"],["instTopologicalSpaceProd"],["MonoidWithZero","toMulZeroOneClass"],["SProd","sprod"],["instHDiv"],["Function","hasUncurryBase"],["HDiv","hDiv"],["Filter","instSProd"],["GroupWithZero","toMonoidWithZero"],["Zero","toOfNat0"],["ContinuousAt"],["Filter","instTop"],["Eq"],["nhds"],["Filter","Tendsto"],["InvOneClass","toInv"],["ContinuousInv₀"],["GroupWithZero"],["Filter"],["DivInvMonoid","toDiv"],["OfNat","ofNat"],["TopologicalSpace"],["Prod"],["Continuous"],["DivInvOneMonoid","toInvOneClass"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Top","top"],["Function","HasUncurry","uncurry"],["Ne"],["ContinuousMul"],["DivisionMonoid","toDivInvOneMonoid"],["Function","hasUncurryInduction"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["HDiv","hDiv"],["Continuous"],["ContinuousAt","comp_div_cases"],["Continuous","continuousAt"],["continuous_iff_continuousAt"],["GroupWithZero","toDivInvMonoid"],["Iff","mpr"],["ContinuousAt"],["instHDiv"],["DivInvMonoid","toDiv"]]},{"isProp":true,"kind":"theorem","name":["Homeomorph","mulRight₀_symm_apply"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.245463609._hygCtx._hyg.5 : TopologicalSpace.{u_1} α] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.245463609._hygCtx._hyg.8 : GroupWithZero.{u_1} α] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.245463609._hygCtx._hyg.11 : SeparatelyContinuousMul.{u_1} α inst._@.Mathlib.Topology.Algebra.GroupWithZero.245463609._hygCtx._hyg.5 (MulZeroClass.toMul.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Topology.Algebra.GroupWithZero.245463609._hygCtx._hyg.8))))] (c : α) (hc : Ne.{succ u_1} α c (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Topology.Algebra.GroupWithZero.245463609._hygCtx._hyg.8))))))), Eq.{succ u_1} (α -> α) (DFunLike.coe.{succ u_1, succ u_1, succ u_1} (Homeomorph.{u_1, u_1} α α inst._@.Mathlib.Topology.Algebra.GroupWithZero.245463609._hygCtx._hyg.5 inst._@.Mathlib.Topology.Algebra.GroupWithZero.245463609._hygCtx._hyg.5) α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => α) (EquivLike.toFunLike.{succ u_1, succ u_1, succ u_1} (Homeomorph.{u_1, u_1} α α inst._@.Mathlib.Topology.Algebra.GroupWithZero.245463609._hygCtx._hyg.5 inst._@.Mathlib.Topology.Algebra.GroupWithZero.245463609._hygCtx._hyg.5) α α (Homeomorph.instEquivLike.{u_1, u_1} α α inst._@.Mathlib.Topology.Algebra.GroupWithZero.245463609._hygCtx._hyg.5 inst._@.Mathlib.Topology.Algebra.GroupWithZero.245463609._hygCtx._hyg.5)) (Homeomorph.symm.{u_1, u_1} α α inst._@.Mathlib.Topology.Algebra.GroupWithZero.245463609._hygCtx._hyg.5 inst._@.Mathlib.Topology.Algebra.GroupWithZero.245463609._hygCtx._hyg.5 (Homeomorph.mulRight₀.{u_1} α inst._@.Mathlib.Topology.Algebra.GroupWithZero.245463609._hygCtx._hyg.5 inst._@.Mathlib.Topology.Algebra.GroupWithZero.245463609._hygCtx._hyg.8 inst._@.Mathlib.Topology.Algebra.GroupWithZero.245463609._hygCtx._hyg.11 c hc))) (fun (x._@.Mathlib.Topology.Algebra.GroupWithZero.245463609._hygCtx._hyg.38 : α) => HMul.hMul.{u_1, u_1, u_1} α α α (instHMul.{u_1} α (MulZeroClass.toMul.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Topology.Algebra.GroupWithZero.245463609._hygCtx._hyg.8))))) x._@.Mathlib.Topology.Algebra.GroupWithZero.245463609._hygCtx._hyg.38 (Inv.inv.{u_1} α (InvOneClass.toInv.{u_1} α (DivInvOneMonoid.toInvOneClass.{u_1} α (DivisionMonoid.toDivInvOneMonoid.{u_1} α (GroupWithZero.toDivisionMonoid.{u_1} α inst._@.Mathlib.Topology.Algebra.GroupWithZero.245463609._hygCtx._hyg.8)))) c))","typeFull":"∀ {α : Type u_1} [inst : TopologicalSpace α] [inst_1 : GroupWithZero α] [inst_2 : SeparatelyContinuousMul α] (c : α)\n  (hc : c ≠ 0), ⇑(Homeomorph.mulRight₀ c hc).symm = fun x => x * c⁻¹","typeReadable":"∀ {α : Type u_1} [inst : TopologicalSpace α] [inst_1 : GroupWithZero α] [inst_2 : SeparatelyContinuousMul α] (c : α)\n  (hc : c ≠ 0), ⇑(Homeomorph.mulRight₀ c hc).symm = fun x => x * c⁻¹","typeReferences":[["MulZeroClass","toMul"],["HMul","hMul"],["MonoidWithZero","toMulZeroOneClass"],["DFunLike","coe"],["SeparatelyContinuousMul"],["Homeomorph","symm"],["GroupWithZero","toMonoidWithZero"],["EquivLike","toFunLike"],["Zero","toOfNat0"],["Eq"],["Inv","inv"],["InvOneClass","toInv"],["Homeomorph","instEquivLike"],["GroupWithZero"],["Homeomorph"],["Homeomorph","mulRight₀"],["OfNat","ofNat"],["TopologicalSpace"],["DivInvOneMonoid","toInvOneClass"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["instHMul"],["Ne"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["rfl"],["Homeomorph","instEquivLike"],["Homeomorph","symm"],["EquivLike","toFunLike"],["Homeomorph"],["Homeomorph","mulRight₀"],["DFunLike","coe"]]},{"isProp":true,"kind":"theorem","name":["ContinuousAt","div₀"],"typeFallback":"forall {α : Type.{u_1}} {G₀ : Type.{u_3}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828828._hygCtx._hyg.5 : GroupWithZero.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828828._hygCtx._hyg.8 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828828._hygCtx._hyg.11 : ContinuousInv₀.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828828._hygCtx._hyg.5)))) (InvOneClass.toInv.{u_3} G₀ (DivInvOneMonoid.toInvOneClass.{u_3} G₀ (DivisionMonoid.toDivInvOneMonoid.{u_3} G₀ (GroupWithZero.toDivisionMonoid.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828828._hygCtx._hyg.5)))) inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828828._hygCtx._hyg.8] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828828._hygCtx._hyg.14 : ContinuousMul.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828828._hygCtx._hyg.8 (MulZeroClass.toMul.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828828._hygCtx._hyg.5))))] {f : α -> G₀} {g : α -> G₀} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828828._hygCtx._hyg.23 : TopologicalSpace.{u_1} α] {a : α}, (ContinuousAt.{u_1, u_3} α G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828828._hygCtx._hyg.23 inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828828._hygCtx._hyg.8 f a) -> (ContinuousAt.{u_1, u_3} α G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828828._hygCtx._hyg.23 inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828828._hygCtx._hyg.8 g a) -> (Ne.{succ u_3} G₀ (g a) (OfNat.ofNat.{u_3} G₀ 0 (Zero.toOfNat0.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ 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[inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828829._hygCtx._hyg.8 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828829._hygCtx._hyg.11 : ContinuousInv₀.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828829._hygCtx._hyg.5)))) (InvOneClass.toInv.{u_3} G₀ (DivInvOneMonoid.toInvOneClass.{u_3} G₀ (DivisionMonoid.toDivInvOneMonoid.{u_3} G₀ (GroupWithZero.toDivisionMonoid.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828829._hygCtx._hyg.5)))) inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828829._hygCtx._hyg.8] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828829._hygCtx._hyg.14 : ContinuousMul.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828829._hygCtx._hyg.8 (MulZeroClass.toMul.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828829._hygCtx._hyg.5))))] {f : α -> G₀} {g : α -> G₀} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828829._hygCtx._hyg.23 : TopologicalSpace.{u_1} α] {s : Set.{u_1} α}, (ContinuousOn.{u_1, u_3} α G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828829._hygCtx._hyg.23 inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828829._hygCtx._hyg.8 f s) -> (ContinuousOn.{u_1, u_3} α G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828829._hygCtx._hyg.23 inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828829._hygCtx._hyg.8 g s) -> (forall (x : α), (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s x) -> (Ne.{succ u_3} G₀ (g x) (OfNat.ofNat.{u_3} G₀ 0 (Zero.toOfNat0.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828829._hygCtx._hyg.5)))))))) -> (ContinuousOn.{u_1, u_3} α G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828829._hygCtx._hyg.23 inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828829._hygCtx._hyg.8 (fun (x : α) => HDiv.hDiv.{u_3, u_3, u_3} G₀ G₀ G₀ (instHDiv.{u_3} G₀ (DivInvMonoid.toDiv.{u_3} G₀ (GroupWithZero.toDivInvMonoid.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.3686828829._hygCtx._hyg.5))) (f x) (g x)) s)","typeFull":"∀ {α : Type u_1} {G₀ : Type u_3} [inst : GroupWithZero G₀] [inst_1 : TopologicalSpace G₀] [ContinuousInv₀ G₀]\n  [ContinuousMul G₀] {f g : α → G₀} [inst_4 : TopologicalSpace α] {s : Set α},\n  ContinuousOn f s → ContinuousOn g s → (∀ x ∈ s, g x ≠ 0) → ContinuousOn (fun x => f x / g x) s","typeReadable":"∀ {α : Type u_1} {G₀ : Type u_3} [inst : GroupWithZero G₀] [inst_1 : TopologicalSpace G₀] [ContinuousInv₀ G₀]\n  [ContinuousMul G₀] {f g : α → G₀} [inst_4 : TopologicalSpace α] {s : Set α},\n  ContinuousOn f s → ContinuousOn g s → (∀ x ∈ s, g x ≠ 0) → ContinuousOn (fun x => f x / g x) s","typeReferences":[["InvOneClass","toInv"],["Set"],["GroupWithZero","toDivInvMonoid"],["Membership","mem"],["ContinuousInv₀"],["MulZeroClass","toMul"],["ContinuousOn"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["instHDiv"],["DivInvMonoid","toDiv"],["OfNat","ofNat"],["Set","instMembership"],["HDiv","hDiv"],["TopologicalSpace"],["DivInvOneMonoid","toInvOneClass"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["GroupWithZero","toMonoidWithZero"],["Zero","toOfNat0"],["Ne"],["DivisionMonoid","toDivInvOneMonoid"],["ContinuousMul"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["ContinuousOn","div"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_mul_iff_of_ne_zero"],"typeFallback":"forall {α : Type.{u_1}} {G₀ : Type.{u_3}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2697587295._hygCtx._hyg.5 : GroupWithZero.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2697587295._hygCtx._hyg.8 : TopologicalSpace.{u_3} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2697587295._hygCtx._hyg.11 : ContinuousInv₀.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2697587295._hygCtx._hyg.5)))) (InvOneClass.toInv.{u_3} G₀ (DivInvOneMonoid.toInvOneClass.{u_3} G₀ (DivisionMonoid.toDivInvOneMonoid.{u_3} G₀ (GroupWithZero.toDivisionMonoid.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2697587295._hygCtx._hyg.5)))) inst._@.Mathlib.Topology.Algebra.GroupWithZero.2697587295._hygCtx._hyg.8] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2697587295._hygCtx._hyg.14 : ContinuousMul.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2697587295._hygCtx._hyg.8 (MulZeroClass.toMul.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2697587295._hygCtx._hyg.5))))] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.2697587295._hygCtx._hyg.23 : T1Space.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2697587295._hygCtx._hyg.8] {f : α -> G₀} {g : α -> G₀} {l : Filter.{u_1} α} {x : G₀} {y : G₀}, (Filter.Tendsto.{u_1, u_3} α G₀ g l (nhds.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2697587295._hygCtx._hyg.8 y)) -> (Ne.{succ u_3} G₀ y (OfNat.ofNat.{u_3} G₀ 0 (Zero.toOfNat0.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2697587295._hygCtx._hyg.5))))))) -> (Iff (Filter.Tendsto.{u_1, u_3} α G₀ (fun (n : α) => HMul.hMul.{u_3, u_3, u_3} G₀ G₀ G₀ (instHMul.{u_3} G₀ (MulZeroClass.toMul.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2697587295._hygCtx._hyg.5))))) (f n) (g n)) l (nhds.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2697587295._hygCtx._hyg.8 (HMul.hMul.{u_3, u_3, u_3} G₀ G₀ G₀ (instHMul.{u_3} G₀ (MulZeroClass.toMul.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2697587295._hygCtx._hyg.5))))) x y))) (Filter.Tendsto.{u_1, u_3} α G₀ f l (nhds.{u_3} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.2697587295._hygCtx._hyg.8 x)))","typeFull":"∀ {α : Type u_1} {G₀ : Type u_3} [inst : GroupWithZero G₀] [inst_1 : TopologicalSpace G₀] [ContinuousInv₀ G₀]\n  [ContinuousMul G₀] [T1Space G₀] {f g : α → G₀} {l : Filter α} {x y : G₀},\n  Filter.Tendsto g l (nhds y) →\n    y ≠ 0 → (Filter.Tendsto (fun n => f n * g n) l (nhds (x * y)) ↔ Filter.Tendsto f l (nhds x))","typeReadable":"∀ {α : Type u_1} {G₀ : Type u_3} [inst : GroupWithZero G₀] [inst_1 : TopologicalSpace G₀] [ContinuousInv₀ G₀]\n  [ContinuousMul G₀] [T1Space G₀] {f g : α → G₀} {l : Filter α} {x y : G₀},\n  Filter.Tendsto g l (nhds y) →\n    y ≠ 0 → (Filter.Tendsto (fun n => f n * g n) l (nhds (x * y)) ↔ Filter.Tendsto f l (nhds x))","typeReferences":[["InvOneClass","toInv"],["ContinuousInv₀"],["MulZeroClass","toMul"],["HMul","hMul"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["OfNat","ofNat"],["Filter"],["TopologicalSpace"],["DivInvOneMonoid","toInvOneClass"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Iff"],["GroupWithZero","toMonoidWithZero"],["instHMul"],["T1Space"],["Zero","toOfNat0"],["Ne"],["nhds"],["Filter","Tendsto"],["DivisionMonoid","toDivInvOneMonoid"],["ContinuousMul"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["Filter","Tendsto","congr'"],["mul_div_cancel_right₀"],["GroupWithZero","toDivInvMonoid"],["MulZeroClass","toMul"],["Filter","Tendsto","div"],["HMul","hMul"],["MonoidWithZero","toMulZeroOneClass"],["Filter","Tendsto","eventually_ne"],["instHDiv"],["Filter","Tendsto","mul"],["Iff","intro"],["congrArg"],["HDiv","hDiv"],["GroupWithZero","toMonoidWithZero"],["Eq","symm"],["Zero","toOfNat0"],["Eq"],["nhds"],["Filter","Tendsto"],["Pi","instDiv"],["DivInvMonoid","toDiv"],["OfNat","ofNat"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["GroupWithZero","toMulDivCancelClass"],["id"],["instHMul"],["Eq","mpr"],["Ne"],["Filter","Eventually","mono"]]},{"isProp":false,"kind":"definition","name":["ContinuousInv₀","casesOn"],"typeFallback":"forall {G₀ : Type.{u_4}} [inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.9 : Zero.{u_4} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.12 : Inv.{u_4} G₀] [inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.15 : TopologicalSpace.{u_4} G₀] {motive : (ContinuousInv₀.{u_4} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.9 inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.12 inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.15) -> Sort.{u}} (t : ContinuousInv₀.{u_4} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.9 inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.12 inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.15), (forall (continuousAt_inv₀ : forall {{x : G₀}}, (Ne.{succ u_4} G₀ x (OfNat.ofNat.{u_4} G₀ 0 (Zero.toOfNat0.{u_4} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.9))) -> (ContinuousAt.{u_4, u_4} G₀ G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.15 inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.15 (Inv.inv.{u_4} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.12) x)), motive (ContinuousInv₀.mk.{u_4} G₀ inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.9 inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.12 inst._@.Mathlib.Topology.Algebra.GroupWithZero.847197903._hygCtx._hyg.15 continuousAt_inv₀)) -> (motive t)","typeFull":"{G₀ : Type u_4} →\n  [inst : Zero G₀] →\n    [inst_1 : Inv G₀] →\n      [inst_2 : TopologicalSpace G₀] →\n        {motive : ContinuousInv₀ G₀ → Sort u} →\n          (t : ContinuousInv₀ G₀) →\n            ((continuousAt_inv₀ : ∀ ⦃x : G₀⦄, x ≠ 0 → ContinuousAt Inv.inv x) → motive ⋯) → motive t","typeReadable":"{G₀ : Type u_4} →\n  [inst : Zero G₀] →\n    [inst_1 : Inv G₀] →\n      [inst_2 : TopologicalSpace G₀] →\n        {motive : ContinuousInv₀ G₀ → Sort u} →\n          (t : ContinuousInv₀ G₀) →\n            ((continuousAt_inv₀ : ∀ ⦃x : G₀⦄, x ≠ 0 → ContinuousAt Inv.inv x) → motive ⋯) → motive t","typeReferences":[["TopologicalSpace"],["Inv","inv"],["Inv"],["ContinuousInv₀"],["ContinuousAt"],["Zero","toOfNat0"],["Ne"],["Zero"],["ContinuousInv₀","mk"],["OfNat","ofNat"]],"valueReferences":[["ContinuousInv₀","rec"]]}]