diff --git "a/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.LocalizedModule.Basic.sym.json" "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.LocalizedModule.Basic.sym.json"
new file mode 100644--- /dev/null
+++ "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.LocalizedModule.Basic.sym.json"
@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","mk'_add"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.811411590._hygCtx._hyg.3 : CommSemiring.{u_1} R] {S : 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.Algebra.Module.LocalizedModule.Basic.811411590._hygCtx._hyg.3))))} {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.811411590._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.811411590._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.811411590._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.811411590._hygCtx._hyg.3) 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.811411590._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.811411590._hygCtx._hyg.40) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.811411590._hygCtx._hyg.71 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.811411590._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.811411590._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.811411590._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.811411590._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.811411590._hygCtx._hyg.40 S f] (m₁ : M) (m₂ : M) (s : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.811411590._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.811411590._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.811411590._hygCtx._hyg.40 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.811411590._hygCtx._hyg.71 m₁ s) (IsLocalizedModule.mk'.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.811411590._hygCtx._hyg.3 S M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.811411590._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.811411590._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.811411590._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.811411590._hygCtx._hyg.40 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.811411590._hygCtx._hyg.71 m₂ s))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (m₁ m₂ : M) (s : ↥S),\n  IsLocalizedModule.mk' f (m₁ + m₂) s = IsLocalizedModule.mk' f m₁ s + IsLocalizedModule.mk' f m₂ s","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (m₁ m₂ : M) (s : ↥S),\n  IsLocalizedModule.mk' f (m₁ + m₂) s = IsLocalizedModule.mk' f m₁ s + IsLocalizedModule.mk' f m₂ s","typeReferences":[["IsLocalizedModule"],["SetLike","instMembership"],["Subtype"],["instHAdd"],["Submonoid","instSetLike"],["Module"],["CommSemiring"],["CommSemiring","toSemiring"],["Membership","mem"],["MulZeroOneClass","toMulOneClass"],["LinearMap"],["HAdd","hAdd"],["Submonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["AddCommMonoid","toAddCommSemigroup"],["RingHom","id"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["IsLocalizedModule","mk'"]],"valueReferences":[["Subtype"],["IsLocalizedModule","mk'_add_mk'"],["Membership","mem"],["Submonoid","smul"],["HMul","hMul"],["MulZeroOneClass","toMulOneClass"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["congrArg"],["Semiring","toNonAssocSemiring"],["Submonoid","distribMulAction"],["IsLocalizedModule","mk'_cancel_left"],["MonoidWithZero","toMonoid"],["Eq","symm"],["instHSMul"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Eq"],["Submonoid","toMonoid"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["IsLocalizedModule","mk'"],["SetLike","instMembership"],["instHAdd"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["Submonoid"],["HAdd","hAdd"],["Module","toDistribMulAction"],["Submonoid","mul"],["AddCommMonoid","toAddCommSemigroup"],["Eq","refl"],["HSMul","hSMul"],["id"],["smul_add"],["instHMul"],["Eq","mpr"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","instSemiring","_proof_17"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 : CommSemiring.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 : Semiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29] {S : 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.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3))))}, autoParam.{0} (forall (x : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (Monoid.npow.{max u_2 u_1} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (inferInstance.{max (succ u_1) (succ u_2)} (Monoid.{max u_2 u_1} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32))) (LocalizedModule.instMonoid.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 S)) (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) x) (OfNat.ofNat.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A 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(OreLocalization.instOne.{u_1, u_2} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3) S) A (DistribMulAction.toMulAction.{u_1, u_2} R A (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29)))) (Module.toDistribMulAction.{u_1, u_2} R A (CommSemiring.toSemiring.{u_1} R 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autoParam (∀ (x : LocalizedModule S A), Monoid.npow 0 x = 1) Monoid.npow_zero._autoParam","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A] {S : Submonoid R},\n  autoParam (∀ (x : LocalizedModule S A), Monoid.npow 0 x = 1) Monoid.npow_zero._autoParam","typeReferences":[["OreLocalization","oreSetComm"],["OreLocalization","instOne"],["CommSemiring"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["instOfNatNat"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["Monoid","npow"],["NonAssocSemiring","toMulZeroOneClass"],["LocalizedModule","instMonoid"],["LocalizedModule"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Monoid","npow_zero","_autoParam"],["Semiring","toMonoidWithZero"],["CommSemiring","toCommMonoid"],["OfNat","ofNat"],["Algebra"],["Submonoid"],["Module","toDistribMulAction"],["Nat"],["One","toOfNat1"],["AddMonoidWithOne","toOne"],["Monoid"],["inferInstance"],["autoParam"],["Semiring"]],"valueReferences":[["LocalizedModule","instMonoid"],["Monoid","npow_zero"],["Semiring","toNonAssocSemiring"],["LocalizedModule"],["Algebra","toModule"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Monoid"],["inferInstance"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","mkLinearMap","_proof_3"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2851436865._hygCtx._hyg.3 : CommSemiring.{u_1} R] (M : Type.{u_2}) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2851436865._hygCtx._hyg.9 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2851436865._hygCtx._hyg.12 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2851436865._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2851436865._hygCtx._hyg.9], IsScalarTower.{u_1, u_1, u_2} R R M (SemigroupAction.toSMul.{u_1, u_1} R R (Monoid.toSemigroup.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2851436865._hygCtx._hyg.3)))) (MulAction.toSemigroupAction.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2851436865._hygCtx._hyg.3))) (Monoid.toMulAction.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2851436865._hygCtx._hyg.3)))))) (SemigroupAction.toSMul.{u_1, u_2} R M (Monoid.toSemigroup.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2851436865._hygCtx._hyg.3)))) (MulAction.toSemigroupAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2851436865._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2851436865._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2851436865._hygCtx._hyg.9) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2851436865._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2851436865._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2851436865._hygCtx._hyg.12)))) (SemigroupAction.toSMul.{u_1, u_2} R M (Monoid.toSemigroup.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2851436865._hygCtx._hyg.3)))) (MulAction.toSemigroupAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2851436865._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2851436865._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2851436865._hygCtx._hyg.9) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2851436865._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2851436865._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2851436865._hygCtx._hyg.12))))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (M : Type u_2) [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n  IsScalarTower R R M","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (M : Type u_2) [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n  IsScalarTower R R M","typeReferences":[["Module"],["IsScalarTower"],["CommSemiring"],["CommSemiring","toSemiring"],["SemigroupAction","toSMul"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"],["Monoid","toMulAction"],["Module","toDistribMulAction"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Monoid","toSemigroup"],["MulAction","toSemigroupAction"]],"valueReferences":[["Module","toDistribMulAction"],["IsScalarTower","left"],["CommSemiring","toSemiring"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","instSemiring","_proof_9"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 : CommSemiring.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 : Semiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29] {S : 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.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3))))} (a : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (b : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (c : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)), Eq.{succ (max u_1 u_2)} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 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R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (MulOneClass.toMulOne.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (Monoid.toMulOneClass.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (instHAdd.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (AddSemigroup.toAdd.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (AddMonoid.toAddSemigroup.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (AddCommMonoid.toAddMonoid.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_2} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3) S) A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Module.toDistribMulAction.{u_1, u_2} R A (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32))))))) a b) c) (HAdd.hAdd.{max u_1 u_2, max u_1 u_2, max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (Monoid.toMulOneClass.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 : CommSemiring.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 : Semiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29] {S : 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.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3))))}, autoParam.{0} (Eq.{succ (max u_1 u_2)} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 S) (LocalizedModule.instSemiring._proof_10.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 S) (LocalizedModule.instSemiring._proof_11.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 S)) (LocalizedModule.instSemiring._proof_12.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 S))))))))) AddMonoidWithOne.natCast_zero._autoParam","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A] {S : Submonoid R},\n  autoParam (↑0 = 0) AddMonoidWithOne.natCast_zero._autoParam","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A] {S : Submonoid R},\n  autoParam (↑0 = 0) AddMonoidWithOne.natCast_zero._autoParam","typeReferences":[["MulOneClass","toMulOne"],["LocalizedModule","instSemiring","_proof_9"],["AddMonoid","toZero"],["NonUnitalNonAssocSemiring","mk"],["OreLocalization","oreSetComm"],["CommSemiring"],["LocalizedModule","instSemiring","_proof_8"],["LocalizedModule","instSemiring","_proof_11"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["MulOne","toMul"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["LocalizedModule","instSemiring","_proof_12"],["instOfNatNat"],["Algebra","toModule"],["NatCast","natCast"],["AddMonoidWithOne","natCast_zero","_autoParam"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["Zero","toOfNat0"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["AddSemigroup","toAdd"],["LocalizedModule","instMonoid"],["NonUnitalSemiring","mk"],["LocalizedModule"],["OreLocalization","instAddCommMonoidOreLocalization"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Monoid","toOne"],["Semiring","toMonoidWithZero"],["OfNat","ofNat"],["CommSemiring","toCommMonoid"],["Algebra"],["Submonoid"],["NatCast","mk"],["Nat"],["Module","toDistribMulAction"],["AddMonoid","toAddSemigroup"],["LocalizedModule","instSemiring","_proof_10"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["Monoid"],["Nat","unaryCast"],["inferInstance"],["autoParam"],["Semiring"]],"valueReferences":[["LocalizedModule","instSemiring","_proof_6"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","mk'_cancel_right"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3618768629._hygCtx._hyg.3 : CommSemiring.{u_1} R] {S : 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.Algebra.Module.LocalizedModule.Basic.3618768629._hygCtx._hyg.3))))} {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3618768629._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3618768629._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3618768629._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3618768629._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3618768629._hygCtx._hyg.11] 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(Submonoid.mul.{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.Algebra.Module.LocalizedModule.Basic.3618768629._hygCtx._hyg.3)))) S)) s₁ s₂)) (IsLocalizedModule.mk'.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3618768629._hygCtx._hyg.3 S M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3618768629._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3618768629._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3618768629._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3618768629._hygCtx._hyg.40 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3618768629._hygCtx._hyg.71 m s₁)","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (m : M) (s₁ s₂ : ↥S),\n  IsLocalizedModule.mk' f (s₂ • m) (s₁ * s₂) = IsLocalizedModule.mk' f m s₁","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (m : M) (s₁ s₂ : ↥S),\n  IsLocalizedModule.mk' f (s₂ • m) (s₁ * s₂) = IsLocalizedModule.mk' f m 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3 : CommSemiring.{u_1} R] {S : 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.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3))))} {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.29 : Semiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.32 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.29] (r : R) (x : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S : 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.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3))))) {M : Type.{u_2}} {M'' : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 : AddCommMonoid.{u_4} M''] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44 : Module.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17], (LinearMap.{u_1, u_1, u_2, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3))) M M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44) -> (forall (x : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3))))) R (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.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3)))))) S x)), IsUnit.{u_4} (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44) (Module.End.instMonoid.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44) (DFunLike.coe.{max (succ u_1) (succ u_4), succ u_1, succ u_4} (RingHom.{u_1, u_4} R (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 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(CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44) (RingHom.instFunLike.{u_1, u_4} R (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44) (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3)) (Semiring.toNonAssocSemiring.{u_4} (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44) (Module.End.instSemiring.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44))) (algebraMap.{u_1, u_4} R (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3 (Module.End.instSemiring.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44) (Module.End.instAlgebra.{u_1, u_1, u_4} R R M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44 (smulCommClass_self.{u_1, u_4} R M'' (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_4} R M'' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_4} M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17) (Module.toDistribMulAction.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44))) (Algebra.toSMul.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_4} R M'' (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R 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(CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3))) (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.36) M'' (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_2} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.11 (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.36)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 (OreLocalization.instModuleOfIsScalarTower.{u_1, u_2, u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.36 R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.36 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.11) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) 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M''","typeReferences":[["RingHom"],["smulCommClass_self"],["Subtype"],["CommMonoid","toMonoid"],["OreLocalization","oreSetComm"],["Module"],["Membership","mem"],["CommSemiring"],["RingHom","instFunLike"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["Subtype","val"],["Algebra","id"],["DFunLike","coe"],["AddCommMonoid"],["Module","End","instAlgebra"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["Algebra","toSMul"],["DistribMulAction","toMulAction"],["IsScalarTower","right"],["Semiring","toModule"],["NonAssocSemiring","toMulZeroOneClass"],["Module","End"],["SetLike","instMembership"],["Module","End","instSemiring"],["OreLocalization","instAddCommMonoidOreLocalization"],["LocalizedModule"],["IsScalarTower","left"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["OreLocalization","instModuleOfIsScalarTower"],["Semiring","toMonoidWithZero"],["IsUnit"],["LinearMap"],["CommSemiring","toCommMonoid"],["Submonoid"],["Module","toDistribMulAction"],["Module","End","instMonoid"],["algebraMap"]],"valueReferences":[["LocalizedModule"],["OreLocalization","instAddCommMonoidOreLocalization"],["OreLocalization","oreSetComm"],["LinearMap","mk"],["AddHom","mk"],["CommSemiring","toSemiring"],["OreLocalization","instModuleOfIsScalarTower"],["Semiring","toMonoidWithZero"],["LocalizedModule","lift'_add"],["LocalizedModule","lift'"],["CommSemiring","toCommMonoid"],["Module","toDistribMulAction"],["LocalizedModule","mkLinearMap","_proof_3"],["Semiring","toNonAssocSemiring"],["AddCommMonoid","toAddCommSemigroup"],["RingHom","id"],["MonoidWithZero","toMonoid"],["LocalizedModule","algebra'","_proof_4"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["LocalizedModule","lift","_proof_1"],["Semiring","toModule"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","iso","congr_simp"],"typeFallback":"forall 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2012850423._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2012850423._hygCtx._hyg.40 S f] {m : M}, Iff (Eq.{succ u_3} M' (DFunLike.coe.{max (succ u_2) (succ u_3), succ u_2, succ u_3} (LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2012850423._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2012850423._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2012850423._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2012850423._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2012850423._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2012850423._hygCtx._hyg.36 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CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M') [IsLocalizedModule S f]\n  {m : M}, f m = 0 ↔ ∃ s', s' • m = 0","typeReferences":[["IsLocalizedModule"],["Subtype"],["LinearMap","instFunLike"],["Module"],["Submonoid","smul"],["CommSemiring"],["Membership","mem"],["SMulZeroClass","toSMul"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["instHSMul"],["Zero","toOfNat0"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["Exists"],["SetLike","instMembership"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["LinearMap"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Submonoid"],["Module","toDistribMulAction"],["Iff"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Subtype"],["LinearMap","instFunLike"],["Submonoid","smul"],["Membership","mem"],["SMulZeroClass","toSMul"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["instHSMul"],["IsLocalizedModule","mk'_eq_zero"],["Zero","toOfNat0"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["IsLocalizedModule","mk'"],["Iff","trans"],["SetLike","instMembership"],["Exists"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["IsLocalizedModule","mk'_eq_zero'"],["AddZeroClass","toAddZero"],["LinearMap"],["Iff","symm"],["OfNat","ofNat"],["Submonoid","one"],["Submonoid"],["Module","toDistribMulAction"],["One","toOfNat1"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","lift_unique"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S : 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.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.3))))) {M : Type.{u_2}} {M' : Type.{u_3}} {M'' : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.17 : AddCommMonoid.{u_4} M''] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.40 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.14] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.44 : Module.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.17] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.40) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.71 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.40 S f] (g : LinearMap.{u_1, u_1, u_2, u_4} R R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.44 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.44 (smulCommClass_self.{u_1, u_4} R M'' (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_4} R M'' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_4} M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.17) (Module.toDistribMulAction.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.3) 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.3))))) R (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.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.3)))))) S x) x))) (l : LinearMap.{u_1, u_1, u_3, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.3))) M' M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.44), (Eq.{max (succ u_2) (succ u_4)} (LinearMap.{u_1, u_1, u_2, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.3))) M M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.44) (LinearMap.comp.{u_1, u_1, u_1, u_2, u_3, u_4} R R R M 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(RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.3))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.3))) (RingHomCompTriple.ids.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.3)))) l f) g) -> (Eq.{max (succ u_3) (succ u_4)} (LinearMap.{u_1, u_1, u_3, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106595._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R 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: AddCommMonoid M''] [inst_4 : Module R M]\n  [inst_5 : Module R M'] [inst_6 : Module R M''] (f : M →ₗ[R] M') [inst_7 : IsLocalizedModule S f] (g : M →ₗ[R] M'')\n  (h : ∀ (x : ↥S), IsUnit ((algebraMap R (Module.End R M'')) ↑x)) (l : M' →ₗ[R] M''),\n  l ∘ₗ f = g → IsLocalizedModule.lift S f g h = 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{R : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3 : CommSemiring.{u_3} R] (S : Submonoid.{u_3} R (MulZeroOneClass.toMulOneClass.{u_3} R (NonAssocSemiring.toMulZeroOneClass.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3))))) {M : Type.{u_2}} {M'' : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.17 : AddCommMonoid.{u_1} M''] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.36 : Module.{u_3, u_2} R M (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.44 : Module.{u_3, u_1} R M'' (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.17] (g : LinearMap.{u_3, u_3, u_2, u_1} R R (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3) (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3))) M M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.44) (h : forall (x : Subtype.{succ u_3} R (fun (x : R) => Membership.mem.{u_3, u_3} R (Submonoid.{u_3} R (MulZeroOneClass.toMulOneClass.{u_3} R (NonAssocSemiring.toMulZeroOneClass.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3))))) (SetLike.instMembership.{u_3, u_3} (Submonoid.{u_3} R (MulZeroOneClass.toMulOneClass.{u_3} R (NonAssocSemiring.toMulZeroOneClass.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3))))) R (Submonoid.instSetLike.{u_3} R (MulZeroOneClass.toMulOneClass.{u_3} R (NonAssocSemiring.toMulZeroOneClass.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3)))))) S x)), IsUnit.{u_1} (Module.End.{u_3, u_1} R M'' (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.44) (Module.End.instMonoid.{u_3, u_1} R M'' (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.44) (DFunLike.coe.{max (succ u_3) (succ u_1), succ u_3, succ u_1} (RingHom.{u_3, u_1} R (Module.End.{u_3, u_1} R M'' (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.17 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.44) (Module.End.instSemiring.{u_3, u_1} R M'' (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.44))) (algebraMap.{u_3, u_1} R (Module.End.{u_3, u_1} R M'' (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.44) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3 (Module.End.instSemiring.{u_3, u_1} R M'' (CommSemiring.toSemiring.{u_3} R 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.366197653._hygCtx._hyg.3 : CommSemiring.{u_1} R] {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.366197653._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.366197653._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.366197653._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.366197653._hygCtx._hyg.11] {M₁ : Type.{u_7}} {M₂ : Type.{u_6}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.366197653._hygCtx._hyg.77 : AddCommMonoid.{u_7} M₁] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.366197653._hygCtx._hyg.80 : AddCommMonoid.{u_6} M₂] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.366197653._hygCtx._hyg.83 : Module.{u_1, u_7} R M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.366197653._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.366197653._hygCtx._hyg.77] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.366197653._hygCtx._hyg.87 : Module.{u_1, u_6} R M₂ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.366197653._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.366197653._hygCtx._hyg.80] (S₁ : 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.Algebra.Module.LocalizedModule.Basic.366197653._hygCtx._hyg.3))))) (S₂ : 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.Algebra.Module.LocalizedModule.Basic.366197653._hygCtx._hyg.3))))) (h : LE.le.{u_1} 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[inst_7 : IsLocalizedModule S₁ f₁]\n  [inst_8 : IsLocalizedModule S₂ f₂] (m : M) (s : ↥S₁),\n  (IsLocalizedModule.liftOfLE S₁ S₂ h f₁ f₂) (IsLocalizedModule.mk' f₁ m s) = IsLocalizedModule.mk' f₂ m ⟨↑s, 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3 : CommSemiring.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.29 : Semiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.32 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.29], IsScalarTower.{u_1, u_1, u_2} R R A (SemigroupAction.toSMul.{u_1, u_1} R R (Monoid.toSemigroup.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3)))) (MulAction.toSemigroupAction.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.29)))) (Module.toDistribMulAction.{u_1, u_2} R A (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.32))))) (SemigroupAction.toSMul.{u_1, u_2} R A (Monoid.toSemigroup.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3)))) (MulAction.toSemigroupAction.{u_1, u_2} R A (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_2} R A (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.29)))) (Module.toDistribMulAction.{u_1, u_2} R A (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.32)))))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A],\n  IsScalarTower R R A","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A],\n  IsScalarTower R R A","typeReferences":[["IsScalarTower"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring"],["CommSemiring","toSemiring"],["SemigroupAction","toSMul"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"],["Monoid","toMulAction"],["Algebra"],["Module","toDistribMulAction"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Monoid","toSemigroup"],["MulAction","toSemigroupAction"],["Semiring"]],"valueReferences":[["Module","toDistribMulAction"],["Semiring","toNonAssocSemiring"],["IsScalarTower","left"],["Algebra","toModule"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","Module","LocalizedModule","Basic",0,"LocalizedModule","r","isEquiv","match_1_7"],"typeFallback":"forall {R : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3 : CommSemiring.{u_2} R] (S : Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3))))) (M : Type.{u_1}) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.9 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.12 : Module.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.9] (x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.68 : Prod.{u_1, u_2} M (Subtype.{succ u_2} R (fun (x : R) => Membership.mem.{u_2, u_2} R (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3))))) (SetLike.instMembership.{u_2, u_2} (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3))))) R (Submonoid.instSetLike.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3)))))) S x))) (m1 : M) (s1 : Subtype.{succ u_2} R (fun (x : R) => Membership.mem.{u_2, u_2} R (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3))))) (SetLike.instMembership.{u_2, u_2} (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3))))) R (Submonoid.instSetLike.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3)))))) S x)) (motive : forall (x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx.67.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.99 : Prod.{u_1, u_2} M (Subtype.{succ u_2} R (fun (x : R) => Membership.mem.{u_2, u_2} R (Submonoid.{u_2} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.12 x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx.67.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.99 x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.68) -> (LocalizedModule.r.{u_2, u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.12 (Prod.mk.{u_1, u_2} M (Subtype.{succ u_2} R (fun (x : R) => Membership.mem.{u_2, u_2} R (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3))))) (SetLike.instMembership.{u_2, u_2} (Submonoid.{u_2} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.12 x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx.67.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.99 x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.68) (x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.69 : LocalizedModule.r.{u_2, u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.12 (Prod.mk.{u_1, u_2} M (Subtype.{succ u_2} R (fun (x : R) => Membership.mem.{u_2, u_2} R (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R 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x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.68) (x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.69 : LocalizedModule.r.{u_2, u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.12 (Prod.mk.{u_1, u_2} M (Subtype.{succ u_2} R (fun (x : R) => Membership.mem.{u_2, u_2} R (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3))))) (SetLike.instMembership.{u_2, u_2} (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3))))) R (Submonoid.instSetLike.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3)))))) S x)) m2 s2) x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.70 x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.69) -> (motive x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx.67.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.99 x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.70 x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.69)","typeFull":"∀ {R : Type u_2} [inst : CommSemiring R] (S : Submonoid R) (M : Type u_1) [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] (x : M × ↥S) (m1 : M) (s1 : ↥S)\n  (motive : (x_1 : M × ↥S) → LocalizedModule.r S M x_1 x → LocalizedModule.r S M (m1, s1) x_1 → Prop) (x_1 : M × ↥S)\n  (x_2 : LocalizedModule.r S M x_1 x) (x_3 : LocalizedModule.r S M (m1, s1) x_1),\n  (∀ (m2 : M) (s2 : ↥S) (x : LocalizedModule.r S M (m2, s2) x) (x_4 : LocalizedModule.r S M (m1, s1) (m2, s2)),\n      motive (m2, s2) x x_4) →\n    motive x_1 x_2 x_3","typeReadable":"∀ {R : Type u_2} [inst : CommSemiring R] (S : Submonoid R) (M : Type u_1) [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] (x : M × ↥S) (m1 : M) (s1 : ↥S)\n  (motive : (x_1 : M × ↥S) → LocalizedModule.r S M x_1 x → LocalizedModule.r S M (m1, s1) x_1 → Prop) (x_1 : M × ↥S)\n  (x_2 : LocalizedModule.r S M x_1 x) (x_3 : LocalizedModule.r S M (m1, s1) x_1),\n  (∀ (m2 : M) (s2 : ↥S) (x : LocalizedModule.r S M (m2, s2) x) (x_4 : LocalizedModule.r S M (m1, s1) (m2, s2)),\n      motive (m2, s2) x x_4) →\n    motive x_1 x_2 x_3","typeReferences":[["Subtype"],["SetLike","instMembership"],["Prod","mk"],["Module"],["Submonoid","instSetLike"],["CommSemiring"],["CommSemiring","toSemiring"],["Membership","mem"],["MulZeroOneClass","toMulOneClass"],["Submonoid"],["Prod"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["LocalizedModule","r"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["Submonoid"],["Prod","casesOn"],["SetLike","instMembership"],["Semiring","toNonAssocSemiring"],["Subtype"],["Prod","mk"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["Membership","mem"],["LocalizedModule","r"],["MulZeroOneClass","toMulOneClass"],["NonAssocSemiring","toMulZeroOneClass"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","module","_proof_1"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1565340096._hygCtx._hyg.162 : CommSemiring.{u_1} R] 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1565340096._hygCtx._hyg.162))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1565340096._hygCtx._hyg.174) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1565340096._hygCtx._hyg.162) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1565340096._hygCtx._hyg.174 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1565340096._hygCtx._hyg.177))))","typeFull":"∀ {R : Type u_1} {M : Type u_2} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n  IsScalarTower R R M","typeReadable":"∀ {R : Type u_1} {M : Type u_2} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n  IsScalarTower R R M","typeReferences":[["Module"],["IsScalarTower"],["CommSemiring"],["CommSemiring","toSemiring"],["SemigroupAction","toSMul"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"],["Monoid","toMulAction"],["Module","toDistribMulAction"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Monoid","toSemigroup"],["MulAction","toSemigroupAction"]],"valueReferences":[["Module","toDistribMulAction"],["IsScalarTower","left"],["CommSemiring","toSemiring"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","fromLocalizedModule_mk"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S : 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.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3))))) {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.40 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.14] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.40) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.71 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3 M M' 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3))) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.40) (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.36) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.36) => M') (LinearMap.instFunLike.{u_1, u_1, max u_1 u_2, u_3} R R (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.36) M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_2} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.11 (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.36)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.14 (OreLocalization.instModuleOfIsScalarTower.{u_1, u_2, u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.36 R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.36 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.11) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.36))) (IsScalarTower.right.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3))) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.40 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3)))) (IsLocalizedModule.fromLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3 S M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.40 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.71) (LocalizedModule.mk.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.36 m s)) (DFunLike.coe.{succ u_3, succ u_3, succ u_3} (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.40) M' (fun 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.40) (Module.End.instMonoid.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.40) (Inv.inv.{u_3} (Units.{u_3} (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.40) (Module.End.instMonoid.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.40)) (Units.instInv.{u_3} (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.40) (Module.End.instMonoid.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.40)) (IsUnit.unit.{u_3} (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.40) (Module.End.instMonoid.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.40) (DFunLike.coe.{max (succ u_1) (succ u_3), succ u_1, succ u_3} (RingHom.{u_1, u_3} R (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.40) (Semiring.toNonAssocSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.40) (RingHom.instFunLike.{u_1, u_3} R (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.40) (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3)) (Semiring.toNonAssocSemiring.{u_3} (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.40) (Module.End.instSemiring.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.40))) (algebraMap.{u_1, u_3} R (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.40) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3 (Module.End.instSemiring.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.40) (Module.End.instAlgebra.{u_1, u_1, u_3} R R M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.40 (smulCommClass_self.{u_1, u_3} R M' (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_3} R M' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.14) (Module.toDistribMulAction.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.40))) (Algebra.toSMul.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_3} R M' (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2112693474._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_3} R M' 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3022597841._hygCtx._hyg.40) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3022597841._hygCtx._hyg.71 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3022597841._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3022597841._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3022597841._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3022597841._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3022597841._hygCtx._hyg.40 S f], Function.Surjective.{max (succ u_1) (succ u_2), succ u_3} (Prod.{u_2, u_1} M (Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3022597841._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3022597841._hygCtx._hyg.40 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3022597841._hygCtx._hyg.71))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f], Function.Surjective (Function.uncurry (IsLocalizedModule.mk' f))","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f], Function.Surjective (Function.uncurry (IsLocalizedModule.mk' 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(RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3))) M M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.83) (f₂ : LinearMap.{u_1, u_1, u_2, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3))) M M₂ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.87) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.118 : IsLocalizedModule.{u_1, u_2, u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3 M M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.83 S₁ f₁] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.122 : IsLocalizedModule.{u_1, u_2, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3 M M₂ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.87 S₂ f₂], LinearMap.{u_1, u_1, u_6, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3))) M₁ M₂ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.87)","typeFull":"{R : Type u_1} →\n  [inst : CommSemiring R] →\n    {M : Type u_2} →\n      [inst_1 : AddCommMonoid M] →\n        [inst_2 : Module R M] →\n          {M₁ : Type u_6} →\n            {M₂ : Type u_7} →\n              [inst_3 : AddCommMonoid M₁] →\n                [inst_4 : AddCommMonoid M₂] →\n                  [inst_5 : Module R M₁] →\n                    [inst_6 : Module R M₂] →\n                      (S₁ S₂ : Submonoid R) →\n                        S₁ ≤ S₂ →\n                          (f₁ : M →ₗ[R] M₁) →\n                            (f₂ : M →ₗ[R] M₂) → [IsLocalizedModule S₁ f₁] → [IsLocalizedModule S₂ f₂] → M₁ →ₗ[R] M₂","typeReadable":"{R : Type u_1} →\n  [inst : CommSemiring R] →\n    {M : Type u_2} →\n      [inst_1 : AddCommMonoid M] →\n        [inst_2 : Module R M] →\n          {M₁ : Type u_6} →\n            {M₂ : Type u_7} →\n              [inst_3 : AddCommMonoid M₁] →\n                [inst_4 : AddCommMonoid M₂] →\n                  [inst_5 : Module R M₁] →\n                    [inst_6 : Module R M₂] →\n                      (S₁ S₂ : Submonoid R) →\n                        S₁ ≤ S₂ →\n                          (f₁ : M →ₗ[R] M₁) →\n                            (f₂ : M →ₗ[R] M₂) → [IsLocalizedModule S₁ f₁] → [IsLocalizedModule S₂ f₂] → M₁ →ₗ[R] M₂","typeReferences":[["IsLocalizedModule"],["PartialOrder","toPreorder"],["Module"],["CommSemiring"],["CommSemiring","toSemiring"],["MulZeroOneClass","toMulOneClass"],["LinearMap"],["Submonoid"],["Submonoid","instPartialOrder"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["LE","le"],["Preorder","toLE"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["IsLocalizedModule","liftOfLE","_proof_2"],["IsLocalizedModule","lift"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","eq_zero_of_smul_eq_zero"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.3 : CommSemiring.{u} R] {S : Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.3))))} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.9 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.12 : Module.{u, v} R M (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.9] (r : R), (Membership.mem.{u, u} R (Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.3))))) (SetLike.instMembership.{u, u} (Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.3))))) R (Submonoid.instSetLike.{u} R 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(DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.9)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.9) (Module.toDistribMulAction.{u, v} R M (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.12)))) (Algebra.toSMul.{u, u} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.3 (CommSemiring.toSemiring.{u} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.12)))))) -> (Eq.{max (succ u) (succ v)} (LocalizedModule.{u, v} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.12) x (OfNat.ofNat.{max u v} (LocalizedModule.{u, v} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.12) 0 (Zero.toOfNat0.{max u v} (LocalizedModule.{u, v} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.12) (OreLocalization.instZero.{u, v} R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.3) S) M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.9)))) (DistribMulAction.toMulAction.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.9) (Module.toDistribMulAction.{u, v} R M (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1535028910._hygCtx._hyg.12)))))))","typeFull":"∀ {R : Type u} [inst : CommSemiring R] {S : Submonoid R} {M : Type v} [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n  ∀ r ∈ S, ∀ (x : LocalizedModule S M), r • x = 0 → x = 0","typeReadable":"∀ {R : Type u} [inst : CommSemiring R] {S : Submonoid R} {M : Type v} [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n  ∀ r ∈ S, ∀ (x : LocalizedModule S M), r • x = 0 → x = 0","typeReferences":[["Module"],["OreLocalization","oreSetComm"],["Membership","mem"],["CommSemiring"],["MulZeroOneClass","toMulOneClass"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["OreLocalization","instZero"],["Algebra","id"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["Algebra","toSMul"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["IsScalarTower","right"],["instHSMul"],["Zero","toOfNat0"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["IsScalarTower","left"],["LocalizedModule"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["OreLocalization","instSMulOfIsScalarTower"],["CommSemiring","toCommMonoid"],["Submonoid"],["Module","toDistribMulAction"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Localization"],["Membership","mem"],["SMulZeroClass","toSMul"],["Algebra","id"],["Semiring","toNonAssocSemiring"],["DistribMulAction","toMulAction"],["IsScalarTower","right"],["NonAssocSemiring","toAddCommMonoidWithOne"],["Localization","mk"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["LocalizedModule"],["Submonoid","instSetLike"],["OreLocalization","instMonoid"],["DistribMulAction","toDistribSMul"],["AddZeroClass","toAddZero"],["CommSemiring","toCommMonoid"],["OreLocalization","instSMulOfIsScalarTower"],["Submonoid"],["Eq","refl"],["AddMonoidWithOne","toOne"],["HSMul","hSMul"],["id"],["Eq","mpr"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["smul_zero"],["CommMonoid","toMonoid"],["OreLocalization","oreSetComm"],["Eq","mp"],["OreLocalization","instSMul"],["AddCommMonoid","toAddMonoid"],["OreLocalization","instZero"],["congrArg"],["MonoidWithZero","toMonoid"],["Monoid","toMulOneClass"],["Algebra","toSMul"],["instHSMul"],["LocalizedModule","smul_eq_iff_of_mem"],["Zero","toOfNat0"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["propext"],["OreLocalization","instDistribMulAction"],["IsScalarTower","left"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["OfNat","ofNat"],["Module","toDistribMulAction"],["One","toOfNat1"],["Subtype","mk"],["OreLocalization","instAddMonoid"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","instRing","_proof_6"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 : CommSemiring.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29 : Ring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.32 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29)] {S : 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.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3))))}, autoParam.{0} (forall (n : Nat) (a : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A 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{S : Submonoid R},\n  autoParam\n    (∀ (n : ℕ) (a : LocalizedModule S A), SubNegMonoid.zsmul (Int.negSucc n) a = -SubNegMonoid.zsmul (↑n.succ) a)\n    SubNegMonoid.zsmul_neg'._autoParam","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Ring A] [inst_2 : Algebra R A] {S : Submonoid R},\n  autoParam\n    (∀ (n : ℕ) (a : LocalizedModule S A), SubNegMonoid.zsmul (Int.negSucc n) a = -SubNegMonoid.zsmul (↑n.succ) a)\n    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{R : Type.{u}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 : CommSemiring.{u} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29 : Ring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.32 : Algebra.{u, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29)] {S : Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3))))}, Ring.{max u_2 u} (LocalizedModule.{u, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} 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{F : Sort.{u_1}} {α : Sort.{u_2}} {β : Sort.{u_3}} [inst._@.Mathlib.Data.FunLike.Embedding.1058169014._hygCtx._hyg.6 : FunLike.{u_1, u_2, u_3} F α β] [i : EmbeddingLike.{u_1, u_2, u_3} F α β inst._@.Mathlib.Data.FunLike.Embedding.1058169014._hygCtx._hyg.6] (f : F) {x : α} {y : α}, Eq.{1} Prop (Eq.{u_3} β (DFunLike.coe.{u_1, u_2, u_3} F α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => β) inst._@.Mathlib.Data.FunLike.Embedding.1058169014._hygCtx._hyg.6 f x) (DFunLike.coe.{u_1, u_2, u_3} F α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => β) inst._@.Mathlib.Data.FunLike.Embedding.1058169014._hygCtx._hyg.6 f y)) (Eq.{u_2} α x y)","typeFull":"∀ {F : Sort u_1} {α : Sort u_2} {β : Sort u_3} [inst : FunLike F α β] [i : EmbeddingLike F α β] (f : F) {x y : α},\n  (f x = f y) = (x = y)","typeReadable":"∀ {F : Sort u_1} {α : Sort u_2} {β : Sort u_3} [inst : FunLike F α β] [i : EmbeddingLike F α β] (f : F) {x y : α},\n  (f x = f y) = (x = 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.141 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.141 (smulCommClass_self.{u_1, u_10} R M₂' (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_10} R M₂' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_10} M₂' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.134) (Module.toDistribMulAction.{u_1, u_10} R M₂' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.134 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.141))))) (LinearMap.{u_1, u_1, u_6, u_8} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) M₀ M₂ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.131 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.79 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.137) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : LinearMap.{u_1, u_1, u_6, u_8} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) M₀ M₂ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.131 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.79 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.137) => LinearMap.{u_1, u_1, u_9, u_10} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) M₀' M₂' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.76 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.134 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.141) (LinearMap.instFunLike.{u_1, u_1, max u_6 u_8, max u_9 u_10} R R (LinearMap.{u_1, u_1, u_6, u_8} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) M₀' M₂' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.76 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.134 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.141) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (LinearMap.addCommMonoid.{u_1, u_1, u_6, u_8} R R M₀ M₂ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.131 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.79 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.137 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3)))) (LinearMap.addCommMonoid.{u_1, u_1, u_9, u_10} R R M₀' M₂' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.76 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.134 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.141 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3)))) (LinearMap.module.{u_1, u_1, u_1, u_6, u_8} R R R M₀ M₂ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.131 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.79 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.137 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.137 (smulCommClass_self.{u_1, u_8} R M₂ (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_8} R M₂ (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_8} M₂ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.131) (Module.toDistribMulAction.{u_1, u_8} R M₂ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.131 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.137)))) (LinearMap.module.{u_1, u_1, u_1, u_9, u_10} R R R M₀' M₂' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.76 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.134 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.141 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.141 (smulCommClass_self.{u_1, u_10} R M₂' (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_10} R M₂' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_10} M₂' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.134) (Module.toDistribMulAction.{u_1, u_10} R M₂' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.134 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.141)))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3)))) (IsLocalizedModule.map.{u_1, u_6, u_9, u_8, u_10} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3 S M₀ M₀' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.76 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.79 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.83 f₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.96 M₂ M₂' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.131 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.134 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.137 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.141 f₂ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.154) (LinearMap.comp.{u_1, u_1, u_1, u_6, u_7, u_8} R R R M₀ M₁ M₂ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.102 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.131 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.79 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.108 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.137 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) (RingHomCompTriple.ids.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3)))) h g)) (LinearMap.comp.{u_1, u_1, u_1, u_9, u_11, u_10} R R R M₀' 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.131 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.108 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.137) (LinearMap.{u_1, u_1, u_11, u_10} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) M₁' M₂' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.134 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.112 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.141) (LinearMap.addCommMonoid.{u_1, u_1, u_7, u_8} R R M₁ M₂ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.102 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.131 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.108 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.137 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3)))) (LinearMap.addCommMonoid.{u_1, u_1, u_11, u_10} R R M₁' M₂' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.134 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.112 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.141 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3)))) (LinearMap.module.{u_1, u_1, u_1, u_7, u_8} R R R M₁ M₂ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.102 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.131 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.108 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.137 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.137 (smulCommClass_self.{u_1, u_8} R M₂ (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_8} R M₂ (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_8} M₂ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.131) (Module.toDistribMulAction.{u_1, u_8} R M₂ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.131 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.137)))) (LinearMap.module.{u_1, u_1, u_1, u_11, u_10} R R R M₁' M₂' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.134 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.112 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.141 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.141 (smulCommClass_self.{u_1, u_10} R M₂' (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_10} R M₂' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_10} M₂' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.134) (Module.toDistribMulAction.{u_1, u_10} R M₂' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.134 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.141))))) (LinearMap.{u_1, u_1, u_7, u_8} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) M₁ M₂ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.102 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.131 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.108 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.137) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : LinearMap.{u_1, u_1, u_7, u_8} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) M₁ M₂ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.102 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.131 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.108 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.137) => LinearMap.{u_1, u_1, u_11, u_10} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) M₁' M₂' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.134 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.112 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.141) (LinearMap.instFunLike.{u_1, u_1, max u_7 u_8, max u_11 u_10} R R (LinearMap.{u_1, u_1, u_7, u_8} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) M₁ M₂ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.102 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.131 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.108 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.137) (LinearMap.{u_1, u_1, u_11, u_10} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) M₁' M₂' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.134 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.112 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.141) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (LinearMap.addCommMonoid.{u_1, u_1, u_7, u_8} R R M₁ M₂ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.102 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.131 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.108 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.137 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3)))) (LinearMap.addCommMonoid.{u_1, u_1, u_11, u_10} R R M₁' M₂' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.134 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.112 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.141 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3)))) (LinearMap.module.{u_1, u_1, u_1, u_7, u_8} R R R M₁ M₂ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.102 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.131 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.108 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.137 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.137 (smulCommClass_self.{u_1, u_8} R M₂ (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_8} R M₂ (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_8} M₂ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.131) (Module.toDistribMulAction.{u_1, u_8} R M₂ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.131 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.137)))) (LinearMap.module.{u_1, u_1, u_1, u_11, u_10} R R R M₁' M₂' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.134 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.112 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.141 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.141 (smulCommClass_self.{u_1, u_10} R M₂' (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_10} R M₂' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_10} M₂' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.134) (Module.toDistribMulAction.{u_1, u_10} R M₂' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.134 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.141)))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3)))) (IsLocalizedModule.map.{u_1, u_7, u_11, u_8, u_10} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3 S M₁ M₁' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.102 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.108 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.112 f₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.125 M₂ M₂' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.131 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.134 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.137 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.141 f₂ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.154) h) (DFunLike.coe.{max (max (max (succ u_6) (succ u_9)) (succ u_7)) (succ u_11), max (succ u_6) (succ u_7), max (succ u_9) (succ u_11)} (LinearMap.{u_1, u_1, max u_7 u_6, max u_11 u_9} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) (LinearMap.{u_1, u_1, u_6, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) M₀ M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.102 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.79 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.108) (LinearMap.{u_1, u_1, u_9, u_11} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) M₀' M₁' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.76 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.112) (LinearMap.addCommMonoid.{u_1, u_1, u_6, u_7} R R M₀ M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.102 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.79 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.108 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3)))) (LinearMap.addCommMonoid.{u_1, u_1, u_9, u_11} R R M₀' M₁' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.76 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.112 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3)))) (LinearMap.module.{u_1, u_1, u_1, u_6, u_7} R R R M₀ M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.102 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.79 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.108 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.108 (smulCommClass_self.{u_1, u_7} R M₁ (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_7} R M₁ (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_7} M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.102) (Module.toDistribMulAction.{u_1, u_7} R M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.102 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.108)))) (LinearMap.module.{u_1, u_1, u_1, u_9, u_11} R R R M₀' M₁' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.76 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.112 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.112 (smulCommClass_self.{u_1, u_11} R M₁' (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_11} R M₁' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_11} M₁' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.105) (Module.toDistribMulAction.{u_1, u_11} R M₁' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.112))))) (LinearMap.{u_1, u_1, u_6, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) M₀ M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.102 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.79 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.108) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : LinearMap.{u_1, u_1, u_6, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) M₀ M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.102 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.79 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.108) => LinearMap.{u_1, u_1, u_9, u_11} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) M₀' M₁' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.76 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.112) (LinearMap.instFunLike.{u_1, u_1, max u_6 u_7, max u_9 u_11} R R (LinearMap.{u_1, u_1, u_6, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) M₀ M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.102 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.79 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.108) (LinearMap.{u_1, u_1, u_9, u_11} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) M₀' M₁' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.76 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.112) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (LinearMap.addCommMonoid.{u_1, u_1, u_6, u_7} R R M₀ M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.102 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.79 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.108 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3)))) (LinearMap.addCommMonoid.{u_1, u_1, u_9, u_11} R R M₀' M₁' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.76 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.112 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3)))) (LinearMap.module.{u_1, u_1, u_1, u_6, u_7} R R R M₀ M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.102 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.79 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.108 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.108 (smulCommClass_self.{u_1, u_7} R M₁ (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_7} R M₁ (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_7} M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.102) (Module.toDistribMulAction.{u_1, u_7} R M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.102 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.108)))) (LinearMap.module.{u_1, u_1, u_1, u_9, u_11} R R R M₀' M₁' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.76 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.112 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.112 (smulCommClass_self.{u_1, u_11} R M₁' (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_11} R M₁' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_11} M₁' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.105) (Module.toDistribMulAction.{u_1, u_11} R M₁' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.112)))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3)))) (IsLocalizedModule.map.{u_1, u_6, u_9, u_7, u_11} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.3 S M₀ M₀' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.76 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.79 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.83 f₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.96 M₁ M₁' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.102 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.108 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.112 f₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3465103411._hygCtx._hyg.125) g))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M₀ : Type u_6} {M₀' : Type u_9} [inst_1 : AddCommMonoid M₀]\n  [inst_2 : AddCommMonoid M₀'] [inst_3 : Module R M₀] [inst_4 : Module R M₀'] (f₀ : M₀ →ₗ[R] M₀')\n  [inst_5 : IsLocalizedModule S f₀] {M₁ : Type u_7} {M₁' : Type u_11} [inst_6 : AddCommMonoid M₁]\n  [inst_7 : AddCommMonoid M₁'] [inst_8 : Module R M₁] [inst_9 : Module R M₁'] (f₁ : M₁ →ₗ[R] M₁')\n  [inst_10 : IsLocalizedModule S f₁] {M₂ : Type u_8} {M₂' : Type u_10} [inst_11 : AddCommMonoid M₂]\n  [inst_12 : AddCommMonoid M₂'] [inst_13 : Module R M₂] [inst_14 : Module R M₂'] (f₂ : M₂ →ₗ[R] M₂')\n  [inst_15 : IsLocalizedModule S f₂] (g : M₀ →ₗ[R] M₁) (h : M₁ →ₗ[R] M₂),\n  (IsLocalizedModule.map S f₀ f₂) (h ∘ₗ g) = (IsLocalizedModule.map S f₁ f₂) h ∘ₗ (IsLocalizedModule.map S f₀ f₁) g","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M₀ : Type u_6} {M₀' : Type u_9} [inst_1 : AddCommMonoid M₀]\n  [inst_2 : AddCommMonoid M₀'] [inst_3 : Module R M₀] [inst_4 : Module R M₀'] (f₀ : M₀ →ₗ[R] M₀')\n  [inst_5 : IsLocalizedModule S f₀] {M₁ : Type u_7} {M₁' : Type u_11} [inst_6 : AddCommMonoid M₁]\n  [inst_7 : AddCommMonoid M₁'] [inst_8 : Module R M₁] [inst_9 : Module R M₁'] (f₁ : M₁ →ₗ[R] M₁')\n  [inst_10 : IsLocalizedModule S f₁] {M₂ : Type u_8} {M₂' : Type u_10} [inst_11 : AddCommMonoid M₂]\n  [inst_12 : AddCommMonoid M₂'] [inst_13 : Module R M₂] [inst_14 : Module R M₂'] (f₂ : M₂ →ₗ[R] M₂')\n  [inst_15 : IsLocalizedModule S f₂] (g : M₀ →ₗ[R] M₁) (h : M₁ →ₗ[R] M₂),\n  (IsLocalizedModule.map S f₀ f₂) (h ∘ₗ g) = (IsLocalizedModule.map S f₁ f₂) h ∘ₗ (IsLocalizedModule.map S f₀ f₁) g","typeReferences":[["IsLocalizedModule"],["smulCommClass_self"],["CommMonoid","toMonoid"],["LinearMap","instFunLike"],["Module"],["RingHomCompTriple","ids"],["CommSemiring"],["LinearMap","comp"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["DistribMulAction","toMulAction"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["IsLocalizedModule","map"],["CommSemiring","toSemiring"],["LinearMap","module"],["LinearMap"],["CommSemiring","toCommMonoid"],["Submonoid"],["LinearMap","addCommMonoid"],["Module","toDistribMulAction"]],"valueReferences":[["smulCommClass_self"],["Subtype"],["Eq","trans"],["CommMonoid","toMonoid"],["LinearMap","instFunLike"],["Prod","mk"],["RingHomCompTriple","ids"],["Membership","mem"],["LinearMap","comp"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["congrArg"],["Prod","casesOn"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["congr"],["LinearMap","ext"],["DistribMulAction","toMulAction"],["Eq","ndrec"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["IsLocalizedModule","mk'"],["Function","uncurry"],["IsLocalizedModule","map"],["SetLike","instMembership"],["True"],["Submonoid","instSetLike"],["LinearMap","module"],["CommSemiring","toSemiring"],["LinearMap"],["IsLocalizedModule","map_mk'"],["CommSemiring","toCommMonoid"],["LinearMap","addCommMonoid"],["Exists","casesOn"],["Prod"],["Submonoid"],["eq_self"],["Module","toDistribMulAction"],["of_eq_true"],["IsLocalizedModule","mk'_surjective"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","is_universal"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S : 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.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3))))) {M : Type.{u_2}} {M' : Type.{u_3}} {M'' : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.17 : AddCommMonoid.{u_4} M''] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.40 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.14] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.44 : Module.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.17] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.40) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.71 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.40 S f] (g : LinearMap.{u_1, u_1, u_2, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3))) M M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.44), (forall (x : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3))))) R (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.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3)))))) S x)), IsUnit.{u_4} (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.44) (Module.End.instMonoid.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.44) (DFunLike.coe.{max (succ u_1) (succ u_4), succ u_1, succ u_4} (RingHom.{u_1, u_4} R (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.44) (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3)) (Semiring.toNonAssocSemiring.{u_4} (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.44) (Module.End.instSemiring.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.44))) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.44) (RingHom.instFunLike.{u_1, u_4} R (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.44) (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3)) (Semiring.toNonAssocSemiring.{u_4} (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.44) (Module.End.instSemiring.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.44))) (algebraMap.{u_1, u_4} R (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.44) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3 (Module.End.instSemiring.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.44) (Module.End.instAlgebra.{u_1, u_1, u_4} R R M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.44 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.44 (smulCommClass_self.{u_1, u_4} R M'' (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_4} R M'' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_4} M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1219585497._hygCtx._hyg.17) (Module.toDistribMulAction.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R 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[inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.71 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 S f] {N : Type.{u_6}} {N' : Type.{u_7}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 : AddCommMonoid.{u_6} N] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 : AddCommMonoid.{u_7} N'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 : Module.{u_1, u_6} R N (CommSemiring.toSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87))))) (IsLocalizedModule.map.{u_1, u_2, u_3, u_6, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3 S M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.71 N N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) M N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83) (LinearMap.{u_1, u_1, u_3, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) M' N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87) (LinearMap.addCommMonoid.{u_1, u_1, u_2, u_6} R R M N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 (LocalizedModule.lift'._proof_2.{u_1, u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3 N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83)) (LinearMap.module.{u_1, u_1, u_1, u_3, u_7} R R R M' N' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) M N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83) (LinearMap.{u_1, u_1, u_3, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) M' N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87) (AddCommMagma.toAdd.{max u_2 u_6} (LinearMap.{u_1, u_1, u_2, u_6} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) M N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83) (AddCommSemigroup.toAddCommMagma.{max u_2 u_6} (LinearMap.{u_1, u_1, u_2, u_6} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) M N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83) (AddCommMonoid.toAddCommSemigroup.{max u_2 u_6} (LinearMap.{u_1, u_1, u_2, u_6} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) M N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83) (LinearMap.addCommMonoid.{u_1, u_1, u_2, u_6} R R M N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))))))) (AddCommMagma.toAdd.{max u_3 u_7} (LinearMap.{u_1, u_1, u_3, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) M' N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87) (AddCommSemigroup.toAddCommMagma.{max u_3 u_7} (LinearMap.{u_1, u_1, u_3, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) M' N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87) (AddCommMonoid.toAddCommSemigroup.{max u_3 u_7} (LinearMap.{u_1, u_1, u_3, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) M' N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87) (LinearMap.addCommMonoid.{u_1, u_1, u_3, u_7} R R M' N' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))))))) (fun (h : LinearMap.{u_1, u_1, u_2, u_6} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) M N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83) => IsLocalizedModule.lift.{u_1, u_2, u_3, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3 S M M' N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.71 (LinearMap.comp.{u_1, u_1, u_1, u_2, u_6, u_7} R R R M N N' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) (IsLocalizedModule.lift._proof_1.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) g h) (IsLocalizedModule.map_units.{u_1, u_6, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3 N N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87 S g inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.100)) (fun (h₁ : LinearMap.{u_1, u_1, u_2, u_6} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) M N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83) (h₂ : LinearMap.{u_1, u_1, u_2, u_6} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) M N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83) => IsLocalizedModule.map._proof_2.{u_3, u_7, u_1, u_2, u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3 S M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.71 N N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87 g inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.100 h₁ h₂)) (fun (r : R) (h : LinearMap.{u_1, u_1, u_2, u_6} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) M N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83) => IsLocalizedModule.map._proof_3.{u_3, u_7, u_1, u_2, u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3 S M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.71 N N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87 g inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.100 r h))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] {N : Type u_6} {N' : Type u_7} [inst_6 : AddCommMonoid N] [inst_7 : AddCommMonoid N']\n  [inst_8 : Module R N] [inst_9 : Module R N'] (g : N →ₗ[R] N') [inst_10 : IsLocalizedModule S g],\n  IsLocalizedModule.map S f g =\n    { toFun := fun h => IsLocalizedModule.lift S f (g ∘ₗ h) ⋯, map_add' := ⋯, map_smul' := ⋯ }","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] {N : Type u_6} {N' : Type u_7} [inst_6 : AddCommMonoid N] [inst_7 : AddCommMonoid N']\n  [inst_8 : Module R N] [inst_9 : Module R N'] (g : N →ₗ[R] N') [inst_10 : IsLocalizedModule S g],\n  IsLocalizedModule.map S f g =\n    { toFun := fun h => IsLocalizedModule.lift S f (g ∘ₗ h) ⋯, map_add' := ⋯, map_smul' := ⋯ 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3389458913._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3389458913._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3389458913._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3389458913._hygCtx._hyg.40) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3389458913._hygCtx._hyg.71 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3389458913._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3389458913._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3389458913._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3389458913._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3389458913._hygCtx._hyg.40 S f] (g : LinearMap.{u_1, u_1, u_2, u_4} R R (CommSemiring.toSemiring.{u_1} R 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[inst_4 : Module R M]\n  [inst_5 : Module R M'] [inst_6 : Module R M''] (f : M →ₗ[R] M') [inst_7 : IsLocalizedModule S f] (g : M →ₗ[R] M'')\n  (h : ∀ (x : ↥S), IsUnit ((algebraMap R (Module.End R M'')) ↑x)), IsLocalizedModule.lift S f g h ∘ₗ f = g","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} {M'' : Type u_4}\n  [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M'] [inst_3 : AddCommMonoid M''] [inst_4 : Module R M]\n  [inst_5 : Module R M'] [inst_6 : Module R M''] (f : M →ₗ[R] M') [inst_7 : IsLocalizedModule S f] (g : M →ₗ[R] M'')\n  (h : ∀ (x : ↥S), IsUnit ((algebraMap R (Module.End R M'')) ↑x)), IsLocalizedModule.lift S f g h ∘ₗ f = 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.614465900._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S : 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.Algebra.Module.LocalizedModule.Basic.614465900._hygCtx._hyg.3))))) {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.614465900._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.614465900._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.614465900._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.614465900._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.614465900._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.614465900._hygCtx._hyg.40 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[inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f], ↑(IsLocalizedModule.iso S f).symm ∘ₗ f = LocalizedModule.mkLinearMap S 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2712534485._hygCtx._hyg.3 : CommSemiring.{u_1} R] {S : 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.Algebra.Module.LocalizedModule.Basic.2712534485._hygCtx._hyg.3))))} {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2712534485._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2712534485._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2712534485._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2712534485._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2712534485._hygCtx._hyg.11] 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2712534485._hygCtx._hyg.40) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2712534485._hygCtx._hyg.71 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2712534485._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2712534485._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2712534485._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2712534485._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2712534485._hygCtx._hyg.40 S f] (s : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.2712534485._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.2712534485._hygCtx._hyg.3))))) R (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.Algebra.Module.LocalizedModule.Basic.2712534485._hygCtx._hyg.3)))))) S x)), Eq.{succ u_3} M' (IsLocalizedModule.mk'.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2712534485._hygCtx._hyg.3 S M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2712534485._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2712534485._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2712534485._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2712534485._hygCtx._hyg.40 f 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Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (s : ↥S), IsLocalizedModule.mk' f 0 s = 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.3 : CommSemiring.{u_1} R] {S : 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.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.3))))} {M : Type.{u_6}} {M' : Type.{u_7}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.77 : AddCommGroup.{u_6} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.80 : SubtractionCommMonoid.{u_7} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.83 : Module.{u_1, u_6} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_6} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.77)] 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.80) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.87) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.100 : IsLocalizedModule.{u_1, u_6, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.3 M M' (AddCommGroup.toAddCommMonoid.{u_6} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.77) (SubtractionCommMonoid.toAddCommMonoid.{u_7} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.80) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.87 S f] (m₁ : M) (m₂ : M) (s : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.3))))) R (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.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.3)))))) S x)), Eq.{succ u_7} M' (IsLocalizedModule.mk'.{u_1, u_6, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.3 S M M' (AddCommGroup.toAddCommMonoid.{u_6} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.77) (SubtractionCommMonoid.toAddCommMonoid.{u_7} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.80) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.87 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.100 (HSub.hSub.{u_6, u_6, u_6} M M M (instHSub.{u_6} M (SubNegMonoid.toSub.{u_6} M (AddGroup.toSubNegMonoid.{u_6} M (AddCommGroup.toAddGroup.{u_6} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.77)))) m₁ m₂) s) (HSub.hSub.{u_7, u_7, u_7} M' M' M' (instHSub.{u_7} M' (SubNegMonoid.toSub.{u_7} M' (SubtractionMonoid.toSubNegMonoid.{u_7} M' (SubtractionCommMonoid.toSubtractionMonoid.{u_7} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.80)))) (IsLocalizedModule.mk'.{u_1, u_6, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.3 S M M' (AddCommGroup.toAddCommMonoid.{u_6} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.77) (SubtractionCommMonoid.toAddCommMonoid.{u_7} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.80) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.87 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.100 m₁ s) (IsLocalizedModule.mk'.{u_1, u_6, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.3 S M M' (AddCommGroup.toAddCommMonoid.{u_6} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.77) (SubtractionCommMonoid.toAddCommMonoid.{u_7} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.80) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.87 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2897001000._hygCtx._hyg.100 m₂ s))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_6} {M' : Type u_7} [inst_1 : AddCommGroup M]\n  [inst_2 : SubtractionCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (m₁ m₂ : M) (s : ↥S),\n  IsLocalizedModule.mk' f (m₁ - m₂) s = IsLocalizedModule.mk' f m₁ s - IsLocalizedModule.mk' f m₂ s","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_6} {M' : Type u_7} [inst_1 : AddCommGroup M]\n  [inst_2 : SubtractionCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (m₁ m₂ : M) (s : ↥S),\n  IsLocalizedModule.mk' f (m₁ - m₂) s = IsLocalizedModule.mk' f m₁ s - IsLocalizedModule.mk' f m₂ s","typeReferences":[["IsLocalizedModule"],["Subtype"],["Module"],["AddCommGroup","toAddGroup"],["CommSemiring"],["Membership","mem"],["SubtractionCommMonoid","toSubtractionMonoid"],["MulZeroOneClass","toMulOneClass"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["SubtractionCommMonoid","toAddCommMonoid"],["SubNegMonoid","toSub"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["IsLocalizedModule","mk'"],["SetLike","instMembership"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["SubtractionCommMonoid"],["AddCommGroup"],["LinearMap"],["Submonoid"],["SubtractionMonoid","toSubNegMonoid"],["AddCommGroup","toAddCommMonoid"],["instHSub"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["IsLocalizedModule","mk'_neg"],["AddCommGroup","toAddGroup"],["SubtractionCommMonoid","toSubtractionMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["congrArg"],["sub_eq_add_neg"],["IsLocalizedModule","mk'_add"],["SubtractionCommMonoid","toAddCommMonoid"],["SubNegMonoid","toSub"],["HSub","hSub"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["AddGroup","toSubNegMonoid"],["Eq"],["IsLocalizedModule","mk'"],["instHAdd"],["Neg","neg"],["SubNegMonoid","toNeg"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["HAdd","hAdd"],["NegZeroClass","toNeg"],["AddCommGroup","toDivisionAddCommMonoid"],["AddCommMonoid","toAddCommSemigroup"],["SubNegMonoid","toAddMonoid"],["SubtractionMonoid","toSubNegMonoid"],["Eq","refl"],["AddCommGroup","toAddCommMonoid"],["id"],["Eq","mpr"],["instHSub"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Module","LocalizedModule","Basic",0,"LocalizedModule","example_oreLocalizationInstMonoid_eq_localizedModuleInstMonoid"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1592591021._hygCtx._hyg.3 : CommSemiring.{u} R] {S : Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1592591021._hygCtx._hyg.3))))}, Eq.{succ u} (Monoid.{u} (OreLocalization.{u, u} R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1592591021._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1592591021._hygCtx._hyg.3) S) R (Monoid.toMulAction.{u} R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1592591021._hygCtx._hyg.3)))))) (OreLocalization.instMonoid.{u} R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1592591021._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1592591021._hygCtx._hyg.3) S)) (LocalizedModule.instMonoid.{u, u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1592591021._hygCtx._hyg.3 R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1592591021._hygCtx._hyg.3) (Algebra.id.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1592591021._hygCtx._hyg.3) S)","typeFull":"∀ {R : Type u} [inst : CommSemiring R] {S : Submonoid R}, OreLocalization.instMonoid = LocalizedModule.instMonoid","typeReadable":"∀ {R : Type u} [inst : CommSemiring R] {S : Submonoid R}, OreLocalization.instMonoid = LocalizedModule.instMonoid","typeReferences":[["LocalizedModule","instMonoid"],["OreLocalization","oreSetComm"],["OreLocalization","instMonoid"],["CommSemiring"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["MulZeroOneClass","toMulOneClass"],["Algebra","id"],["Monoid","toMulAction"],["CommSemiring","toCommMonoid"],["Submonoid"],["Semiring","toNonAssocSemiring"],["OreLocalization"],["MonoidWithZero","toMonoid"],["Monoid"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["OreLocalization"],["OreLocalization","oreSetComm"],["OreLocalization","instMonoid"],["Eq","refl"],["CommSemiring","toSemiring"],["MonoidWithZero","toMonoid"],["Monoid"],["Semiring","toMonoidWithZero"],["Monoid","toMulAction"],["CommSemiring","toCommMonoid"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","algebra'","_proof_4"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3 : CommSemiring.{u_1} R], IsScalarTower.{u_1, u_1, u_1} R R R (Algebra.toSMul.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3)) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3)))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3)))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3)))))) (Algebra.toSMul.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R], IsScalarTower R R R","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R], IsScalarTower R R R","typeReferences":[["IsScalarTower"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["CommSemiring"],["CommSemiring","toSemiring"],["SMulZeroClass","toSMul"],["instDistribSMul"],["AddZeroClass","toAddZero"],["AddMonoidWithOne","toAddMonoid"],["Algebra","id"],["Semiring","toNonAssocSemiring"],["Algebra","toSMul"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["AddZero","toZero"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["CommSemiring","toSemiring"],["IsScalarTower","right"],["Algebra","id"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","liftOn","_proof_2"],"typeFallback":"forall {R : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.584353708._hygCtx._hyg.3 : CommSemiring.{u_2} R] {S : Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.584353708._hygCtx._hyg.3))))} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.584353708._hygCtx._hyg.9 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.584353708._hygCtx._hyg.12 : Module.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.584353708._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.584353708._hygCtx._hyg.9] {α : Type.{u_3}} (f : (Prod.{u_1, u_2} M (Subtype.{succ u_2} R (fun (x : R) => Membership.mem.{u_2, u_2} R (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.584353708._hygCtx._hyg.3))))) (SetLike.instMembership.{u_2, u_2} (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R 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{R : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.325308076._hygCtx._hyg.29 : CommRing.{u_2} R] {S : Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.325308076._hygCtx._hyg.29)))))}, Eq.{succ u_2} (CommRing.{u_2} (OreLocalization.{u_2, u_2} R (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.325308076._hygCtx._hyg.29)))) S (OreLocalization.oreSetComm.{u_2} R (CommSemiring.toCommMonoid.{u_2} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.325308076._hygCtx._hyg.29)) S) R (DistribMulAction.toMulAction.{u_2, u_2} R R (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.325308076._hygCtx._hyg.29)))) (AddCommMonoid.toAddMonoid.{u_2} R (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.Algebra.Module.LocalizedModule.Basic.325308076._hygCtx._hyg.29)))))) (Module.toDistribMulAction.{u_2, u_2} R R (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.325308076._hygCtx._hyg.29)) (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.Algebra.Module.LocalizedModule.Basic.325308076._hygCtx._hyg.29))))) (Semiring.toModule.{u_2} R (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.325308076._hygCtx._hyg.29))))))) (OreLocalization.instCommRing.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.325308076._hygCtx._hyg.29 S (OreLocalization.oreSetComm.{u_2} R (CommSemiring.toCommMonoid.{u_2} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.325308076._hygCtx._hyg.29)) S)) (LocalizedModule.instCommRing.{u_2, u_2} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.325308076._hygCtx._hyg.29) R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.325308076._hygCtx._hyg.29 (Algebra.id.{u_2} R (CommRing.toCommSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.325308076._hygCtx._hyg.29)) S)","typeFull":"∀ {R : Type u_2} [inst : CommRing R] {S : Submonoid R}, OreLocalization.instCommRing = LocalizedModule.instCommRing","typeReadable":"∀ {R : Type u_2} [inst : CommRing R] {S : Submonoid R}, OreLocalization.instCommRing = LocalizedModule.instCommRing","typeReferences":[["OreLocalization","oreSetComm"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["CommRing","toNonUnitalCommRing"],["Algebra","id"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["OreLocalization"],["Semiring","toNonAssocSemiring"],["OreLocalization","instCommRing"],["MonoidWithZero","toMonoid"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["DistribMulAction","toMulAction"],["Eq"],["Semiring","toModule"],["NonAssocSemiring","toMulZeroOneClass"],["CommRing","toCommSemiring"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Semiring","toMonoidWithZero"],["CommRing"],["CommSemiring","toCommMonoid"],["Submonoid"],["Module","toDistribMulAction"],["LocalizedModule","instCommRing"]],"valueReferences":[["CommRing","toCommSemiring"],["OreLocalization","oreSetComm"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Semiring","toMonoidWithZero"],["CommRing","toNonUnitalCommRing"],["AddCommMonoid","toAddMonoid"],["CommRing"],["CommSemiring","toCommMonoid"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Module","toDistribMulAction"],["OreLocalization","instCommRing"],["OreLocalization"],["Eq","refl"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Semiring","toModule"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","lift_iso"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S : 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.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.3))))) {M : Type.{u_2}} {M' : Type.{u_3}} {M'' : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.17 : AddCommMonoid.{u_4} M''] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.40 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.14] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.44 : Module.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.17] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.40) (g : LinearMap.{u_1, u_1, u_2, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.3))) M M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.44) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.71 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.40 S f] (h : forall (x : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.44) (Module.End.instMonoid.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.44) (DFunLike.coe.{max (succ u_1) (succ u_4), succ u_1, succ u_4} (RingHom.{u_1, u_4} R (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.44) (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.44) (RingHom.instFunLike.{u_1, u_4} R (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.44) (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.3)) (Semiring.toNonAssocSemiring.{u_4} (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3594686664._hygCtx._hyg.44) 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Type u_4}\n  [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M'] [inst_3 : AddCommMonoid M''] [inst_4 : Module R M]\n  [inst_5 : Module R M'] [inst_6 : Module R M''] (f : M →ₗ[R] M') (g : M →ₗ[R] M'') [inst_7 : IsLocalizedModule S f]\n  (h : ∀ (x : ↥S), IsUnit ((algebraMap R (Module.End R M'')) ↑x)) (x : LocalizedModule S M),\n  (IsLocalizedModule.lift S f g h) ((IsLocalizedModule.iso S f) x) = (LocalizedModule.lift S g h) x","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} {M'' : Type u_4}\n  [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M'] [inst_3 : AddCommMonoid M''] [inst_4 : Module R M]\n  [inst_5 : Module R M'] [inst_6 : Module R M''] (f : M →ₗ[R] M') (g : M →ₗ[R] M'') [inst_7 : IsLocalizedModule S f]\n  (h : ∀ (x : ↥S), IsUnit ((algebraMap R (Module.End R M'')) ↑x)) (x : LocalizedModule S M),\n  (IsLocalizedModule.lift S f g h) ((IsLocalizedModule.iso S f) x) = (LocalizedModule.lift S g h) x","typeReferences":[["smulCommClass_self"],["RingHom"],["IsLocalizedModule"],["Membership","mem"],["Algebra","id"],["Subtype","val"],["LocalizedModule","lift"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["DistribMulAction","toMulAction"],["IsScalarTower","right"],["IsLocalizedModule","lift"],["Semiring","toModule"],["NonAssocSemiring","toMulZeroOneClass"],["Module","End","instSemiring"],["SetLike","instMembership"],["LocalizedModule"],["Submonoid","instSetLike"],["LinearMap"],["CommSemiring","toCommMonoid"],["Submonoid"],["IsLocalizedModule","iso"],["RingHomInvPair","ids"],["algebraMap"],["Subtype"],["CommMonoid","toMonoid"],["LinearMap","instFunLike"],["Module"],["OreLocalization","oreSetComm"],["CommSemiring"],["RingHom","instFunLike"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["LinearEquiv"],["DFunLike","coe"],["Module","End","instAlgebra"],["Algebra","toSMul"],["MonoidWithZero","toMonoid"],["EquivLike","toFunLike"],["Eq"],["LinearEquiv","instEquivLike"],["Module","End"],["IsScalarTower","left"],["OreLocalization","instAddCommMonoidOreLocalization"],["CommSemiring","toSemiring"],["OreLocalization","instModuleOfIsScalarTower"],["Semiring","toMonoidWithZero"],["IsUnit"],["Module","toDistribMulAction"],["Module","End","instMonoid"]],"valueReferences":[["LinearEquiv","symm_apply_apply"],["IsLocalizedModule","iso","_proof_1"],["Eq","trans"],["LinearMap","instFunLike"],["OreLocalization","oreSetComm"],["AddCommMonoid","toAddMonoid"],["LinearEquiv"],["DFunLike","coe"],["Algebra","id"],["congrArg"],["LocalizedModule","lift"],["LocalizedModule","mkLinearMap","_proof_3"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["EquivLike","toFunLike"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["IsScalarTower","right"],["congrFun'"],["LinearEquiv","instEquivLike"],["IsLocalizedModule","lift"],["Eq"],["Semiring","toModule"],["True"],["OreLocalization","instAddCommMonoidOreLocalization"],["LocalizedModule"],["IsScalarTower","left"],["CommSemiring","toSemiring"],["OreLocalization","instModuleOfIsScalarTower"],["Semiring","toMonoidWithZero"],["LinearMap"],["LinearEquiv","symm"],["CommSemiring","toCommMonoid"],["eq_self"],["Module","toDistribMulAction"],["of_eq_true"],["IsLocalizedModule","iso"],["RingHomInvPair","ids"],["LocalizedModule","algebra'","_proof_4"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","of_restrictScalars"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.3 : CommSemiring.{u_1} R] {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] {A : Type.{u_5}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.21 : CommSemiring.{u_5} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.24 : Algebra.{u_1, u_5} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_5} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.21)] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.11] (S : 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.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.3))))) {N : Type.{u_6}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.74 : AddCommMonoid.{u_6} N] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.77 : Module.{u_1, u_6} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.74] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.81 : Module.{u_5, u_2} A M (CommSemiring.toSemiring.{u_5} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.21) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.85 : Module.{u_5, u_6} A N (CommSemiring.toSemiring.{u_5} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.21) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.74] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.89 : IsScalarTower.{u_1, u_5, u_2} R A M (Algebra.toSMul.{u_1, u_5} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_5} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.21) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.24) (SMulZeroClass.toSMul.{u_5, u_2} A M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_5, u_2} A M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_5, u_2} A M (MonoidWithZero.toMonoid.{u_5} A (Semiring.toMonoidWithZero.{u_5} A (CommSemiring.toSemiring.{u_5} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.21))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.11) (Module.toDistribMulAction.{u_5, u_2} A M (CommSemiring.toSemiring.{u_5} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.21) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.81)))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.11) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.36))))] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.94 : IsScalarTower.{u_1, u_5, u_6} R A N (Algebra.toSMul.{u_1, u_5} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_5} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.21) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.24) (SMulZeroClass.toSMul.{u_5, u_6} A N (AddZero.toZero.{u_6} N (AddZeroClass.toAddZero.{u_6} N (AddMonoid.toAddZeroClass.{u_6} N (AddCommMonoid.toAddMonoid.{u_6} N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.74)))) (DistribSMul.toSMulZeroClass.{u_5, u_6} A N (AddMonoid.toAddZeroClass.{u_6} N (AddCommMonoid.toAddMonoid.{u_6} N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.74)) (DistribMulAction.toDistribSMul.{u_5, u_6} A N (MonoidWithZero.toMonoid.{u_5} A (Semiring.toMonoidWithZero.{u_5} A (CommSemiring.toSemiring.{u_5} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.21))) (AddCommMonoid.toAddMonoid.{u_6} N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.74) (Module.toDistribMulAction.{u_5, u_6} A N (CommSemiring.toSemiring.{u_5} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.21) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.74 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.85)))) (SMulZeroClass.toSMul.{u_1, u_6} R N (AddZero.toZero.{u_6} N (AddZeroClass.toAddZero.{u_6} N (AddMonoid.toAddZeroClass.{u_6} N (AddCommMonoid.toAddMonoid.{u_6} N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.74)))) (DistribSMul.toSMulZeroClass.{u_1, u_6} R N (AddMonoid.toAddZeroClass.{u_6} N (AddCommMonoid.toAddMonoid.{u_6} N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.74)) 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IsLocalizedModule.{u_5, u_2, u_6} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.21 M N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.74 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.81 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.85 (Algebra.algebraMapSubmonoid.{u_1, u_5} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.3 A (CommSemiring.toSemiring.{u_5} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.21) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2138442654._hygCtx._hyg.24 S) f","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {M : Type u_2} [inst_1 : AddCommMonoid M] {A : Type u_5}\n  [inst_2 : CommSemiring A] [inst_3 : Algebra R A] [inst_4 : Module R M] (S : Submonoid R) {N : Type u_6}\n  [inst_5 : AddCommMonoid N] [inst_6 : Module R N] [inst_7 : Module A M] [inst_8 : Module A N]\n  [inst_9 : IsScalarTower R A M] [inst_10 : IsScalarTower R A N] (f : M →ₗ[A] N) [IsLocalizedModule S (↑R f)],\n  IsLocalizedModule (Algebra.algebraMapSubmonoid A S) f","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {M : Type u_2} [inst_1 : AddCommMonoid M] {A : Type u_5}\n  [inst_2 : CommSemiring A] [inst_3 : Algebra R A] [inst_4 : Module R M] (S : Submonoid R) {N : Type u_6}\n  [inst_5 : AddCommMonoid N] [inst_6 : Module R N] [inst_7 : Module A M] [inst_8 : Module A N]\n  [inst_9 : IsScalarTower R A M] [inst_10 : IsScalarTower R A N] (f : M →ₗ[A] N) [IsLocalizedModule S (↑R f)],\n  IsLocalizedModule (Algebra.algebraMapSubmonoid A S) 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(CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3)))))) S x)) m3 s3))))) u2 hu2)) -> (motive x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx.70.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.147)","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) (M : Type u_2) [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] (m2 : M) (s2 : ↥S) (m3 : M) (s3 : ↥S) (motive : LocalizedModule.r S M (m2, s2) (m3, s3) → Prop)\n  (x : LocalizedModule.r S M (m2, s2) (m3, s3)),\n  (∀ (u2 : ↥S) (hu2 : u2 • (m3, s3).2 • (m2, s2).1 = u2 • (m2, s2).2 • (m3, s3).1), motive ⋯) → motive x","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) (M : Type u_2) [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] (m2 : M) (s2 : ↥S) (m3 : M) (s3 : ↥S) (motive : LocalizedModule.r S M (m2, s2) (m3, s3) → Prop)\n  (x : LocalizedModule.r S M (m2, s2) (m3, s3)),\n  (∀ (u2 : ↥S) (hu2 : u2 • (m3, s3).2 • (m2, s2).1 = u2 • (m2, s2).2 • (m3, s3).1), motive ⋯) → motive x","typeReferences":[["Subtype"],["Module"],["Prod","mk"],["Membership","mem"],["CommSemiring"],["Submonoid","smul"],["Exists","intro"],["MulZeroOneClass","toMulOneClass"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Prod","fst"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["Prod","snd"],["AddZeroClass","toAddZero"],["Submonoid"],["Module","toDistribMulAction"],["HSMul","hSMul"],["LocalizedModule","r"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Subtype"],["Prod","mk"],["Membership","mem"],["Submonoid","smul"],["MulZeroOneClass","toMulOneClass"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Prod","fst"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["Prod","snd"],["AddZeroClass","toAddZero"],["Submonoid"],["Exists","casesOn"],["Module","toDistribMulAction"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"definition","name":["LocalizedModule"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1673475284._hygCtx._hyg.3 : CommSemiring.{u} R], (Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1673475284._hygCtx._hyg.3))))) -> (forall (M : Type.{v}) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1673475284._hygCtx._hyg.9 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1673475284._hygCtx._hyg.12 : Module.{u, v} R M (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1673475284._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1673475284._hygCtx._hyg.9], Type.{max u v})","typeFull":"{R : Type u} →\n  [inst : CommSemiring R] → Submonoid R → (M : Type v) → [inst_1 : AddCommMonoid M] → [Module R M] → Type (max u v)","typeReadable":"{R : Type u} →\n  [inst : CommSemiring R] → Submonoid R → (M : Type v) → [inst_1 : AddCommMonoid M] → [Module R M] → Type (max u v)","typeReferences":[["Submonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["Module"],["CommSemiring","toSemiring"],["CommSemiring"],["MulZeroOneClass","toMulOneClass"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["Module","toDistribMulAction"],["OreLocalization"],["OreLocalization","oreSetComm"],["CommSemiring","toSemiring"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"],["CommSemiring","toCommMonoid"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","linearMap_ext"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S : 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.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.3))))) {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.40 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.14] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.40) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.71 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.40 S f] {N : Type.{u_6}} {N' : Type.{u_7}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.77 : AddCommMonoid.{u_6} N] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.80 : Module.{u_1, u_6} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.77] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.84 : AddCommMonoid.{u_7} N'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.87 : Module.{u_1, u_7} R N' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.84] (f' : LinearMap.{u_1, u_1, u_6, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.3))) N N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.84 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.87) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.100 : IsLocalizedModule.{u_1, u_6, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.3 N N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.84 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.87 S f'] {{g : LinearMap.{u_1, u_1, u_3, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.3))) M' N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.84 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.87}} {{g' : LinearMap.{u_1, u_1, u_3, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.3))) M' N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.84 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.87}}, (Eq.{max (succ u_2) (succ u_7)} (LinearMap.{u_1, u_1, u_2, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.3))) M N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.84 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.87) (LinearMap.comp.{u_1, u_1, u_1, u_2, u_3, 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.84 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.87 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1257460159._hygCtx._hyg.3))) (RingHom.id.{u_1} R 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[IsLocalizedModule S f'] ⦃g g' : M' →ₗ[R] N'⦄, g ∘ₗ f = g' ∘ₗ f → g = g'","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M') [IsLocalizedModule S f]\n  {N : Type u_6} {N' : Type u_7} [inst_6 : AddCommMonoid N] [inst_7 : Module R N] [inst_8 : AddCommMonoid N']\n  [inst_9 : Module R N'] (f' : N →ₗ[R] N') [IsLocalizedModule S f'] ⦃g g' : M' →ₗ[R] N'⦄, g ∘ₗ f = g' ∘ₗ f → g = g'","typeReferences":[["IsLocalizedModule"],["RingHomCompTriple","ids"],["Module"],["CommSemiring"],["CommSemiring","toSemiring"],["LinearMap","comp"],["MulZeroOneClass","toMulOneClass"],["LinearMap"],["Submonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["rfl"],["IsLocalizedModule","map_units"],["Semiring","toNonAssocSemiring"],["RingHomCompTriple","ids"],["RingHom","id"],["IsLocalizedModule","is_universal"],["CommSemiring","toSemiring"],["LinearMap","comp"],["ExistsUnique","unique"],["Eq"],["LinearMap"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","instSemiring","_proof_2"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 : CommSemiring.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 : Semiring.{u_2} A] 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.36)) (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_2} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.11 (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.36)) 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3)))) (IsLocalizedModule.iso.{u_1, u_2, max u_1 u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3 S M (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.36) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.11 (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_2} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.11 (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.36)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.36 (OreLocalization.instModuleOfIsScalarTower.{u_1, u_2, u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.36 R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.36 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.11) 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.36)) (LinearEquiv.refl.{u_1, max u_2 u_1} R (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.36) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3) (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_2} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.11 (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.36)) (OreLocalization.instModuleOfIsScalarTower.{u_1, u_2, u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.36 R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.36 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.11) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.36))) (IsScalarTower.right.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.257542238._hygCtx._hyg.3))))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) (M : Type u_2) [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M],\n  IsLocalizedModule.iso S (LocalizedModule.mkLinearMap S M) = LinearEquiv.refl R (LocalizedModule S M)","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) (M : Type u_2) [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M],\n  IsLocalizedModule.iso S (LocalizedModule.mkLinearMap S M) = LinearEquiv.refl R (LocalizedModule S M)","typeReferences":[["Module"],["OreLocalization","oreSetComm"],["CommSemiring"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["LinearEquiv"],["Algebra","id"],["LocalizedModule","mkLinearMap"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["IsScalarTower","right"],["Eq"],["Semiring","toModule"],["NonAssocSemiring","toMulZeroOneClass"],["LocalizedModule"],["OreLocalization","instAddCommMonoidOreLocalization"],["IsScalarTower","left"],["CommSemiring","toSemiring"],["OreLocalization","instModuleOfIsScalarTower"],["Semiring","toMonoidWithZero"],["localizedModuleIsLocalizedModule"],["CommSemiring","toCommMonoid"],["Submonoid"],["Module","toDistribMulAction"],["IsLocalizedModule","iso"],["RingHomInvPair","ids"],["LinearEquiv","refl"]],"valueReferences":[["OreLocalization","oreSetComm"],["RingHomCompTriple","ids"],["Iff","mp"],["LinearMap","comp"],["AddCommMonoid","toAddMonoid"],["LinearEquiv"],["Algebra","id"],["congrArg"],["LocalizedModule","mkLinearMap"],["IsLocalizedModule","map_units"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["IsLocalizedModule","is_universal"],["MonoidWithZero","toMonoid"],["Eq","symm"],["DistribMulAction","toMulAction"],["IsScalarTower","right"],["Semiring","toModule"],["Eq"],["propext"],["OreLocalization","instAddCommMonoidOreLocalization"],["LocalizedModule"],["IsScalarTower","left"],["CommSemiring","toSemiring"],["OreLocalization","instModuleOfIsScalarTower"],["And"],["Semiring","toMonoidWithZero"],["LinearEquiv","toLinearMap"],["localizedModuleIsLocalizedModule"],["LinearMap"],["LinearEquiv","symm"],["LinearEquiv","toLinearMap_inj"],["CommSemiring","toCommMonoid"],["Exists","casesOn"],["eq_self"],["Module","toDistribMulAction"],["of_eq_true"],["IsLocalizedModule","iso"],["LinearEquiv","eq_toLinearMap_symm_comp"],["IsLocalizedModule","iso_symm_comp"],["id"],["RingHomInvPair","ids"],["Eq","mpr"],["And","casesOn"],["LinearEquiv","refl"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","fromLocalizedModule","bij"],"typeFallback":"forall 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) [self : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109 S f] (x : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))))) (SetLike.instMembership.{u_1, u_1} (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)))))) S x) x))","typeFull":"∀ {R : Type u_1} {inst : CommSemiring R} {M : Type u_2} {M' : Type u_3} {inst_1 : AddCommMonoid M}\n  {inst_2 : AddCommMonoid M'} {inst_3 : Module R M} {inst_4 : Module R M'} {S : Submonoid R} (f : M →ₗ[R] M')\n  [self : IsLocalizedModule S f] (x : ↥S), IsUnit ((algebraMap R (Module.End R M')) ↑x)","typeReadable":"∀ {R : Type u_1} {inst : CommSemiring R} {M : Type u_2} {M' : Type u_3} {inst_1 : AddCommMonoid M}\n  {inst_2 : AddCommMonoid M'} {inst_3 : Module R M} {inst_4 : Module R M'} {S : Submonoid R} (f : M →ₗ[R] M')\n  [self : IsLocalizedModule S f] (x : ↥S), IsUnit ((algebraMap R (Module.End R M')) ↑x)","typeReferences":[["IsLocalizedModule"],["RingHom"],["smulCommClass_self"],["Subtype"],["CommMonoid","toMonoid"],["Module"],["Membership","mem"],["CommSemiring"],["RingHom","instFunLike"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["Subtype","val"],["Algebra","id"],["DFunLike","coe"],["AddCommMonoid"],["Module","End","instAlgebra"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["Algebra","toSMul"],["DistribMulAction","toMulAction"],["NonAssocSemiring","toMulZeroOneClass"],["Module","End"],["SetLike","instMembership"],["Module","End","instSemiring"],["IsScalarTower","left"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["IsUnit"],["LinearMap"],["CommSemiring","toCommMonoid"],["Submonoid"],["Module","toDistribMulAction"],["Module","End","instMonoid"],["algebraMap"]],"valueReferences":[]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","instMonoid","_proof_6"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 : CommSemiring.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 : Semiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29] {S : 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.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3))))} (n : Nat) (x : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (MulOneClass.toMulOne.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (Monoid.toMulOneClass.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (NonUnitalNonAssocSemiring.toMul.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 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(OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3) S) A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Module.toDistribMulAction.{u_1, u_2} R A (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 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(Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (Monoid.toMulOneClass.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (LocalizedModule.instMonoid.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 S)))) (LocalizedModule.instSemiring._proof_8.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 S) (LocalizedModule.instSemiring._proof_9.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 S) (LocalizedModule.instSemiring._proof_10.{u_1, 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (instHMul.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 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(NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_2} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3) S) A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (MulOneClass.toMulOne.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (Monoid.toMulOneClass.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (LocalizedModule.instMonoid.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 A 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 S) (LocalizedModule.instSemiring._proof_11.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 S)))) b c))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A] {S : Submonoid R}\n  (a b c : LocalizedModule S A), a * b * c = a * (b * c)","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A] {S : Submonoid R}\n  (a b c : LocalizedModule S A), a * b * c = a * (b * c)","typeReferences":[["MulOneClass","toMulOne"],["LocalizedModule","instSemiring","_proof_9"],["NonUnitalNonAssocSemiring","toMul"],["NonUnitalNonAssocSemiring","mk"],["OreLocalization","oreSetComm"],["CommSemiring"],["LocalizedModule","instSemiring","_proof_8"],["HMul","hMul"],["LocalizedModule","instSemiring","_proof_11"],["MulZeroOneClass","toMulOneClass"],["MulOne","toMul"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["LocalizedModule","instMonoid"],["LocalizedModule"],["OreLocalization","instAddCommMonoidOreLocalization"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["CommSemiring","toCommMonoid"],["Algebra"],["Submonoid"],["Module","toDistribMulAction"],["LocalizedModule","instSemiring","_proof_10"],["instHMul"],["Semiring"]],"valueReferences":[["LocalizedModule","instMonoid"],["Semiring","toNonAssocSemiring"],["LocalizedModule"],["Algebra","toModule"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Semigroup","mul_assoc"],["Monoid"],["inferInstance"],["Monoid","toSemigroup"]]},{"isProp":true,"kind":"theorem","name":["isLocalizedModule_iff"],"typeFallback":"forall 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (DFunLike.coe.{max (succ u_1) (succ u_3), succ u_1, succ u_3} (RingHom.{u_1, u_3} R (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)) (Semiring.toNonAssocSemiring.{u_3} (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 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R M] →\n                [inst_4 : Module R M'] → (f : M →ₗ[R] M') → [IsLocalizedModule S f] → LocalizedModule S M →ₗ[R] 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{R : Type.{u}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.3 : CommSemiring.{u} R] {S : Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.3))))} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.9 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.12 : Module.{u, v} R M (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.9] (T : Type.{u_1}) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.17 : CommSemiring.{u_1} T] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.20 : Algebra.{u, u_1} R T 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.17) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.20) (LocalizedModule.smulOfIsLocalization.{u, v, u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.12 T inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.24) (OreLocalization.instSMulOfIsScalarTower.{u, u, v} R R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.3) S) (DistribMulAction.toMulAction.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.9) (Module.toDistribMulAction.{u, v} R M (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.12)) (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.9)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.9)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.9) (Module.toDistribMulAction.{u, v} R M (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.12)))) (Algebra.toSMul.{u, u} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.3 (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.772748180._hygCtx._hyg.3) 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AddCommMonoid M'] →\n                      [inst_6 : Module R M'] →\n                        [IsLocalization S A] → (f : M →ₗ[R] M') → [IsLocalizedModule S f] → Module A M'","typeReadable":"{R : Type u_6} →\n  {A : Type u_7} →\n    {M : Type u_8} →\n      {M' : Type u_9} →\n        [inst : CommSemiring R] →\n          [inst_1 : CommSemiring A] →\n            [inst_2 : Algebra R A] →\n              (S : Submonoid R) →\n                [inst_3 : AddCommMonoid M] →\n                  [inst_4 : Module R M] →\n                    [inst_5 : AddCommMonoid M'] →\n                      [inst_6 : Module R M'] →\n                        [IsLocalization S A] → (f : M →ₗ[R] M') → [IsLocalizedModule S f] → Module A 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(DistribMulAction.toMulAction.{u_1, u_2} R M'' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_2} M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.17) (Module.toDistribMulAction.{u_1, u_2} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.44)))) (SemigroupAction.toSMul.{u_1, u_2} R M'' (Monoid.toSemigroup.{u_1} R (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3))) (MulAction.toSemigroupAction.{u_1, u_2} R M'' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3)) (DistribMulAction.toMulAction.{u_1, u_2} R M'' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_2} M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.17) (Module.toDistribMulAction.{u_1, u_2} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.44))))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {M'' : Type u_2} [inst_1 : AddCommMonoid M''] [inst_2 : Module R M''],\n  SMulCommClass R R M''","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {M'' : Type u_2} [inst_1 : AddCommMonoid M''] [inst_2 : Module R M''],\n  SMulCommClass R R M''","typeReferences":[["CommMonoid","toMonoid"],["Module"],["CommSemiring"],["CommSemiring","toSemiring"],["SemigroupAction","toSMul"],["AddCommMonoid","toAddMonoid"],["CommSemiring","toCommMonoid"],["SMulCommClass"],["AddCommMonoid"],["Module","toDistribMulAction"],["DistribMulAction","toMulAction"],["Monoid","toSemigroup"],["MulAction","toSemigroupAction"]],"valueReferences":[["smulCommClass_self"],["Module","toDistribMulAction"],["CommMonoid","toMonoid"],["CommSemiring","toSemiring"],["DistribMulAction","toMulAction"],["AddCommMonoid","toAddMonoid"],["CommSemiring","toCommMonoid"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","lift'_smul"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S : 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.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3))))) {M : Type.{u_2}} {M'' : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.17 : AddCommMonoid.{u_4} M''] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.44 : Module.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.17] (g : LinearMap.{u_1, u_1, u_2, 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.44 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.44 (smulCommClass_self.{u_1, u_4} R M'' (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_4} R M'' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_4} M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.17) (Module.toDistribMulAction.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.44))) (Algebra.toSMul.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_4} R M'' (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_4} R M'' (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_4} M'' 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(DistribSMul.toSMulZeroClass.{u_1, u_4} R M'' (AddMonoid.toAddZeroClass.{u_4} M'' (AddCommMonoid.toAddMonoid.{u_4} M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.17)) (DistribMulAction.toDistribSMul.{u_1, u_4} R M'' (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_4} M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.17) (Module.toDistribMulAction.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.44))))) r (LocalizedModule.lift'.{u_1, u_2, u_4} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3 S M M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.44 g h m)) (LocalizedModule.lift'.{u_1, u_2, u_4} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3 S M M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.44 g h (HSMul.hSMul.{u_1, max u_1 u_2, max u_1 u_2} R (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3 S M 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(OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3) S) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.11) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.36)) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.11) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.36)))) (Algebra.toSMul.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.11) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.36))) (IsScalarTower.right.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3401544047._hygCtx._hyg.3)))) r m))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M'' : Type u_4} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M''] [inst_3 : Module R M] [inst_4 : Module R M''] (g : M →ₗ[R] M'')\n  (h : ∀ (x : ↥S), IsUnit ((algebraMap R (Module.End R M'')) ↑x)) (r : R) (m : LocalizedModule S M),\n  r • LocalizedModule.lift' S g h m = LocalizedModule.lift' S g h (r • m)","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M'' : Type u_4} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M''] [inst_3 : Module R M] [inst_4 : Module R M''] (g : M →ₗ[R] M'')\n  (h : ∀ (x : ↥S), IsUnit ((algebraMap R (Module.End R M'')) ↑x)) (r : R) (m : LocalizedModule S M),\n  r • LocalizedModule.lift' S g h m = LocalizedModule.lift' S g h (r • m)","typeReferences":[["smulCommClass_self"],["RingHom"],["Membership","mem"],["SMulZeroClass","toSMul"],["Algebra","id"],["Subtype","val"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["DistribMulAction","toMulAction"],["IsScalarTower","right"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["Module","End","instSemiring"],["SetLike","instMembership"],["LocalizedModule"],["Submonoid","instSetLike"],["DistribMulAction","toDistribSMul"],["LinearMap"],["AddZeroClass","toAddZero"],["LocalizedModule","lift'"],["CommSemiring","toCommMonoid"],["OreLocalization","instSMulOfIsScalarTower"],["Submonoid"],["HSMul","hSMul"],["AddZero","toZero"],["algebraMap"],["AddMonoid","toAddZeroClass"],["Subtype"],["CommMonoid","toMonoid"],["Module"],["OreLocalization","oreSetComm"],["CommSemiring"],["RingHom","instFunLike"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["Module","End","instAlgebra"],["Algebra","toSMul"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["Module","End"],["IsScalarTower","left"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["IsUnit"],["Module","toDistribMulAction"],["Module","End","instMonoid"]],"valueReferences":[["IsUnit","unit"],["RingHom"],["smulCommClass_self"],["Membership","mem"],["SMulZeroClass","toSMul"],["LocalizedModule","mk"],["map_smul"],["Algebra","id"],["Subtype","val"],["LinearMap","semilinearMapClass"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Eq","symm"],["DistribMulAction","toMulAction"],["IsScalarTower","right"],["NonAssocSemiring","toMulZeroOneClass"],["Units"],["DistribSMul","toSMulZeroClass"],["LinearMap","map_smul"],["SemilinearMapClass","toMulActionSemiHomClass"],["SetLike","instMembership"],["Module","End","instSemiring"],["LocalizedModule"],["Submonoid","instSetLike"],["DistribMulAction","toDistribSMul"],["LocalizedModule","induction_on"],["LinearMap"],["AddZeroClass","toAddZero"],["LocalizedModule","lift'"],["CommSemiring","toCommMonoid"],["OreLocalization","instSMulOfIsScalarTower"],["Submonoid"],["Units","val"],["Eq","refl"],["HSMul","hSMul"],["id"],["Eq","mpr"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["algebraMap"],["LinearMap","instFunLike"],["CommMonoid","toMonoid"],["OreLocalization","oreSetComm"],["Units","instInv"],["RingHom","instFunLike"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["congrArg"],["Module","End","instAlgebra"],["LocalizedModule","smul'_mk"],["Algebra","toSMul"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["Module","End"],["Inv","inv"],["IsScalarTower","left"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["LocalizedModule","lift'_mk"],["Module","toDistribMulAction"],["Module","End","instMonoid"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","divBy_mul_by"],"typeFallback":"forall 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(CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.12)))) (SemigroupAction.toSMul.{u, u} R R (Monoid.toSemigroup.{u} R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3)))) (MulAction.toSemigroupAction.{u, u} R R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u, u} R R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.12))) (SemigroupAction.toSMul.{u, v} R M (Monoid.toSemigroup.{u} R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3)))) (MulAction.toSemigroupAction.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.9) (Module.toDistribMulAction.{u, v} R M (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.12)))) (SemigroupAction.toSMul.{u, u} R R (Monoid.toSemigroup.{u} R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3)))) (MulAction.toSemigroupAction.{u, u} R R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u, u} R R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3 (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3) (Algebra.id.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3))) (Algebra.toSMul.{u, u} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3 (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3) (Algebra.id.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3)) (OreLocalization.instIsScalarTower.{u, u, u, v} R R R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3) S) (DistribMulAction.toMulAction.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.9) (Module.toDistribMulAction.{u, v} R M (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.12)) (SemigroupAction.toSMul.{u, v} R M (Monoid.toSemigroup.{u} R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3)))) (MulAction.toSemigroupAction.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R 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(MulAction.toSemigroupAction.{u, u} R R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u, u} R R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3))))) (Module.toDistribMulAction.{u, u} R R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3)))) (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3)))))) (IsScalarTower.right.{u, u} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3 (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3) (Algebra.id.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3)) (IsScalarTower.left.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.9) (Module.toDistribMulAction.{u, v} R M (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.12))) (SemigroupAction.toSMul.{u, v} R M (Monoid.toSemigroup.{u} R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3)))) (MulAction.toSemigroupAction.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.9) (Module.toDistribMulAction.{u, v} R M (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.12)))) (SemigroupAction.toSMul.{u, u} R R (Monoid.toSemigroup.{u} R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3)))) (MulAction.toSemigroupAction.{u, u} R R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R 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Membership.mem.{u, u} R (Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3))))) (SetLike.instMembership.{u, u} (Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3))))) R (Submonoid.instSetLike.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3434122011._hygCtx._hyg.3)))))) S x) s)) p)) p","typeFull":"∀ {R : Type u} [inst : CommSemiring R] {S : Submonoid R} {M : Type v} [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  (s : ↥S) (p : LocalizedModule S M),\n  (LocalizedModule.divBy s) (((algebraMap R (Module.End R (LocalizedModule S M))) ↑s) p) = p","typeReadable":"∀ {R : Type u} [inst : CommSemiring R] {S : Submonoid R} {M : Type v} [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  (s : ↥S) (p : LocalizedModule S M),\n  (LocalizedModule.divBy s) (((algebraMap R (Module.End R (LocalizedModule S M))) ↑s) p) = 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 : CommSemiring.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 : Semiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29] {S : 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.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3))))}, Eq.{succ (max u_1 u_2)} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1925582909._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1925582909._hygCtx._hyg.24 x)) s))","typeFull":"∀ {R : Type u} [inst : CommSemiring R] {S : Submonoid R} {M : Type v} [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  (T : Type u_1) [inst_3 : CommSemiring T] [inst_4 : Algebra R T] [inst_5 : IsLocalization S T] (x : T) (m : M)\n  (s : ↥S),\n  x • LocalizedModule.mk m s = LocalizedModule.mk ((IsLocalization.sec S x).1 • m) ((IsLocalization.sec S x).2 * s)","typeReadable":"∀ {R : Type u} [inst : CommSemiring R] {S : Submonoid R} {M : Type v} [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  (T : Type u_1) [inst_3 : CommSemiring T] [inst_4 : Algebra R T] [inst_5 : IsLocalization S T] (x : T) (m : M)\n  (s : ↥S),\n  x • LocalizedModule.mk m s = LocalizedModule.mk ((IsLocalization.sec S x).1 • m) ((IsLocalization.sec S x).2 * s)","typeReferences":[["Subtype"],["Module"],["Membership","mem"],["CommSemiring"],["HMul","hMul"],["MulZeroOneClass","toMulOneClass"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["LocalizedModule","mk"],["Prod","fst"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["LocalizedModule","smulOfIsLocalization"],["SetLike","instMembership"],["LocalizedModule"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["Prod","snd"],["AddZeroClass","toAddZero"],["IsLocalization"],["Algebra"],["Submonoid"],["Module","toDistribMulAction"],["Submonoid","mul"],["HSMul","hSMul"],["instHMul"],["AddZero","toZero"],["IsLocalization","sec"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["rfl"],["LocalizedModule"],["HSMul","hSMul"],["instHSMul"],["LocalizedModule","mk"],["LocalizedModule","smulOfIsLocalization"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","instSemiring","_proof_4"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 : CommSemiring.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 : Semiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29] {S : 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.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3))))} (a : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3389458912._hygCtx._hyg.3)))) (LocalizedModule.lift.{u_1, u_2, u_4} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3389458912._hygCtx._hyg.3 S M M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3389458912._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3389458912._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3389458912._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3389458912._hygCtx._hyg.44 g h) (LocalizedModule.mkLinearMap.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3389458912._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3389458912._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3389458912._hygCtx._hyg.36)) g","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M'' : Type u_4} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M''] [inst_3 : 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S : 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.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.3))))) {M : Type.{u_2}} {M' : Type.{u_3}} {M'' : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.17 : AddCommMonoid.{u_4} M''] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.40 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.14] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.44 : Module.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.17] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.40) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.71 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.40 S f] (g : LinearMap.{u_1, u_1, u_2, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.3))) M M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.361390238._hygCtx._hyg.44) (h : forall (x : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R 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inst._@.Mathlib.Algebra.GroupWithZero.Action.Defs.3291880512._hygCtx._hyg.13))) b₁ b₂))","typeFull":"∀ {M : Type u_1} {A : Type u_7} [inst : AddZeroClass A] [inst_1 : DistribSMul M A] (a : M) (b₁ b₂ : A),\n  a • b₁ + a • b₂ = a • (b₁ + b₂)","typeReadable":"∀ {M : Type u_1} {A : Type u_7} [inst : AddZeroClass A] [inst_1 : DistribSMul M A] (a : M) (b₁ b₂ : A),\n  a • b₁ + a • b₂ = a • (b₁ + b₂)","typeReferences":[["HAdd","hAdd"],["AddZeroClass"],["instHAdd"],["HSMul","hSMul"],["instHSMul"],["SMulZeroClass","toSMul"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["Eq"],["AddZero","toZero"],["DistribSMul"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["HAdd","hAdd"],["instHAdd"],["HSMul","hSMul"],["Eq","symm"],["smul_add"],["instHSMul"],["SMulZeroClass","toSMul"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"]]},{"isProp":false,"kind":"definition","name":["LocalizedModule","mul"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.3 : CommSemiring.{u} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.29 : Semiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.32 : Algebra.{u, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.29] {S : Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.3))))}, (LocalizedModule.{u, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.29))) (Algebra.toModule.{u, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.32)) -> (LocalizedModule.{u, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.29))) (Algebra.toModule.{u, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.32)) -> (LocalizedModule.{u, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.29))) (Algebra.toModule.{u, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.32))","typeFull":"{R : Type u} →\n  [inst : CommSemiring R] →\n    {A : Type u_2} →\n      [inst_1 : Semiring A] →\n        [inst_2 : Algebra R A] → {S : Submonoid R} → LocalizedModule S A → LocalizedModule S A → LocalizedModule S A","typeReadable":"{R : Type u} →\n  [inst : CommSemiring R] →\n    {A : Type u_2} →\n      [inst_1 : Semiring A] →\n        [inst_2 : Algebra R A] → {S : Submonoid R} → LocalizedModule S A → LocalizedModule S A → LocalizedModule S A","typeReferences":[["Submonoid"],["Semiring","toNonAssocSemiring"],["LocalizedModule"],["Algebra","toModule"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["CommSemiring"],["MulZeroOneClass","toMulOneClass"],["NonAssocSemiring","toMulZeroOneClass"],["Algebra"],["Semiring"]],"valueReferences":[["LocalizedModule","liftOn₂"],["LocalizedModule","mul","_proof_1"],["SetLike","instMembership"],["Subtype"],["LocalizedModule"],["NonUnitalNonAssocSemiring","toDistrib"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["Membership","mem"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Distrib","toMul"],["HMul","hMul"],["MulZeroOneClass","toMulOneClass"],["Prod","snd"],["LocalizedModule","mk"],["Prod","fst"],["Submonoid"],["Semiring","toNonAssocSemiring"],["Submonoid","mul"],["Algebra","toModule"],["instHMul"],["NonAssocSemiring","toMulZeroOneClass"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","instSemiring","_proof_14"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 : CommSemiring.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 : Semiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29] {S : 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.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3))))} (a : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)), Eq.{succ (max u_1 u_2)} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_2} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3) S) A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Module.toDistribMulAction.{u_1, u_2} R A (CommSemiring.toSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 S) (LocalizedModule.instSemiring._proof_9.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 S) (LocalizedModule.instSemiring._proof_10.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 S) (LocalizedModule.instSemiring._proof_11.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 : CommSemiring.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 : Semiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29] {S : 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.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3))))}, autoParam.{0} (forall (x : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A 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(LocalizedModule.{u, v} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.12) (LocalizedModule.{u, v} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.12) (instHSMul.{u, max u v} R (LocalizedModule.{u, v} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.12) (OreLocalization.instSMulOfIsScalarTower.{u, u, v} R R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.3 (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.3) (Algebra.id.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.3)) (IsScalarTower.left.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.9) (Module.toDistribMulAction.{u, v} R M (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.12))) (IsScalarTower.right.{u, u} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.3 (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.3) (Algebra.id.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.3)))) r (LocalizedModule.mk.{u, v} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.12 m s)) (LocalizedModule.mk.{u, v} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.12 (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.9)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.9)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.9) (Module.toDistribMulAction.{u, v} R M (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2636407509._hygCtx._hyg.12))))) r m) s)","typeFull":"∀ {R : Type u} [inst : CommSemiring R] {S : Submonoid R} {M : Type v} [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  (r : R) (s : ↥S) (m : M), r • LocalizedModule.mk m s = LocalizedModule.mk (r • m) s","typeReadable":"∀ {R : Type u} [inst : CommSemiring R] {S : Submonoid R} {M : Type v} [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  (r : R) (s : ↥S) (m : M), r • LocalizedModule.mk m s = LocalizedModule.mk (r • m) s","typeReferences":[["Subtype"],["Module"],["OreLocalization","oreSetComm"],["Membership","mem"],["CommSemiring"],["MulZeroOneClass","toMulOneClass"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["LocalizedModule","mk"],["Algebra","id"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["Algebra","toSMul"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["IsScalarTower","right"],["instHSMul"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["IsScalarTower","left"],["LocalizedModule"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["OreLocalization","instSMulOfIsScalarTower"],["CommSemiring","toCommMonoid"],["Submonoid"],["Module","toDistribMulAction"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Eq","trans"],["Membership","mem"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["LocalizedModule","mk"],["Algebra","id"],["Semiring","toNonAssocSemiring"],["OreLocalization"],["DistribMulAction","toMulAction"],["IsScalarTower","right"],["Monoid","toSemigroup"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["OreLocalization","oreDenom"],["SetLike","instMembership"],["MulOne","toOne"],["LocalizedModule"],["Submonoid","instSetLike"],["DistribMulAction","toDistribSMul"],["AddZeroClass","toAddZero"],["CommSemiring","toCommMonoid"],["OreLocalization","instSMulOfIsScalarTower"],["Submonoid"],["OreLocalization","smul_oreDiv"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["MulOneClass","toMulOne"],["Subtype"],["OreLocalization","oreSetComm"],["OreLocalization","oreNum"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["congrArg"],["congr"],["Algebra","toSMul"],["MonoidWithZero","toMonoid"],["Monoid","toMulOneClass"],["instHSMul"],["congrFun'"],["Eq"],["IsScalarTower","left"],["True"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["mul_one"],["OfNat","ofNat"],["eq_self"],["Module","toDistribMulAction"],["One","toOfNat1"],["of_eq_true"],["OreLocalization","oreDiv"],["MulAction","toSemigroupAction"]]},{"isProp":true,"kind":"theorem","name":["isLocalizedModule_id"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S : 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.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.3))))) (M : Type.{u_2}) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.11] (R' : Type.{u_6}) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.72 : CommSemiring.{u_6} R'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.75 : Algebra.{u_1, u_6} R R' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_6} R' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.72)] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.79 : IsLocalization.{u_1, u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.3 S R' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.72 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.75] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.83 : Module.{u_6, u_2} R' M (CommSemiring.toSemiring.{u_6} R' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.87 : IsScalarTower.{u_1, u_6, u_2} R R' M (Algebra.toSMul.{u_1, u_6} R R' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_6} R' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.75) (SMulZeroClass.toSMul.{u_6, u_2} R' M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_6, u_2} R' M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_6, u_2} R' M (MonoidWithZero.toMonoid.{u_6} R' (Semiring.toMonoidWithZero.{u_6} R' (CommSemiring.toSemiring.{u_6} R' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.72))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.11) (Module.toDistribMulAction.{u_6, u_2} R' M (CommSemiring.toSemiring.{u_6} R' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.83)))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.11) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.36))))], IsLocalizedModule.{u_1, u_2, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.3 M M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.36 S (LinearMap.id.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1085218831._hygCtx._hyg.36)","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) (M : Type u_2) [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] (R' : Type u_6) [inst_3 : CommSemiring R'] [inst_4 : Algebra R R'] [IsLocalization S R']\n  [inst_6 : Module R' M] [IsScalarTower R R' M], IsLocalizedModule S LinearMap.id","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) (M : Type u_2) [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] (R' : Type u_6) [inst_3 : CommSemiring R'] [inst_4 : Algebra R R'] [IsLocalization S R']\n  [inst_6 : Module R' M] [IsScalarTower R R' M], IsLocalizedModule S 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-↑(n + 1)","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Ring A] [inst_2 : Algebra R A] {S : Submonoid R}\n  (n : ℕ), (Int.negSucc n).castDef = -↑(n + 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{α : Type.{u_9}} {β : Type.{u_10}} {inst._@.Mathlib.Algebra.Group.Action.Defs.3323368930._hygCtx._hyg.20 : Semigroup.{u_9} α} [self : SemigroupAction.{u_9, u_10} α β inst._@.Mathlib.Algebra.Group.Action.Defs.3323368930._hygCtx._hyg.20] (x : α) (y : α) (b : β), Eq.{succ u_10} β (HSMul.hSMul.{u_9, u_10, u_10} α β β (instHSMul.{u_9, u_10} α β (SemigroupAction.toSMul.{u_9, u_10} α β inst._@.Mathlib.Algebra.Group.Action.Defs.3323368930._hygCtx._hyg.20 self)) x (HSMul.hSMul.{u_9, u_10, u_10} α β β (instHSMul.{u_9, u_10} α β (SemigroupAction.toSMul.{u_9, u_10} α β inst._@.Mathlib.Algebra.Group.Action.Defs.3323368930._hygCtx._hyg.20 self)) y b)) (HSMul.hSMul.{u_9, u_10, u_10} α β β (instHSMul.{u_9, u_10} α β (SemigroupAction.toSMul.{u_9, u_10} α β inst._@.Mathlib.Algebra.Group.Action.Defs.3323368930._hygCtx._hyg.20 self)) (HMul.hMul.{u_9, u_9, u_9} α α α (instHMul.{u_9} α (Semigroup.toMul.{u_9} α inst._@.Mathlib.Algebra.Group.Action.Defs.3323368930._hygCtx._hyg.20)) x y) b)","typeFull":"∀ {α : 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(CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))))) {M : Type.{u_4}} {M' : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 : AddCommMonoid.{u_4} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 : AddCommMonoid.{u_1} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 : Module.{u_3, u_4} R M (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 : Module.{u_3, u_1} R M' (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14] (f : LinearMap.{u_3, u_3, u_4, u_1} 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 S f] {N : Type.{u_5}} {N' : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 : AddCommMonoid.{u_5} N] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 : AddCommMonoid.{u_2} N'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 : Module.{u_3, u_5} R N (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87 : Module.{u_3, u_2} R N' 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3)))) (LinearMap.module.{u_3, u_3, u_3, u_1, u_2} R R R M' N' (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) r) (IsLocalizedModule.lift.{u_3, u_4, u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3 S M M' N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.71 (LinearMap.comp.{u_3, u_3, u_3, u_4, u_5, u_2} R R R M N N' (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R (CommSemiring.toSemiring.{u_3} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87 S g inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.100)))","typeFull":"∀ {R : Type u_3} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_4} {M' : Type u_1} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] {N : Type u_5} {N' : Type u_2} [inst_6 : AddCommMonoid N] [inst_7 : AddCommMonoid N']\n  [inst_8 : Module R N] [inst_9 : Module R N'] (g : N →ₗ[R] N') [inst_10 : IsLocalizedModule S g] (r : R)\n  (h : M →ₗ[R] N),\n  IsLocalizedModule.lift S f (g ∘ₗ (r • h)) ⋯ = (RingHom.id R) r • IsLocalizedModule.lift S f (g ∘ₗ h) ⋯","typeReadable":"∀ {R : Type u_3} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_4} {M' : Type u_1} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] {N : Type u_5} {N' : Type u_2} [inst_6 : AddCommMonoid N] [inst_7 : AddCommMonoid N']\n  [inst_8 : Module R N] [inst_9 : Module R N'] (g : N →ₗ[R] N') [inst_10 : IsLocalizedModule S g] (r : R)\n  (h : M →ₗ[R] N),\n  IsLocalizedModule.lift S f (g ∘ₗ (r • h)) ⋯ = (RingHom.id R) r • IsLocalizedModule.lift S f (g ∘ₗ h) ⋯","typeReferences":[["RingHom"],["smulCommClass_self"],["IsLocalizedModule"],["CommMonoid","toMonoid"],["Module"],["RingHomCompTriple","ids"],["CommSemiring"],["RingHom","instFunLike"],["LinearMap","comp"],["MulZeroOneClass","toMulOneClass"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["IsLocalizedModule","map_units"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["IsLocalizedModule","lift"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["LinearMap","module"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["LinearMap"],["AddZeroClass","toAddZero"],["CommSemiring","toCommMonoid"],["Submonoid"],["LinearMap","addCommMonoid"],["Module","toDistribMulAction"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["smulCommClass_self"],["RingHom"],["Eq","trans"],["RingHomCompTriple","ids"],["SMulZeroClass","toSMul"],["Algebra","id"],["Semiring","toNonAssocSemiring"],["LinearMap","comp","congr_simp"],["RingHom","id"],["DistribMulAction","toMulAction"],["IsLocalizedModule","lift"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["LinearMap","module"],["DistribMulAction","toDistribSMul"],["AddZeroClass","toAddZero"],["LinearMap"],["CommSemiring","toCommMonoid"],["Submonoid"],["Eq","refl"],["HSMul","hSMul"],["IsLocalizedModule","lift","congr_simp"],["LinearMap","comp_smul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["CommMonoid","toMonoid"],["LinearMap","comp"],["RingHom","instFunLike"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["congrArg"],["IsLocalizedModule","lift_comp"],["IsLocalizedModule","map_units"],["congr"],["MonoidWithZero","toMonoid"],["Algebra","toSMul"],["instHSMul"],["Eq"],["True"],["IsScalarTower","left"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["IsLocalizedModule","ext"],["LinearMap","addCommMonoid"],["eq_self"],["Module","toDistribMulAction"],["of_eq_true"],["LinearMap","IsScalarTower","compatibleSMul"],["LinearMap","instSMul"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","map_surjective"],"typeFallback":"forall 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.40) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.71 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.40 S f] {N : Type.{u_6}} {N' : Type.{u_7}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.77 : AddCommMonoid.{u_6} N] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.80 : AddCommMonoid.{u_7} N'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.83 : Module.{u_1, u_6} R N (CommSemiring.toSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.83) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : LinearMap.{u_1, u_1, u_2, u_6} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3))) M N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.83) => LinearMap.{u_1, u_1, u_3, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3))) M' N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.87) (LinearMap.instFunLike.{u_1, u_1, max u_2 u_6, max u_3 u_7} R R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3))) M' N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.87) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3) (LinearMap.addCommMonoid.{u_1, u_1, u_2, u_6} R R M N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3) 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.87 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3)))) (LinearMap.module.{u_1, u_1, u_1, u_2, u_6} R R R M N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.83 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.83 (smulCommClass_self.{u_1, u_6} R N (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_6} R N (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_6} N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.77) (Module.toDistribMulAction.{u_1, u_6} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.83)))) (LinearMap.module.{u_1, u_1, u_1, u_3, u_7} R R R M' N' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.87 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R 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[inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (m : M') (a : M) (b : ↥S),\n  b • m = f a → (IsLocalizedModule.iso S f).symm m = LocalizedModule.mk a b","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (m : M') (a : M) (b : ↥S),\n  b • m = f a → (IsLocalizedModule.iso S f).symm m = LocalizedModule.mk a 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.379383624._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S : 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.Algebra.Module.LocalizedModule.Basic.379383624._hygCtx._hyg.3))))) {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.379383624._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.379383624._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.379383624._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.379383624._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.379383624._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.379383624._hygCtx._hyg.40 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(MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.379383624._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.379383624._hygCtx._hyg.11) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.379383624._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.379383624._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.379383624._hygCtx._hyg.36))) (IsScalarTower.right.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.379383624._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.379383624._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.379383624._hygCtx._hyg.3)))) r x))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (r : R) (x : LocalizedModule S M),\n  r • IsLocalizedModule.fromLocalizedModule' S f x = IsLocalizedModule.fromLocalizedModule' S f (r • x)","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (r : R) (x : LocalizedModule S M),\n  r • IsLocalizedModule.fromLocalizedModule' S f x = IsLocalizedModule.fromLocalizedModule' S f (r • 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106594._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S : 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.Algebra.Module.LocalizedModule.Basic.342106594._hygCtx._hyg.3))))) {M : Type.{u_2}} {M'' : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106594._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106594._hygCtx._hyg.17 : AddCommMonoid.{u_4} M''] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106594._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106594._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106594._hygCtx._hyg.11] 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106594._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106594._hygCtx._hyg.36)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106594._hygCtx._hyg.17 (OreLocalization.instModuleOfIsScalarTower.{u_1, u_2, u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106594._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106594._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106594._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106594._hygCtx._hyg.36 R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106594._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.342106594._hygCtx._hyg.36 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R 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u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A]\n  {a₁ a₂ : A} {s₁ s₂ : ↥S}, LocalizedModule.mk a₁ s₁ * LocalizedModule.mk a₂ s₂ = LocalizedModule.mk (a₁ * a₂) (s₂ * s₁)","typeReadable":"∀ {R : Type u} [inst : CommSemiring R] {S : Submonoid R} {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A]\n  {a₁ a₂ : A} {s₁ s₂ : ↥S}, LocalizedModule.mk a₁ s₁ * LocalizedModule.mk a₂ s₂ = LocalizedModule.mk (a₁ * a₂) (s₂ * s₁)","typeReferences":[["MulOneClass","toMulOne"],["Subtype"],["Membership","mem"],["CommSemiring"],["HMul","hMul"],["MulZeroOneClass","toMulOneClass"],["LocalizedModule","mk"],["MulOne","toMul"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["Monoid","toMulOneClass"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["LocalizedModule","instMonoid"],["SetLike","instMembership"],["NonUnitalNonAssocSemiring","toDistrib"],["LocalizedModule"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Distrib","toMul"],["Algebra"],["Submonoid"],["Submonoid","mul"],["instHMul"],["Semiring"]],"valueReferences":[["rfl"],["MulOneClass","toMulOne"],["LocalizedModule","instMonoid"],["LocalizedModule"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["HMul","hMul"],["LocalizedModule","mk"],["MulOne","toMul"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["Monoid","toMulOneClass"],["instHMul"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","restrictScalars","congr_simp"],"typeFallback":"forall (R : Type.{u_1}) {S : Type.{u_5}} {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.33 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.37 : Module.{u_1, u_10} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.71 : Module.{u_5, u_8} S M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.75 : Module.{u_5, u_10} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.79 : LinearMap.CompatibleSMul.{u_8, u_10, u_1, u_5} M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27 R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 (SMulZeroClass.toSMul.{u_1, u_8} R M (AddZero.toZero.{u_8} M (AddZeroClass.toAddZero.{u_8} M (AddMonoid.toAddZeroClass.{u_8} M (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21)))) (DistribSMul.toSMulZeroClass.{u_1, u_8} R M (AddMonoid.toAddZeroClass.{u_8} M (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21)) (DistribMulAction.toDistribSMul.{u_1, u_8} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15)) (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21) (Module.toDistribMulAction.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.33)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.71 (SMulZeroClass.toSMul.{u_1, u_10} R M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27)))) (DistribSMul.toSMulZeroClass.{u_1, u_10} R M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27)) (DistribMulAction.toDistribSMul.{u_1, u_10} R M₂ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15)) (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27) (Module.toDistribMulAction.{u_1, u_10} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.37)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.75] (fₗ : LinearMap.{u_5, u_5, u_8, u_10} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 (RingHom.id.{u_5} S (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.71 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.75) (fₗ_1 : LinearMap.{u_5, u_5, u_8, u_10} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 (RingHom.id.{u_5} S (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.71 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.75), (Eq.{max (succ u_8) (succ u_10)} (LinearMap.{u_5, u_5, u_8, u_10} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 (RingHom.id.{u_5} S (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.71 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.75) fₗ fₗ_1) -> (Eq.{max (succ u_8) (succ u_10)} (LinearMap.{u_1, u_1, u_8, u_10} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.37) (LinearMap.restrictScalars.{u_1, u_5, u_8, u_10} R S M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.71 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.75 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.79 fₗ) (LinearMap.restrictScalars.{u_1, u_5, u_8, u_10} R S M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.71 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.75 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.79 fₗ_1))","typeFull":"∀ (R : Type u_1) {S : Type u_5} {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M] [inst_5 : Module R M₂]\n  [inst_6 : Module S M] [inst_7 : Module S M₂] [inst_8 : LinearMap.CompatibleSMul M M₂ R S] (fₗ fₗ_1 : M →ₗ[S] M���),\n  fₗ = fₗ_1 → ↑R fₗ = ↑R fₗ_1","typeReadable":"∀ (R : Type u_1) {S : Type u_5} {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M] [inst_5 : Module R M₂]\n  [inst_6 : Module S M] [inst_7 : Module S M₂] [inst_8 : LinearMap.CompatibleSMul M M₂ R S] (fₗ fₗ_1 : 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1104725466._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1104725466._hygCtx._hyg.10)) (Eq.{succ u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1104725466._hygCtx._hyg.4)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1104725466._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1104725466._hygCtx._hyg.10) (LinearMap.ker.{u_1, u_1, u_2, max u_1 u_2} R R M (LocalizedModule.{u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1104725466._hygCtx._hyg.4) S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1104725466._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1104725466._hygCtx._hyg.10) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R 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(AddCommMonoid.toAddMonoid.{u_2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29)))) (Module.toDistribMulAction.{u_1, u_2} R A (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.32)))) (MulZeroOneClass.toMulOneClass.{max u_1 u_2} (OreLocalization.{u_1, u_2} R 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(Module.toDistribMulAction.{u_1, u_2} R A (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.32)))) (LocalizedModule.instSemiring.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.32 S))))))))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A],\n  LocalizedModule.mk 1 1 = 1","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A],\n  LocalizedModule.mk 1 1 = 1","typeReferences":[["MulOneClass","toMulOne"],["Subtype"],["OreLocalization","oreSetComm"],["Membership","mem"],["CommSemiring"],["MulZeroOneClass","toMulOneClass"],["LocalizedModule","instSemiring"],["AddCommMonoid","toAddMonoid"],["LocalizedModule","mk"],["OreLocalization"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["SetLike","instMembership"],["MulOne","toOne"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Semiring","toMonoidWithZero"],["Submonoid","one"],["CommSemiring","toCommMonoid"],["OfNat","ofNat"],["Algebra"],["Submonoid"],["Module","toDistribMulAction"],["One","toOfNat1"],["Semiring"]],"valueReferences":[["MulOneClass","toMulOne"],["Subtype"],["Eq","trans"],["OreLocalization","oreSetComm"],["Membership","mem"],["MulZeroOneClass","toMulOneClass"],["LocalizedModule","instSemiring"],["AddCommMonoid","toAddMonoid"],["LocalizedModule","mk"],["congrArg"],["Semiring","toNonAssocSemiring"],["OreLocalization"],["Algebra","toModule"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["NonAssocSemiring","toMulZeroOneClass"],["SetLike","instMembership"],["True"],["MulOne","toOne"],["Submonoid","instSetLike"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["OreLocalization","one_def"],["Semiring","toMonoidWithZero"],["Submonoid","one"],["CommSemiring","toCommMonoid"],["OfNat","ofNat"],["Submonoid"],["eq_self"],["Module","toDistribMulAction"],["of_eq_true"],["One","toOfNat1"],["AddMonoidWithOne","toOne"],["OreLocalization","oreDiv"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Module","LocalizedModule","Basic",0,"LocalizedModule","mul_smul_aux","_simp_1_1"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.RingTheory.Localization.Defs.1346696831._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.1346696831._hygCtx._hyg.3))))} (S : Type.{u_2}) [inst._@.Mathlib.RingTheory.Localization.Defs.1346696831._hygCtx._hyg.9 : CommSemiring.{u_2} S] [inst._@.Mathlib.RingTheory.Localization.Defs.1346696831._hygCtx._hyg.12 : Algebra.{u_1, u_2} R S inst._@.Mathlib.RingTheory.Localization.Defs.1346696831._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} S inst._@.Mathlib.RingTheory.Localization.Defs.1346696831._hygCtx._hyg.9)] [inst._@.Mathlib.RingTheory.Localization.Defs.1346696831._hygCtx._hyg.20 : IsLocalization.{u_1, u_2} R inst._@.Mathlib.RingTheory.Localization.Defs.1346696831._hygCtx._hyg.3 M S inst._@.Mathlib.RingTheory.Localization.Defs.1346696831._hygCtx._hyg.9 inst._@.Mathlib.RingTheory.Localization.Defs.1346696831._hygCtx._hyg.12] (x₁ : R) (x₂ : R) (y₁ : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.1346696831._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (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.1346696831._hygCtx._hyg.3))))) R (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 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inst._@.Mathlib.RingTheory.Localization.Defs.1346696831._hygCtx._hyg.3))))) R (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.1346696831._hygCtx._hyg.3)))))) M x)) (instHMul.{u_1} (Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.1346696831._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (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.1346696831._hygCtx._hyg.3))))) R (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.1346696831._hygCtx._hyg.3)))))) M x)) (Submonoid.mul.{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.1346696831._hygCtx._hyg.3)))) M)) y₁ y₂))","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 = IsLocalization.mk' S (x₁ * x₂) (y₁ * y₂)","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 = IsLocalization.mk' S (x₁ * x₂) (y₁ * y₂)","typeReferences":[["Subtype"],["SetLike","instMembership"],["NonUnitalNonAssocSemiring","toDistrib"],["Submonoid","instSetLike"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Distrib","toMul"],["CommSemiring"],["CommSemiring","toSemiring"],["Membership","mem"],["IsLocalization","mk'"],["HMul","hMul"],["MulZeroOneClass","toMulOneClass"],["Algebra"],["IsLocalization"],["Submonoid"],["Semiring","toNonAssocSemiring"],["Submonoid","mul"],["instHMul"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["SetLike","instMembership"],["Subtype"],["NonUnitalNonAssocSemiring","toDistrib"],["Submonoid","instSetLike"],["Distrib","toMul"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["CommSemiring","toSemiring"],["Membership","mem"],["IsLocalization","mk'"],["HMul","hMul"],["MulZeroOneClass","toMulOneClass"],["IsLocalization","mk'_mul"],["Submonoid"],["Semiring","toNonAssocSemiring"],["Submonoid","mul"],["Eq","symm"],["instHMul"],["NonAssocSemiring","toMulZeroOneClass"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","instSemiring","_proof_13"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 : CommSemiring.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 : Semiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29] {S : 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.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3))))} (a : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_2} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3) S) A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Module.toDistribMulAction.{u_1, u_2} R A (CommSemiring.toSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (MulOneClass.toMulOne.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (Monoid.toMulOneClass.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3 : CommSemiring.{u_1} R] {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.11 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.36 : Module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.11] {M₂ : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.80 : AddCommMonoid.{u_2} M₂] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.87 : Module.{u_1, u_2} R M₂ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.80] (S₁ : Submonoid.{u_1} R 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[IsLocalizedModule S₂ f₂] (x : ↥S₁), IsUnit ((algebraMap R (Module.End R M₂)) ↑⟨↑x, 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{R : Type.{u}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2263982328._hygCtx._hyg.3 : CommSemiring.{u} R] {S : Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2263982328._hygCtx._hyg.3))))} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2263982328._hygCtx._hyg.9 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2263982328._hygCtx._hyg.12 : Module.{u, v} R M (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2263982328._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2263982328._hygCtx._hyg.9] {β : (LocalizedModule.{u, v} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2263982328._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2263982328._hygCtx._hyg.9 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2263982328._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2263982328._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2263982328._hygCtx._hyg.12 m s)) -> (forall (x : LocalizedModule.{u, v} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2263982328._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2263982328._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2263982328._hygCtx._hyg.12), β x)","typeFull":"∀ {R : Type u} [inst : CommSemiring R] {S : Submonoid R} {M : Type v} [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  {β : LocalizedModule S M → Prop}, (∀ (m : M) (s : ↥S), β (LocalizedModule.mk m s)) → ∀ (x : LocalizedModule S M), β x","typeReadable":"∀ {R : Type u} [inst : CommSemiring R] {S : Submonoid R} {M : Type v} [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  {β : LocalizedModule S M → Prop}, (∀ (m : M) (s : ↥S), β (LocalizedModule.mk m s)) → ∀ (x : LocalizedModule S M), β x","typeReferences":[["Subtype"],["SetLike","instMembership"],["LocalizedModule"],["Module"],["Submonoid","instSetLike"],["CommSemiring"],["CommSemiring","toSemiring"],["Membership","mem"],["MulZeroOneClass","toMulOneClass"],["LocalizedModule","mk"],["Submonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["Subtype"],["SetLike","instMembership"],["OreLocalization","oreSetComm"],["Submonoid","instSetLike"],["Membership","mem"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"],["Quot","ind"],["CommSemiring","toCommMonoid"],["Prod","casesOn"],["Prod"],["Submonoid"],["OreLocalization","oreEqv"],["Module","toDistribMulAction"],["Setoid","r"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Quot","mk"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","map","congr_simp"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S : 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.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))))) (S_1 : 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.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))))) (e_S : Eq.{succ u_1} (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.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))))) S S_1) {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R 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(CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 (smulCommClass_self.{u_1, u_6} R N (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_6} R N (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_6} N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77) (Module.toDistribMulAction.{u_1, u_6} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87 (smulCommClass_self.{u_1, u_7} R N' (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_7} R N' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_7} N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80) (Module.toDistribMulAction.{u_1, u_7} R N' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87))))) (IsLocalizedModule.map.{u_1, u_2, u_3, u_6, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3 S M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.71 N N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87 g inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.100) (IsLocalizedModule.map.{u_1, u_2, u_3, u_6, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3 S_1 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 f_1 (Eq.ndrec.{0, succ u_1} (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.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))))) S (fun (S : 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.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))))) => IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 S f_1) (Eq.ndrec.{0, max (succ u_2) (succ u_3)} (LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40) f (fun (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40) => IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 S f) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.71 f_1 e_f) S_1 e_S) N N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87 g_1 (Eq.ndrec.{0, succ u_1} (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) N N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87) g (fun (g : LinearMap.{u_1, u_1, u_6, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) N N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87) => IsLocalizedModule.{u_1, u_6, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3 N N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87 S g) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.100 g_1 e_g) S_1 e_S))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S S_1 : Submonoid R) (e_S : S = S_1) {M : Type u_2} {M' : Type u_3}\n  [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M']\n  (f f_1 : M →ₗ[R] M') (e_f : f = f_1) [inst_5 : IsLocalizedModule S f] {N : Type u_6} {N' : Type u_7}\n  [inst_6 : AddCommMonoid N] [inst_7 : AddCommMonoid N'] [inst_8 : Module R N] [inst_9 : Module R N']\n  (g g_1 : N →ₗ[R] N') (e_g : g = g_1) [inst_10 : IsLocalizedModule S g],\n  IsLocalizedModule.map S f g = IsLocalizedModule.map S_1 f_1 g_1","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S S_1 : Submonoid R) (e_S : S = S_1) {M : Type u_2} {M' : Type u_3}\n  [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M']\n  (f f_1 : M →ₗ[R] M') (e_f : f = f_1) [inst_5 : IsLocalizedModule S f] {N : Type u_6} {N' : Type u_7}\n  [inst_6 : AddCommMonoid N] [inst_7 : AddCommMonoid N'] [inst_8 : Module R N] [inst_9 : Module R N']\n  (g g_1 : N →ₗ[R] N') (e_g : g = g_1) [inst_10 : IsLocalizedModule S g],\n  IsLocalizedModule.map S f g = IsLocalizedModule.map S_1 f_1 g_1","typeReferences":[["smulCommClass_self"],["IsLocalizedModule"],["IsLocalizedModule","map"],["CommMonoid","toMonoid"],["Module"],["LinearMap","module"],["CommSemiring"],["CommSemiring","toSemiring"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["CommSemiring","toCommMonoid"],["Submonoid"],["LinearMap","addCommMonoid"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["DistribMulAction","toMulAction"],["Eq","ndrec"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["smulCommClass_self"],["IsLocalizedModule"],["IsLocalizedModule","map"],["CommMonoid","toMonoid"],["Module"],["LinearMap","module"],["CommSemiring","toSemiring"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["CommSemiring","toCommMonoid"],["Submonoid"],["LinearMap","addCommMonoid"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Eq","refl"],["DistribMulAction","toMulAction"],["Eq","ndrec"],["Eq","rec"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Module","LocalizedModule","Basic",0,"LocalizedModule","instMonoid","_simp_1"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Defs.2875213658._hygCtx._hyg.3 : Semigroup.{u_1} G] (a : G) (b : G) (c : G), Eq.{succ u_1} G (HMul.hMul.{u_1, u_1, u_1} G G G (instHMul.{u_1} G (Semigroup.toMul.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.2875213658._hygCtx._hyg.3)) a (HMul.hMul.{u_1, u_1, u_1} G G G (instHMul.{u_1} G (Semigroup.toMul.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.2875213658._hygCtx._hyg.3)) b c)) (HMul.hMul.{u_1, u_1, u_1} G G G (instHMul.{u_1} G (Semigroup.toMul.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.2875213658._hygCtx._hyg.3)) (HMul.hMul.{u_1, u_1, u_1} G G G (instHMul.{u_1} G (Semigroup.toMul.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.2875213658._hygCtx._hyg.3)) a b) c)","typeFull":"∀ {G : Type u_1} [inst : Semigroup G] (a b c : G), a * (b * c) = a * b * c","typeReadable":"∀ {G : Type u_1} [inst : Semigroup G] (a b c : G), a * (b * c) = a * b * c","typeReferences":[["instHMul"],["HMul","hMul"],["Semigroup"],["Eq"],["Semigroup","toMul"]],"valueReferences":[["Eq","symm"],["instHMul"],["HMul","hMul"],["mul_assoc"],["Semigroup","toMul"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","mk'_eq_zero"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3214136760._hygCtx._hyg.3 : CommSemiring.{u_1} R] {S : 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.Algebra.Module.LocalizedModule.Basic.3214136760._hygCtx._hyg.3))))} {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3214136760._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3214136760._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] 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(CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.3)))))) S x)) M (Submonoid.smul.{u_1, u_2} R M (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.3)))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.11) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.36)))) S)) s₂ m₁))))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (m₁ m₂ : M) (s₁ s₂ : ↥S),\n  (IsLocalizedModule.mk' f m₁ s₁ = IsLocalizedModule.mk' f m₂ s₂) = ∃ s, s • s₁ • m₂ = s • s₂ • m₁","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (m₁ m₂ : M) (s₁ s₂ : ↥S),\n  (IsLocalizedModule.mk' f m₁ s₁ = IsLocalizedModule.mk' f m₂ s₂) = ∃ s, s • s₁ • m₂ = s • s₂ • m₁","typeReferences":[["IsLocalizedModule"],["Subtype"],["Module"],["Membership","mem"],["CommSemiring"],["Submonoid","smul"],["MulZeroOneClass","toMulOneClass"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["IsLocalizedModule","mk'"],["SetLike","instMembership"],["Exists"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["LinearMap"],["AddZeroClass","toAddZero"],["Submonoid"],["Module","toDistribMulAction"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Subtype"],["Membership","mem"],["Submonoid","smul"],["MulZeroOneClass","toMulOneClass"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["propext"],["IsLocalizedModule","mk'"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["Exists"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["IsLocalizedModule","mk'_eq_mk'_iff"],["AddZeroClass","toAddZero"],["Submonoid"],["Module","toDistribMulAction"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","subsingleton_iff"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2601470671._hygCtx._hyg.4 : CommRing.{u_1} R] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2601470671._hygCtx._hyg.7 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2601470671._hygCtx._hyg.10 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2601470671._hygCtx._hyg.4)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2601470671._hygCtx._hyg.7] {S : Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2601470671._hygCtx._hyg.4)))))}, Iff (Subsingleton.{max (succ u_2) (succ u_1)} (LocalizedModule.{u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2601470671._hygCtx._hyg.4) S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2601470671._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2601470671._hygCtx._hyg.10)) (forall (m : M), Exists.{succ u_1} R (fun (r : R) => And (Membership.mem.{u_1, u_1} R (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2601470671._hygCtx._hyg.4)))))) (SetLike.instMembership.{u_1, u_1} (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2601470671._hygCtx._hyg.4)))))) R (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 (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2601470671._hygCtx._hyg.4))))))) S r) (Eq.{succ u_2} M (HSMul.hSMul.{u_1, u_2, u_2} R M M (instHSMul.{u_1, u_2} R M (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2601470671._hygCtx._hyg.7)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2601470671._hygCtx._hyg.7)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R 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(R : Type.{u}) (S : Type.{v}) (A : Type.{w}) [inst._@.Mathlib.Algebra.Algebra.Tower.3506943269._hygCtx._hyg.7 : CommSemiring.{u} R] [inst._@.Mathlib.Algebra.Algebra.Tower.3506943269._hygCtx._hyg.10 : CommSemiring.{v} S] [inst._@.Mathlib.Algebra.Algebra.Tower.3506943269._hygCtx._hyg.13 : Semiring.{w} A] [inst._@.Mathlib.Algebra.Algebra.Tower.3506943269._hygCtx._hyg.19 : Algebra.{u, v} R S inst._@.Mathlib.Algebra.Algebra.Tower.3506943269._hygCtx._hyg.7 (CommSemiring.toSemiring.{v} S inst._@.Mathlib.Algebra.Algebra.Tower.3506943269._hygCtx._hyg.10)] [inst._@.Mathlib.Algebra.Algebra.Tower.3506943269._hygCtx._hyg.23 : Algebra.{v, w} S A inst._@.Mathlib.Algebra.Algebra.Tower.3506943269._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Algebra.Tower.3506943269._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Algebra.Tower.3506943269._hygCtx._hyg.31 : Algebra.{u, w} R A inst._@.Mathlib.Algebra.Algebra.Tower.3506943269._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Algebra.Tower.3506943269._hygCtx._hyg.13] 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.40 (LocalizedModule.lift'._proof_2.{u_2, u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.3 M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.40) (Algebra.toSMul.{u_2, u_2} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.3) (Algebra.id.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.3)) (LocalizedModule.mkLinearMap._proof_3.{u_2, u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.3 M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.40))) (Subtype.val.{succ u_2} R (fun (x : R) => Membership.mem.{u_2, u_2} R (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.3))))) (SetLike.instMembership.{u_2, u_2} (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.3))))) R (Submonoid.instSetLike.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.3)))))) S x) (Prod.snd.{u_3, u_2} M (Subtype.{succ u_2} R (fun (x : R) => Membership.mem.{u_2, u_2} R (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.3))))) (SetLike.instMembership.{u_2, u_2} (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.3))))) R (Submonoid.instSetLike.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.3)))))) S x)) x)))","typeFull":"∀ {R : Type u_2} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_3} {M' : Type u_1} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M') [IsLocalizedModule S f]\n  (x : M × ↥S), IsUnit ((algebraMap R (Module.End R M')) ↑x.2)","typeReadable":"∀ {R : Type u_2} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_3} {M' : Type u_1} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →��[R] M') [IsLocalizedModule S f]\n  (x : M × ↥S), IsUnit ((algebraMap R (Module.End R M')) ↑x.2)","typeReferences":[["IsLocalizedModule"],["RingHom"],["Subtype"],["Module"],["CommSemiring"],["Membership","mem"],["RingHom","instFunLike"],["MulZeroOneClass","toMulOneClass"],["Subtype","val"],["Algebra","id"],["DFunLike","coe"],["AddCommMonoid"],["Module","End","instAlgebra"],["LocalizedModule","mkLinearMap","_proof_3"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Algebra","toSMul"],["NonAssocSemiring","toMulZeroOneClass"],["Module","End"],["SetLike","instMembership"],["Module","End","instSemiring"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["IsUnit"],["Prod","snd"],["LinearMap"],["Prod"],["Submonoid"],["LocalizedModule","lift'","_proof_2"],["Module","End","instMonoid"],["algebraMap"]],"valueReferences":[["Submonoid"],["IsLocalizedModule","map_units"],["SetLike","instMembership"],["Semiring","toNonAssocSemiring"],["Subtype"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["Membership","mem"],["MulZeroOneClass","toMulOneClass"],["Prod","snd"],["NonAssocSemiring","toMulZeroOneClass"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","instCommRing","_proof_1"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3906542404._hygCtx._hyg.3 : CommSemiring.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3906542404._hygCtx._hyg.29 : CommRing.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3906542404._hygCtx._hyg.32 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3906542404._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3906542404._hygCtx._hyg.29))] {S : 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.Algebra.Module.LocalizedModule.Basic.3906542404._hygCtx._hyg.3))))} (a : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3906542404._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A 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(NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} A (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} A (CommRing.toNonUnitalCommRing.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3906542404._hygCtx._hyg.29))))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3906542404._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3906542404._hygCtx._hyg.29)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3906542404._hygCtx._hyg.32)) (Semiring.toNonUnitalSemiring.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3906542404._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} A (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} A (CommRing.toNonUnitalCommRing.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3906542404._hygCtx._hyg.29))))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3906542404._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3906542404._hygCtx._hyg.29)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3906542404._hygCtx._hyg.32)) (Ring.toSemiring.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3906542404._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} A (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} A (CommRing.toNonUnitalCommRing.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3906542404._hygCtx._hyg.29))))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3906542404._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3906542404._hygCtx._hyg.29)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3906542404._hygCtx._hyg.32)) (LocalizedModule.instRing.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3906542404._hygCtx._hyg.3 A (CommRing.toRing.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3906542404._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3906542404._hygCtx._hyg.32 S)))))) b a)","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : CommRing A] [inst_2 : Algebra R A] {S : Submonoid R}\n  (a b : LocalizedModule S A), a * b = b * a","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : CommRing A] [inst_2 : Algebra R A] {S : Submonoid R}\n  (a b : LocalizedModule S A), a * b = b * a","typeReferences":[["NonUnitalNonAssocSemiring","toMul"],["CommSemiring"],["HMul","hMul"],["MulZeroOneClass","toMulOneClass"],["CommRing","toNonUnitalCommRing"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Eq"],["Semiring","toNonUnitalSemiring"],["NonAssocSemiring","toMulZeroOneClass"],["CommRing","toCommSemiring"],["LocalizedModule"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["LocalizedModule","instRing"],["CommRing"],["Algebra"],["Ring","toSemiring"],["Submonoid"],["CommRing","toRing"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["instHMul"]],"valueReferences":[["CommRing","toCommSemiring"],["LocalizedModule","instCommSemiring"],["LocalizedModule"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["CommSemiring"],["CommRing","toNonUnitalCommRing"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Algebra","toModule"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["CommSemiring","mul_comm"],["inferInstance"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","liftOfLE_comp"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2021006756._hygCtx._hyg.3 : CommSemiring.{u_1} R] {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2021006756._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2021006756._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2021006756._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2021006756._hygCtx._hyg.11] {M₁ : Type.{u_7}} {M₂ : Type.{u_6}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2021006756._hygCtx._hyg.77 : AddCommMonoid.{u_7} M₁] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2021006756._hygCtx._hyg.80 : AddCommMonoid.{u_6} M₂] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2021006756._hygCtx._hyg.83 : Module.{u_1, u_7} R M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2021006756._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2021006756._hygCtx._hyg.77] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2021006756._hygCtx._hyg.87 : Module.{u_1, u_6} R M₂ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2021006756._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2021006756._hygCtx._hyg.80] (S₁ : 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.Algebra.Module.LocalizedModule.Basic.2021006756._hygCtx._hyg.3))))) (S₂ : 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.Algebra.Module.LocalizedModule.Basic.2021006756._hygCtx._hyg.3))))) (h : LE.le.{u_1} 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(RingHomCompTriple.ids.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2021006756._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2021006756._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2021006756._hygCtx._hyg.3)))) (IsLocalizedModule.liftOfLE.{u_1, u_2, u_7, u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2021006756._hygCtx._hyg.3 M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2021006756._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2021006756._hygCtx._hyg.36 M₁ M₂ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2021006756._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2021006756._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2021006756._hygCtx._hyg.83 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A)","typeReferences":[["CommRing","toCommSemiring"],["LocalizedModule"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["CommSemiring"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["MulZeroOneClass","toMulOneClass"],["CommRing","toNonUnitalCommRing"],["CommRing"],["Algebra"],["Submonoid"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["CommRing","toCommSemiring"],["LocalizedModule","instCommRing","_proof_1"],["CommRing","mk"],["LocalizedModule"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["LocalizedModule","instRing"],["CommRing","toNonUnitalCommRing"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["CommRing","toRing"],["Algebra","toModule"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","mem_ker_mkLinearMap_iff"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2905973367._hygCtx._hyg.4 : CommRing.{u_1} R] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2905973367._hygCtx._hyg.7 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2905973367._hygCtx._hyg.10 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2905973367._hygCtx._hyg.4)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2905973367._hygCtx._hyg.7] {S : Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2905973367._hygCtx._hyg.4)))))} {m : M}, Iff (Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2905973367._hygCtx._hyg.4)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2905973367._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2905973367._hygCtx._hyg.10) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2905973367._hygCtx._hyg.4)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2905973367._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2905973367._hygCtx._hyg.10) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2905973367._hygCtx._hyg.4)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2905973367._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2905973367._hygCtx._hyg.10)) (LinearMap.ker.{u_1, u_1, u_2, max u_1 u_2} R R 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Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M') [IsLocalizedModule S f],\n  Function.Injective ⇑f ↔ ∀ (c : ↥S), IsSMulRegular M 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1010008619._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1010008619._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1010008619._hygCtx._hyg.44) (Module.End.instSemiring.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1010008619._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1010008619._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1010008619._hygCtx._hyg.44))) (algebraMap.{u_1, u_4} R (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1010008619._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1010008619._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1010008619._hygCtx._hyg.44) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1010008619._hygCtx._hyg.3 (Module.End.instSemiring.{u_1, 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(CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1010008619._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.1010008619._hygCtx._hyg.3))))) R (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.Algebra.Module.LocalizedModule.Basic.1010008619._hygCtx._hyg.3)))))) S x) s)) (h s)))) (DFunLike.coe.{max (succ u_2) (succ u_4), succ u_2, succ u_4} (LinearMap.{u_1, u_1, u_2, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1010008619._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1010008619._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1010008619._hygCtx._hyg.3))) M M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1010008619._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1010008619._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1010008619._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1010008619._hygCtx._hyg.44) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M'') (LinearMap.instFunLike.{u_1, u_1, u_2, u_4} R R M M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1010008619._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1010008619._hygCtx._hyg.3) 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[inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M''] [inst_3 : Module R M] [inst_4 : Module R M''] (g : M →ₗ[R] M'')\n  (h : ∀ (x : ↥S), IsUnit ((algebraMap R (Module.End R M'')) ↑x)) (m : M) (s : ↥S),\n  (LocalizedModule.lift S g h) (LocalizedModule.mk m s) = ↑⋯.unit⁻¹ (g 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{R : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3 : CommSemiring.{u_2} R] (S : Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3))))) {M : Type.{u_3}} {M' : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.11 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.14 : AddCommMonoid.{u_1} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.36 : Module.{u_2, u_3} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.40 : Module.{u_2, u_1} R M' (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.14] (f : LinearMap.{u_2, u_2, u_3, u_1} R R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3) (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.40) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.71 : IsLocalizedModule.{u_2, u_3, u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.40 S f] (r : R) (x : LocalizedModule.{u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.36), Eq.{succ u_1} M' (IsLocalizedModule.fromLocalizedModule'.{u_2, u_3, u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3 S M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.40 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.71 (HSMul.hSMul.{u_2, max u_2 u_3, max u_2 u_3} R (LocalizedModule.{u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.36) (LocalizedModule.{u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3 S M 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(MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_2} R (CommSemiring.toCommMonoid.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.11 (Module.toDistribMulAction.{u_2, u_3} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.36)))) (DistribMulAction.toDistribSMul.{u_2, max u_2 u_3} R (LocalizedModule.{u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.36) (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{max u_2 u_3} (LocalizedModule.{u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.36) (OreLocalization.instAddCommMonoidOreLocalization.{u_2, u_3} R (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_2} R (CommSemiring.toCommMonoid.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.11 (Module.toDistribMulAction.{u_2, u_3} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.36))) (Module.toDistribMulAction.{u_2, max u_2 u_3} R (LocalizedModule.{u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.36) (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3) (OreLocalization.instAddCommMonoidOreLocalization.{u_2, u_3} R (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_2} R (CommSemiring.toCommMonoid.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.11 (Module.toDistribMulAction.{u_2, u_3} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.36)) (OreLocalization.instModuleOfIsScalarTower.{u_2, u_3, u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u_2} R (CommSemiring.toCommMonoid.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.36 R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.36 (Semiring.toModule.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3)) (IsScalarTower.left.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.11) (Module.toDistribMulAction.{u_2, u_3} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.36))) (IsScalarTower.right.{u_2, u_2} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3) (Algebra.id.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3)))))))) r x)) (HSMul.hSMul.{u_2, u_1, u_1} R M' M' (instHSMul.{u_2, u_1} R M' (SMulZeroClass.toSMul.{u_2, u_1} R M' (AddZero.toZero.{u_1} M' (AddZeroClass.toAddZero.{u_1} M' (AddMonoid.toAddZeroClass.{u_1} M' (AddCommMonoid.toAddMonoid.{u_1} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.14)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M' (AddMonoid.toAddZeroClass.{u_1} M' (AddCommMonoid.toAddMonoid.{u_1} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.14)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M' (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_1} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.14) (Module.toDistribMulAction.{u_2, u_1} R M' (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.40))))) (DFunLike.coe.{succ u_2, succ u_2, succ u_2} (RingHom.{u_2, u_2} R R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3)) (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3))) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => R) (RingHom.instFunLike.{u_2, u_2} R R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3)) (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3))) (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3))) r) (IsLocalizedModule.fromLocalizedModule'.{u_2, u_3, u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3 S M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.40 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.71 x))","typeFull":"∀ {R : Type u_2} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_3} {M' : Type u_1} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (r : R) (x : LocalizedModule S M),\n  IsLocalizedModule.fromLocalizedModule' S f (r • x) = (RingHom.id R) r • IsLocalizedModule.fromLocalizedModule' S f x","typeReadable":"∀ {R : Type u_2} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_3} {M' : Type u_1} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (r : R) (x : LocalizedModule S M),\n  IsLocalizedModule.fromLocalizedModule' S f (r • x) = (RingHom.id R) r • IsLocalizedModule.fromLocalizedModule' S f x","typeReferences":[["RingHom"],["IsLocalizedModule"],["IsLocalizedModule","fromLocalizedModule'"],["Module"],["OreLocalization","oreSetComm"],["CommSemiring"],["RingHom","instFunLike"],["MulZeroOneClass","toMulOneClass"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["Algebra","id"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["IsScalarTower","right"],["instHSMul"],["Semiring","toModule"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["IsScalarTower","left"],["LocalizedModule"],["OreLocalization","instAddCommMonoidOreLocalization"],["CommSemiring","toSemiring"],["OreLocalization","instModuleOfIsScalarTower"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["LinearMap"],["AddZeroClass","toAddZero"],["CommSemiring","toCommMonoid"],["Submonoid"],["Module","toDistribMulAction"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["RingHom"],["IsLocalizedModule","fromLocalizedModule'"],["OreLocalization","oreSetComm"],["RingHom","id_apply"],["RingHom","instFunLike"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["Algebra","id"],["congrArg"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Algebra","toSMul"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["IsScalarTower","right"],["Eq"],["Semiring","toModule"],["IsLocalizedModule","fromLocalizedModule'_smul"],["DistribSMul","toSMulZeroClass"],["OreLocalization","instAddCommMonoidOreLocalization"],["LocalizedModule"],["IsScalarTower","left"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["OreLocalization","instModuleOfIsScalarTower"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["OreLocalization","instSMulOfIsScalarTower"],["CommSemiring","toCommMonoid"],["Module","toDistribMulAction"],["Eq","refl"],["HSMul","hSMul"],["id"],["Eq","mpr"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","mk'_smul"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3 : CommSemiring.{u_1} R] {S : 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.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3))))} {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.40 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.14] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.40) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.71 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.40 S f] (r : R) (m : M) (s : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3))))) R (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.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3)))))) S x)), Eq.{succ u_3} M' (IsLocalizedModule.mk'.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3 S M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.40 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.71 (HSMul.hSMul.{u_1, u_2, u_2} R M M (instHSMul.{u_1, u_2} R M (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.11) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.36))))) r m) s) (HSMul.hSMul.{u_1, u_3, u_3} R M' M' (instHSMul.{u_1, u_3} R M' (SMulZeroClass.toSMul.{u_1, u_3} R M' (AddZero.toZero.{u_3} M' (AddZeroClass.toAddZero.{u_3} M' (AddMonoid.toAddZeroClass.{u_3} M' (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.14)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R M' (AddMonoid.toAddZeroClass.{u_3} M' (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.14)) (DistribMulAction.toDistribSMul.{u_1, u_3} R M' (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.14) (Module.toDistribMulAction.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.40))))) r (IsLocalizedModule.mk'.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3 S M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.40 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.71 m s))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (r : R) (m : M) (s : ↥S),\n  IsLocalizedModule.mk' f (r • m) s = r • IsLocalizedModule.mk' f m s","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (r : R) (m : M) (s : ↥S),\n  IsLocalizedModule.mk' f (r • m) s = r • IsLocalizedModule.mk' f m 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.77] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.87 : Module.{u_1, u_7} R N' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.80] (g : LinearMap.{u_1, u_1, u_6, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.3))) N N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.77 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.83) => LinearMap.{u_1, u_1, u_3, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.3))) M' N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.87) (LinearMap.instFunLike.{u_1, u_1, max u_2 u_6, max u_3 u_7} R R (LinearMap.{u_1, u_1, u_2, u_6} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.3))) M N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.83) (LinearMap.{u_1, 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.87 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.3)))) (LinearMap.module.{u_1, u_1, u_1, u_2, u_6} R R R M N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.83 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.83 (smulCommClass_self.{u_1, u_6} R N (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_6} R N (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_6} N 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.87 g inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.100) h) (IsLocalizedModule.mk'.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.3 S M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.40 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.71 x s)) (IsLocalizedModule.mk'.{u_1, u_6, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.3 S N N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.87 g inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.100 (DFunLike.coe.{max (succ u_2) (succ u_6), succ u_2, succ u_6} (LinearMap.{u_1, u_1, u_2, u_6} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2462176139._hygCtx._hyg.3))) M N 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IsLocalizedModule S f] {N : Type u_6} {N' : Type u_7} [inst_6 : AddCommMonoid N] [inst_7 : AddCommMonoid N']\n  [inst_8 : Module R N] [inst_9 : Module R N'] (g : N →ₗ[R] N') [inst_10 : IsLocalizedModule S g] (h : M →ₗ[R] N)\n  (x : M) (s : ↥S), ((IsLocalizedModule.map S f g) h) (IsLocalizedModule.mk' f x s) = IsLocalizedModule.mk' g (h x) 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: Type v} [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  {m m' : M} {s s' : ↥S}, LocalizedModule.mk m s = LocalizedModule.mk m' s' ↔ ∃ u, u • s' • m = u • s • m'","typeReadable":"∀ {R : Type u} [inst : CommSemiring R] {S : Submonoid R} {M : Type v} [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  {m m' : M} {s s' : ↥S}, LocalizedModule.mk m s = LocalizedModule.mk m' s' ↔ ∃ u, u • s' • m = u • s • 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p","typeReferences":[["NonUnitalNonAssocSemiring","toDistrib"],["LocalizedModule"],["Module"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Distrib","toMul"],["CommSemiring"],["CommSemiring","toSemiring"],["HMul","hMul"],["MulZeroOneClass","toMulOneClass"],["Algebra"],["IsLocalization"],["Submonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["HSMul","hSMul"],["instHMul"],["instHSMul"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["LocalizedModule","smulOfIsLocalization"]],"valueReferences":[["Eq","trans"],["Membership","mem"],["NonUnitalSemiring","toSemigroupWithZero"],["HMul","hMul"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["LocalizedModule","mk"],["LocalizedModule","mk'_smul_mk"],["Semiring","toNonAssocSemiring"],["DistribMulAction","toMulAction"],["Eq","symm"],["Monoid","toSemigroup"],["NonAssocSemiring","toMulZeroOneClass"],["LocalizedModule","smulOfIsLocalization"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["LocalizedModule"],["NonUnitalNonAssocSemiring","toDistrib"],["NonUnitalCommSemiring","toNonUnitalSemiring"],["Submonoid","instSetLike"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["LocalizedModule","induction_on"],["Prod","snd"],["AddZeroClass","toAddZero"],["Submonoid"],["HSMul","hSMul"],["id"],["instHMul"],["Submonoid","instSubmonoidClass"],["Eq","mpr"],["IsLocalization","mk'_sec"],["AddZero","toZero"],["IsLocalization","sec"],["AddMonoid","toAddZeroClass"],["Subtype"],["_private","Mathlib","Algebra","Module","LocalizedModule","Basic",0,"LocalizedModule","mul_smul_aux","_simp_1_1"],["mul_assoc"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["Semigroup","toMul"],["Prod","fst"],["congrArg"],["CommSemiring","toNonUnitalCommSemiring"],["congr"],["MonoidWithZero","toMonoid"],["instHSMul"],["congrFun'"],["Eq"],["_private","Mathlib","Algebra","Module","LocalizedModule","Basic",0,"LocalizedModule","smulOfIsLocalization","_simp_1"],["True"],["Distrib","toMul"],["CommSemiring","toSemiring"],["IsLocalization","mk'"],["Semiring","toMonoidWithZero"],["eq_self"],["Module","toDistribMulAction"],["of_eq_true"],["Submonoid","mul"],["SemigroupWithZero","toSemigroup"],["MulMemClass","toSemigroup"],["MulAction","toSemigroupAction"],["SubmonoidClass","toMulMemClass"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","isTorsionFree"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} {A : Type.{u_3}} {N : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.6 : CommRing.{u_1} R] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.9 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.12 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.6)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.9] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.16 : CommRing.{u_3} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.19 : AddCommMonoid.{u_4} N] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.22 : Module.{u_3, u_4} A N (CommSemiring.toSemiring.{u_3} A (CommRing.toCommSemiring.{u_3} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.16)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.19] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.26 : Algebra.{u_1, u_3} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.6) (CommSemiring.toSemiring.{u_3} A (CommRing.toCommSemiring.{u_3} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.16))] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.30 : Module.{u_1, u_4} R N (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.6)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.19] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.34 : IsScalarTower.{u_1, u_3, u_4} R A N (Algebra.toSMul.{u_1, u_3} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.6) (CommSemiring.toSemiring.{u_3} A (CommRing.toCommSemiring.{u_3} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.16)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.26) (SMulZeroClass.toSMul.{u_3, u_4} A N (AddZero.toZero.{u_4} N (AddZeroClass.toAddZero.{u_4} N (AddMonoid.toAddZeroClass.{u_4} N (AddCommMonoid.toAddMonoid.{u_4} N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.19)))) (DistribSMul.toSMulZeroClass.{u_3, u_4} A N (AddMonoid.toAddZeroClass.{u_4} N (AddCommMonoid.toAddMonoid.{u_4} N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.19)) (DistribMulAction.toDistribSMul.{u_3, u_4} A N (MonoidWithZero.toMonoid.{u_3} A (Semiring.toMonoidWithZero.{u_3} A (CommSemiring.toSemiring.{u_3} A (CommRing.toCommSemiring.{u_3} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.16)))) (AddCommMonoid.toAddMonoid.{u_4} N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.19) (Module.toDistribMulAction.{u_3, u_4} A N (CommSemiring.toSemiring.{u_3} A (CommRing.toCommSemiring.{u_3} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.16)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.22)))) (SMulZeroClass.toSMul.{u_1, u_4} R N (AddZero.toZero.{u_4} N (AddZeroClass.toAddZero.{u_4} N (AddMonoid.toAddZeroClass.{u_4} N (AddCommMonoid.toAddMonoid.{u_4} N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.19)))) (DistribSMul.toSMulZeroClass.{u_1, u_4} R N (AddMonoid.toAddZeroClass.{u_4} N (AddCommMonoid.toAddMonoid.{u_4} N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.19)) (DistribMulAction.toDistribSMul.{u_1, u_4} R N (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.6)))) (AddCommMonoid.toAddMonoid.{u_4} N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.19) (Module.toDistribMulAction.{u_1, u_4} R N (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.6)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.30))))] (f : LinearMap.{u_1, u_1, u_2, u_4} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.6)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.6)) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.6)))) M N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.30) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.48 : IsDomain.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.6))] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.51 : Module.IsTorsionFree.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.6)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.12] (S : Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.6)))))) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.57 : IsLocalization.{u_1, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.6) S A (CommRing.toCommSemiring.{u_3} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.16) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.26] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.61 : IsLocalizedModule.{u_1, u_2, u_4} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.6) M N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.30 S f], Module.IsTorsionFree.{u_3, u_4} A N (CommSemiring.toSemiring.{u_3} A (CommRing.toCommSemiring.{u_3} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.16)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1483157487._hygCtx._hyg.22","typeFull":"∀ {R : Type u_1} {M : Type u_2} {A : Type u_3} {N : Type u_4} [inst : CommRing R] [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] [inst_3 : CommRing A] [inst_4 : AddCommMonoid N] [inst_5 : Module A N] [inst_6 : Algebra R A]\n  [inst_7 : Module R N] [IsScalarTower R A N] (f : M →ₗ[R] N) [IsDomain R] [Module.IsTorsionFree R M] (S : Submonoid R)\n  [IsLocalization S A] [IsLocalizedModule S f], Module.IsTorsionFree A N","typeReadable":"∀ {R : Type u_1} {M : Type u_2} {A : Type u_3} {N : Type u_4} [inst : CommRing R] [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] [inst_3 : CommRing A] [inst_4 : AddCommMonoid N] [inst_5 : Module A N] [inst_6 : Algebra R A]\n  [inst_7 : Module R N] [IsScalarTower R A N] (f : M →ₗ[R] N) [IsDomain R] [Module.IsTorsionFree R M] (S : Submonoid R)\n  [IsLocalization S A] [IsLocalizedModule S f], Module.IsTorsionFree A N","typeReferences":[["Module","IsTorsionFree"],["IsLocalizedModule"],["IsScalarTower"],["Module"],["SMulZeroClass","toSMul"],["IsDomain"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Algebra","toSMul"],["MonoidWithZero","toMonoid"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["CommRing","toCommSemiring"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["CommRing"],["LinearMap"],["IsLocalization"],["Algebra"],["Submonoid"],["Module","toDistribMulAction"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["implies_congr"],["imp_self","_simp_1"],["Eq","trans"],["Membership","mem"],["MulZeroClass","toMul"],["MulZeroOneClass","toMulOneClass"],["CommRing","toNonUnitalCommRing"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["forall_congr"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Zero","toOfNat0"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["Not"],["IsDomain","toNontrivial"],["CommRing","toCommSemiring"],["SetLike","instMembership"],["True"],["Submonoid","instSetLike"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["CommSemiring","toSemiring"],["OfNat","ofNat"],["Submonoid"],["implies_true"],["IsLocalizedModule","isTorsionFree_of_forall_isRegular"],["of_eq_true"],["MulZeroClass","toZero"],["Eq","refl"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Ne"],["IsDomain","toIsCancelMulZero"],["_private","Mathlib","Algebra","Module","LocalizedModule","Basic",0,"IsLocalizedModule","isTorsionFree","_simp_1_1"],["IsRegular"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Module","LocalizedModule","Basic",0,"IsLocalizedModule","mkOfAlgebra","_simp_1_3"],"typeFallback":"forall {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 = b)","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["SubNegMonoid","toSub"],["NegZeroClass","toZero"],["HSub","hSub"],["AddGroup"],["Zero","toOfNat0"],["AddGroup","toSubNegMonoid"],["instHSub"],["Eq"],["SubNegZeroMonoid","toNegZeroClass"],["AddGroup","toSubtractionMonoid"],["OfNat","ofNat"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["sub_eq_zero"],["SubNegZeroMonoid","toNegZeroClass"],["OfNat","ofNat"],["AddGroup","toSubtractionMonoid"],["SubNegMonoid","toSub"],["HSub","hSub"],["NegZeroClass","toZero"],["Zero","toOfNat0"],["AddGroup","toSubNegMonoid"],["Eq"],["instHSub"],["propext"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","mul","_proof_1"],"typeFallback":"forall {R : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.3 : CommSemiring.{u_2} R] {A : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.29 : Semiring.{u_1} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.32 : Algebra.{u_2, u_1} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.29] {S : Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.3))))} (p : Prod.{u_1, u_2} A (Subtype.{succ u_2} R (fun (x : R) => Membership.mem.{u_2, u_2} R (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2075127268._hygCtx._hyg.3))))) (SetLike.instMembership.{u_2, u_2} (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.40) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.71 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.40 S f] {N : Type.{u_6}} {N' : Type.{u_7}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.77 : AddCommMonoid.{u_6} N] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.80 : 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.83) => LinearMap.{u_1, u_1, u_3, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.3))) M' N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.80 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.87 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.3)))) (LinearMap.module.{u_1, u_1, u_1, u_2, u_6} R R R M N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.11 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(f p q) (f p' q'))) -> α)","typeFull":"{R : Type u} →\n  [inst : CommSemiring R] →\n    {S : Submonoid R} →\n      {M : Type v} →\n        [inst_1 : AddCommMonoid M] →\n          [inst_2 : Module R M] →\n            {α : Type u_2} →\n              LocalizedModule S M →\n                LocalizedModule S M →\n                  (f : M × ↥S → M × ↥S → α) → (∀ (p q p' q' : M × ↥S), p ≈ p' → q ≈ q' → f p q = f p' q') → α","typeReadable":"{R : Type u} →\n  [inst : CommSemiring R] →\n    {S : Submonoid R} →\n      {M : Type v} →\n        [inst_1 : AddCommMonoid M] →\n          [inst_2 : Module R M] →\n            {α : Type u_2} →\n              LocalizedModule S M →\n                LocalizedModule S M →\n                  (f : M × ↥S → M × ↥S → α) → (∀ (p q p' q' : M × ↥S), p ≈ p' → q ≈ q' → f p q = f p' q') → α","typeReferences":[["LocalizedModule","r","setoid"],["Subtype"],["SetLike","instMembership"],["LocalizedModule"],["Submonoid","instSetLike"],["Module"],["CommSemiring"],["CommSemiring","toSemiring"],["Membership","mem"],["MulZeroOneClass","toMulOneClass"],["Submonoid"],["Prod"],["AddCommMonoid"],["HasEquiv","Equiv"],["Semiring","toNonAssocSemiring"],["instHasEquivOfSetoid"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["Subtype"],["SetLike","instMembership"],["Submonoid","instSetLike"],["OreLocalization","oreSetComm"],["Membership","mem"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"],["CommSemiring","toCommMonoid"],["LocalizedModule","liftOn₂","_proof_1"],["Quotient","liftOn₂"],["Prod"],["Submonoid"],["OreLocalization","oreEqv"],["Module","toDistribMulAction"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","instMonoid","_proof_8"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 : CommSemiring.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 : Semiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29] {S : 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.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3))))} (a : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32 S a._@._internal._hyg.0 a_1._@._internal._hyg.0)) (LocalizedModule.instMonoid._proof_7.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32 S)))) (OfNat.ofNat.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 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a","typeReferences":[["Mul","mk"],["OreLocalization","oreSetComm"],["OreLocalization","instOne"],["CommSemiring"],["HMul","hMul"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["Semigroup","toMul"],["Semiring","toNonAssocSemiring"],["Semigroup","mk"],["Algebra","toModule"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["NonAssocSemiring","toMulZeroOneClass"],["LocalizedModule"],["LocalizedModule","mul"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["CommSemiring","toCommMonoid"],["OfNat","ofNat"],["Algebra"],["Submonoid"],["Module","toDistribMulAction"],["One","toOfNat1"],["AddMonoidWithOne","toOne"],["instHMul"],["LocalizedModule","instMonoid","_proof_7"],["Semiring"]],"valueReferences":[["LocalizedModule","instMonoid","_proof_3"]]},{"isProp":true,"kind":"constructor","name":["IsLocalizedModule","mk"],"typeFallback":"forall 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109)))) S)) (Prod.snd.{u_2, u_1} M (Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))))) (SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))))) R (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.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)))))) S 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)))) f x₁) (DFunLike.coe.{max (succ u_2) (succ u_3), succ u_2, succ u_3} (LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M') (LinearMap.instFunLike.{u_1, u_1, u_2, u_3} R R M M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (Monoid.toMulOneClass.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (LocalizedModule.instMonoid.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (Monoid.toMulOneClass.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (LocalizedModule.instMonoid.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 S))))) a c))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A] {S : Submonoid R}\n  (a b c : LocalizedModule S A), a * (b + c) = a * b + a * c","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A] {S : Submonoid R}\n  (a b c : LocalizedModule S A), a * (b + c) = a * b + a * c","typeReferences":[["MulOneClass","toMulOne"],["OreLocalization","oreSetComm"],["CommSemiring"],["HMul","hMul"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["MulOne","toMul"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["MonoidWithZero","toMonoid"],["Monoid","toMulOneClass"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["AddSemigroup","toAdd"],["LocalizedModule","instMonoid"],["LocalizedModule"],["OreLocalization","instAddCommMonoidOreLocalization"],["instHAdd"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["CommSemiring","toCommMonoid"],["Algebra"],["Submonoid"],["HAdd","hAdd"],["Module","toDistribMulAction"],["AddMonoid","toAddSemigroup"],["instHMul"],["Semiring"]],"valueReferences":[["LocalizedModule","instSemiring","_proof_2"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Module","LocalizedModule","Basic",0,"IsLocalizedModule","injective_of_map_zero","_simp_1_1"],"typeFallback":"forall {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 = b)","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["SubNegMonoid","toSub"],["NegZeroClass","toZero"],["HSub","hSub"],["AddGroup"],["Zero","toOfNat0"],["AddGroup","toSubNegMonoid"],["instHSub"],["Eq"],["SubNegZeroMonoid","toNegZeroClass"],["AddGroup","toSubtractionMonoid"],["OfNat","ofNat"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["sub_eq_zero"],["SubNegZeroMonoid","toNegZeroClass"],["OfNat","ofNat"],["AddGroup","toSubtractionMonoid"],["SubNegMonoid","toSub"],["HSub","hSub"],["NegZeroClass","toZero"],["Zero","toOfNat0"],["AddGroup","toSubNegMonoid"],["Eq"],["instHSub"],["propext"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","divBy","_proof_7"],"typeFallback":"forall {R : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3 : CommSemiring.{u_2} R] {S : Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))))} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12 : Module.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9] (s : Subtype.{succ u_2} R (fun (x : R) => Membership.mem.{u_2, u_2} R (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))))) (SetLike.instMembership.{u_2, u_2} (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12)))) S)) (Prod.snd.{u_1, u_2} M (Subtype.{succ u_2} R (fun (x : R) => Membership.mem.{u_2, u_2} R (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))))) (SetLike.instMembership.{u_2, u_2} (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))))) R (Submonoid.instSetLike.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3)))))) S x)) 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(SetLike.instMembership.{u_2, u_2} (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))))) R (Submonoid.instSetLike.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3)))))) S x)) a b)) s) (Prod.fst.{u_1, u_2} M (Subtype.{succ u_2} R (fun (x : R) => Membership.mem.{u_2, u_2} R (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))))) (SetLike.instMembership.{u_2, u_2} (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))))) R (Submonoid.instSetLike.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3)))))) S x)) (Prod.mk.{u_1, u_2} M (Subtype.{succ u_2} R (fun (x : R) => Membership.mem.{u_2, u_2} R (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))))) (SetLike.instMembership.{u_2, u_2} (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))))) R (Submonoid.instSetLike.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3)))))) S x)) a' b')))))","typeFull":"∀ {R : Type u_2} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_1} [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] (s : ↥S) (a : M) (b : ↥S) (a' : M) (b' c : ↥S),\n  c • (a', b').2 • (a, b).1 = c • (a, b).2 • (a', b').1 →\n    c • ((a', b').2 * s) • (a, b).1 = c • ((a, b).2 * s) • (a', b').1","typeReadable":"∀ {R : Type u_2} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_1} [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] (s : ↥S) (a : M) (b : ↥S) (a' : M) (b' c : ↥S),\n  c • (a', b').2 • (a, b).1 = c • (a, b).2 • (a', b').1 →\n    c • ((a', b').2 * s) • (a, b).1 = c • ((a, b).2 * s) • (a', b').1","typeReferences":[["Subtype"],["Prod","mk"],["Module"],["Membership","mem"],["CommSemiring"],["Submonoid","smul"],["HMul","hMul"],["MulZeroOneClass","toMulOneClass"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Prod","fst"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["Prod","snd"],["AddZeroClass","toAddZero"],["Submonoid"],["Module","toDistribMulAction"],["Submonoid","mul"],["HSMul","hSMul"],["instHMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["smulCommClass_self"],["Prod","mk"],["Submonoid","smul"],["Membership","mem"],["HMul","hMul"],["SemigroupAction","toSMul"],["NonUnitalSemiring","toSemigroupWithZero"],["SMulZeroClass","toSMul"],["SMulCommClass","smul_comm"],["Semiring","toNonAssocSemiring"],["DistribMulAction","toMulAction"],["Monoid","toSemigroup"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["Submonoid","instSetLike"],["NonUnitalCommSemiring","toNonUnitalSemiring"],["DistribMulAction","toDistribSMul"],["Submonoid","mulAction"],["Submonoid","smulCommClass_right"],["Prod","snd"],["AddZeroClass","toAddZero"],["CommSemiring","toCommMonoid"],["Submonoid"],["Eq","refl"],["HSMul","hSMul"],["id"],["instHMul"],["Eq","mpr"],["Submonoid","instSubmonoidClass"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Subtype"],["CommMonoid","toMonoid"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["Prod","fst"],["congrArg"],["Semigroup","toMul"],["CommSemiring","toNonUnitalCommSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["Submonoid","toMonoid"],["SemigroupAction","mul_smul"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["Module","toDistribMulAction"],["Submonoid","smulCommClass_left"],["Submonoid","mul"],["SemigroupWithZero","toSemigroup"],["MulMemClass","toSemigroup"],["MulAction","toSemigroupAction"],["SubmonoidClass","toMulMemClass"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","instRing","_proof_3"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 : CommSemiring.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29 : Ring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.32 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29)] {S : 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.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3))))}, autoParam.{0} (forall (a : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.32)) (Semiring.toNonUnitalSemiring.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonAssocRing.toNonUnitalNonAssocRing.{u_2} A (Ring.toNonAssocRing.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.32)) (LocalizedModule.instSemiring.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 A (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.32 S)))))))) a (Neg.neg.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonAssocRing.toNonUnitalNonAssocRing.{u_2} A (Ring.toNonAssocRing.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.32)) (OreLocalization.instNegOreLocalization.{u_1, u_2} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3) S) A (AddGroupWithOne.toAddGroup.{u_2} A (Ring.toAddGroupWithOne.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29)) (Module.toDistribMulAction.{u_1, u_2} R A (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonAssocRing.toNonUnitalNonAssocRing.{u_2} A (Ring.toNonAssocRing.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.32))) b))) SubNegMonoid.sub_eq_add_neg._autoParam","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Ring A] [inst_2 : Algebra R A] {S : Submonoid R},\n  autoParam (∀ (a b : LocalizedModule S A), a - b = a + -b) SubNegMonoid.sub_eq_add_neg._autoParam","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Ring A] [inst_2 : Algebra R A] {S : Submonoid R},\n  autoParam (∀ (a b : LocalizedModule S A), a - b = a + -b) SubNegMonoid.sub_eq_add_neg._autoParam","typeReferences":[["OreLocalization","instAddGroupOreLocalization"],["Ring","toNonAssocRing"],["OreLocalization","oreSetComm"],["CommSemiring"],["MulZeroOneClass","toMulOneClass"],["LocalizedModule","instSemiring"],["AddCommMonoid","toAddMonoid"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["OreLocalization","instNegOreLocalization"],["Semiring","toNonAssocSemiring"],["Ring","toAddGroupWithOne"],["Algebra","toModule"],["SubNegMonoid","toSub"],["MonoidWithZero","toMonoid"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["Eq"],["Semiring","toNonUnitalSemiring"],["NonAssocSemiring","toMulZeroOneClass"],["NonAssocRing","toNonUnitalNonAssocRing"],["AddSemigroup","toAdd"],["LocalizedModule"],["Neg","neg"],["instHAdd"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["CommSemiring","toCommMonoid"],["Ring","toSemiring"],["Algebra"],["HAdd","hAdd"],["Submonoid"],["Module","toDistribMulAction"],["AddGroupWithOne","toAddGroup"],["AddMonoid","toAddSemigroup"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["autoParam"],["instHSub"],["SubNegMonoid","sub_eq_add_neg","_autoParam"],["Ring"]],"valueReferences":[["Ring","toNonAssocRing"],["LocalizedModule"],["OreLocalization","oreSetComm"],["AddCommGroup","toAddGroup"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["CommSemiring","toCommMonoid"],["Ring","toSemiring"],["Ring","toAddCommGroup"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Module","toDistribMulAction"],["Algebra","toModule"],["MonoidWithZero","toMonoid"],["inferInstance"],["AddGroup","toSubNegMonoid"],["OreLocalization","instAddCommGroup"],["SubNegMonoid","sub_eq_add_neg"],["NonAssocRing","toNonUnitalNonAssocRing"]]},{"isProp":false,"kind":"recursor","name":["IsLocalizedModule","rec"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72 : CommSemiring.{u_1} R] {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83] {S : 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.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))))} {f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109} {motive : (IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109 S f) -> Sort.{u}}, (forall (map_units : forall (x : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))))) (SetLike.instMembership.{u_1, u_1} (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (DFunLike.coe.{max (succ u_1) (succ u_3), succ u_1, succ u_3} (RingHom.{u_1, u_3} R (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)) (Semiring.toNonAssocSemiring.{u_3} (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (Module.End.instSemiring.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109))) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (RingHom.instFunLike.{u_1, u_3} R (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)) (Semiring.toNonAssocSemiring.{u_3} (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (Module.End.instSemiring.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109))) (algebraMap.{u_1, u_3} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109 (smulCommClass_self.{u_1, u_3} R M' (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) (DistribMulAction.toMulAction.{u_1, u_3} R M' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)) (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83) (Module.toDistribMulAction.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M') (LinearMap.instFunLike.{u_1, u_1, u_2, u_3} R R M M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.40 S f] {N : Type.{u_6}} {N' : Type.{u_7}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.77 : AddCommMonoid.{u_6} N] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.80 : AddCommMonoid.{u_7} N'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.83 : Module.{u_1, u_6} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.77] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.87 : Module.{u_1, u_7} R N' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3 N N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.87 S g] (h : LinearMap.{u_1, u_1, u_2, u_6} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3))) M N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.11 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.83) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : LinearMap.{u_1, u_1, u_2, u_6} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3))) M N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.83) => LinearMap.{u_1, u_1, u_3, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3))) M' N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.87) (LinearMap.instFunLike.{u_1, u_1, max u_2 u_6, max u_3 u_7} R R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3))) M' N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.87) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) (LinearMap.addCommMonoid.{u_1, u_1, u_2, u_6} R R M N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.87 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3)))) (LinearMap.module.{u_1, u_1, u_1, u_2, u_6} R R R M N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.36 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.83)))) (LinearMap.module.{u_1, u_1, u_1, u_3, u_7} R R R M' N' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.87 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.87 (smulCommClass_self.{u_1, u_7} R N' (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_7} R N' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_7} N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.80) (Module.toDistribMulAction.{u_1, u_7} R N' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.87)))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3)))) (IsLocalizedModule.map.{u_1, u_2, u_3, u_6, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3 S M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.40 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.71 N N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.87 g inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.100) h) (DFunLike.coe.{max (succ u_2) (succ u_3), succ u_2, succ u_3} (LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.40) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M') (LinearMap.instFunLike.{u_1, u_1, u_2, u_3} R R M M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.40 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3)))) f x)) (DFunLike.coe.{max (succ u_6) (succ u_7), succ u_6, succ u_7} 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.87 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3)))) g (DFunLike.coe.{max (succ u_2) (succ u_6), succ u_2, succ u_6} (LinearMap.{u_1, u_1, u_2, u_6} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3))) M N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.83) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => N) (LinearMap.instFunLike.{u_1, u_1, u_2, u_6} R R M N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.83 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4294455747._hygCtx._hyg.3)))) h x))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] {N : Type u_6} {N' : Type u_7} [inst_6 : AddCommMonoid N] [inst_7 : AddCommMonoid N']\n  [inst_8 : Module R N] [inst_9 : Module R N'] (g : N →ₗ[R] N') [inst_10 : IsLocalizedModule S g] (h : M →ₗ[R] N)\n  (x : M), ((IsLocalizedModule.map S f g) h) (f x) = g (h x)","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] {N : Type u_6} {N' : Type u_7} [inst_6 : AddCommMonoid N] [inst_7 : AddCommMonoid N']\n  [inst_8 : Module R N] [inst_9 : Module R N'] (g : N →ₗ[R] N') [inst_10 : IsLocalizedModule S g] (h : M →ₗ[R] N)\n  (x : M), ((IsLocalizedModule.map S f g) h) (f x) = g (h 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(NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonAssocRing.toNonUnitalNonAssocRing.{u_2} A (Ring.toNonAssocRing.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.32)) (Semiring.toNatCast.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonAssocRing.toNonUnitalNonAssocRing.{u_2} A (Ring.toNonAssocRing.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.32)) (LocalizedModule.instSemiring.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 A (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.32 S)) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))))) AddGroupWithOne.intCast_negSucc._autoParam","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Ring A] [inst_2 : Algebra R A] {S : Submonoid R},\n  autoParam (∀ (n : ℕ), IntCast.intCast (Int.negSucc n) = -↑(n + 1)) AddGroupWithOne.intCast_negSucc._autoParam","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Ring A] [inst_2 : Algebra R A] {S : Submonoid R},\n  autoParam (∀ (n : ℕ), IntCast.intCast (Int.negSucc n) = -↑(n + 1)) AddGroupWithOne.intCast_negSucc._autoParam","typeReferences":[["instAddNat"],["Nat","cast"],["Ring","toNonAssocRing"],["OreLocalization","oreSetComm"],["AddCommGroup","toAddGroup"],["CommSemiring"],["MulZeroOneClass","toMulOneClass"],["LocalizedModule","instSemiring"],["IntCast","intCast"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["OreLocalization","instNegOreLocalization"],["Semiring","toNonAssocSemiring"],["Ring","toAddGroupWithOne"],["Semiring","toNatCast"],["instOfNatNat"],["Algebra","toModule"],["MonoidWithZero","toMonoid"],["AddGroup","toSubNegMonoid"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["NonAssocRing","toNonUnitalNonAssocRing"],["LocalizedModule"],["instHAdd"],["Neg","neg"],["SubNegMonoid","toNeg"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Int","negSucc"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["OfNat","ofNat"],["CommSemiring","toCommMonoid"],["Ring","toSemiring"],["Algebra"],["HAdd","hAdd"],["Ring","toAddCommGroup"],["Submonoid"],["Module","toDistribMulAction"],["Nat"],["AddGroupWithOne","toAddGroup"],["inferInstance"],["autoParam"],["OreLocalization","instAddCommGroup"],["AddGroupWithOne","intCast_negSucc","_autoParam"],["IntCast","mk"],["Int","castDef"],["Ring"],["Semiring"]],"valueReferences":[["LocalizedModule","instRing","_proof_2"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","of_exists_mul_mem"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3473276209._hygCtx._hyg.3 : CommSemiring.{u_1} R] {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3473276209._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3473276209._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3473276209._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3473276209._hygCtx._hyg.11] {N : Type.{u_6}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3473276209._hygCtx._hyg.72 : AddCommMonoid.{u_6} N] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3473276209._hygCtx._hyg.75 : Module.{u_1, u_6} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3473276209._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3473276209._hygCtx._hyg.72] (S : 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.Algebra.Module.LocalizedModule.Basic.3473276209._hygCtx._hyg.3))))) (T : 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.Algebra.Module.LocalizedModule.Basic.3473276209._hygCtx._hyg.3))))), (LE.le.{u_1} (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.Algebra.Module.LocalizedModule.Basic.3473276209._hygCtx._hyg.3))))) (Preorder.toLE.{u_1} (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R 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[inst_1 : AddCommMonoid M] [inst_2 : Module R M] {N : Type u_6}\n  [inst_3 : AddCommMonoid N] [inst_4 : Module R N] (S T : Submonoid R),\n  S ≤ T → (∀ (x : ↥T), ∃ m, m * ↑x ∈ S) → ∀ (f : M →ₗ[R] N) [IsLocalizedModule S f], IsLocalizedModule T 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 : CommSemiring.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 : Semiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29] {S : 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.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3))))} (a : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109} {motive : (IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109 S f) -> Sort.{u}} (t : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109 S f), (forall (map_units : forall (x : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))))) (SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))))) R (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.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)))))) S x)), IsUnit.{u_3} (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (Module.End.instMonoid.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (DFunLike.coe.{max (succ u_1) (succ u_3), succ u_1, succ u_3} (RingHom.{u_1, u_3} R (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)) (Semiring.toNonAssocSemiring.{u_3} (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (Module.End.instSemiring.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109))) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (RingHom.instFunLike.{u_1, u_3} R (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)) (Semiring.toNonAssocSemiring.{u_3} (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (Module.End.instSemiring.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109))) (algebraMap.{u_1, u_3} R (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72 (Module.End.instSemiring.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (Module.End.instAlgebra.{u_1, u_1, u_3} R R M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109 (smulCommClass_self.{u_1, u_3} R M' (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) (DistribMulAction.toMulAction.{u_1, u_3} R M' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)) (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83) (Module.toDistribMulAction.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109))) (Algebra.toSMul.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)) (IsScalarTower.left.{u_1, u_3} R M' (MonoidWithZero.toMonoid.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109)))) S)) (Prod.snd.{u_2, u_1} M (Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))))) (SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))))) R (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.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)))))) S x)) x) y) (DFunLike.coe.{max (succ u_2) (succ u_3), succ u_2, succ u_3} (LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M') (LinearMap.instFunLike.{u_1, u_1, u_2, u_3} R R M M' (CommSemiring.toSemiring.{u_1} 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)))) f x₁) (DFunLike.coe.{max (succ u_2) (succ u_3), succ u_2, succ u_3} (LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M') (LinearMap.instFunLike.{u_1, u_1, u_2, u_3} R R M M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)))) f x₂)) -> (Exists.{succ u_1} (Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))))) (SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))))) R (Submonoid.instSetLike.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R 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(CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))))) R (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.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)))))) S x)) M (Submonoid.smul.{u_1, u_2} R M (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105)))) S)) c x₂)))), motive (IsLocalizedModule.mk.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109 S f map_units surj exists_of_eq)) -> (motive t)","typeFull":"{R : Type u_1} →\n  [inst : CommSemiring R] →\n    {M : Type u_2} →\n      {M' : Type u_3} →\n        [inst_1 : AddCommMonoid M] →\n          [inst_2 : AddCommMonoid M'] →\n            [inst_3 : Module R M] →\n              [inst_4 : Module R M'] →\n                {S : Submonoid R} →\n                  {f : M →ₗ[R] M'} →\n                    {motive : IsLocalizedModule S f → Sort u} →\n                      (t : IsLocalizedModule S f) →\n                        ((map_units : ∀ (x : ↥S), IsUnit ((algebraMap R (Module.End R M')) ↑x)) →\n                            (surj : ∀ (y : M'), ∃ x, x.2 • y = f x.1) →\n                              (exists_of_eq : ∀ {x₁ x₂ : M}, f x₁ = f x₂ → ∃ c, c • x₁ = c • x₂) → motive ⋯) →\n                          motive t","typeReadable":"{R : Type u_1} →\n  [inst : CommSemiring R] →\n    {M : Type u_2} →\n      {M' : Type u_3} →\n        [inst_1 : AddCommMonoid M] →\n          [inst_2 : AddCommMonoid M'] →\n            [inst_3 : Module R M] →\n              [inst_4 : Module R M'] →\n                {S : Submonoid R} →\n                  {f : M →ₗ[R] M'} →\n                    {motive : IsLocalizedModule S f → Sort u} →\n                      (t : IsLocalizedModule S f) →\n                        ((map_units : ∀ (x : ↥S), IsUnit ((algebraMap R (Module.End R M')) ↑x)) →\n                            (surj : ∀ (y : M'), ∃ x, x.2 • y = f x.1) →\n                              (exists_of_eq : ∀ {x₁ x₂ : M}, f x₁ = f x₂ → ∃ c, c • x₁ = c • x₂) → motive ⋯) →\n                          motive 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{R : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3 : CommSemiring.{u_2} R] (S : Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3))))) (M : Type.{u_1}) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.9 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.12 : Module.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.9] (x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.67 : Prod.{u_1, u_2} M (Subtype.{succ u_2} R (fun (x : R) => Membership.mem.{u_2, u_2} R (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3))))) (SetLike.instMembership.{u_2, u_2} (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3))))) R (Submonoid.instSetLike.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3)))))) S x))) (motive : forall (x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx.66.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.82 : Prod.{u_1, u_2} M 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.12 x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx.66.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.82 x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.67) -> Prop) (x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx.66.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.82 : Prod.{u_1, u_2} M (Subtype.{succ u_2} R (fun (x : R) => Membership.mem.{u_2, u_2} R (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3))))) (SetLike.instMembership.{u_2, u_2} (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3))))) R (Submonoid.instSetLike.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3)))))) S x))) (x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.69 : LocalizedModule.r.{u_2, u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.12 x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx.66.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.82 x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.67), (forall (m1 : M) (s1 : Subtype.{succ u_2} R (fun (x : R) => Membership.mem.{u_2, u_2} R (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3))))) (SetLike.instMembership.{u_2, u_2} (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3))))) R (Submonoid.instSetLike.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R 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(NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3))))) R (Submonoid.instSetLike.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3)))))) S x)) m1 s1) x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.69) -> (motive x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx.66.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.82 x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.69)","typeFull":"∀ {R : Type u_2} [inst : CommSemiring R] (S : Submonoid R) (M : Type u_1) [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] (x : M × ↥S) (motive : (x_1 : M × ↥S) → LocalizedModule.r S M x_1 x → Prop) (x_1 : M × ↥S)\n  (x_2 : LocalizedModule.r S M x_1 x),\n  (∀ (m1 : M) (s1 : ↥S) (x : LocalizedModule.r S M (m1, s1) x), motive (m1, s1) x) → motive x_1 x_2","typeReadable":"∀ {R : Type u_2} [inst : CommSemiring R] (S : Submonoid R) (M : Type u_1) [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] (x : M × ↥S) (motive : (x_1 : M × ↥S) → LocalizedModule.r S M x_1 x → Prop) (x_1 : M × ↥S)\n  (x_2 : LocalizedModule.r S M x_1 x),\n  (∀ (m1 : M) (s1 : ↥S) (x : LocalizedModule.r S M (m1, s1) x), motive (m1, s1) x) → motive x_1 x_2","typeReferences":[["Subtype"],["SetLike","instMembership"],["Prod","mk"],["Module"],["Submonoid","instSetLike"],["CommSemiring"],["CommSemiring","toSemiring"],["Membership","mem"],["MulZeroOneClass","toMulOneClass"],["Submonoid"],["Prod"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["LocalizedModule","r"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["Submonoid"],["Prod","casesOn"],["SetLike","instMembership"],["Semiring","toNonAssocSemiring"],["Subtype"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["Membership","mem"],["LocalizedModule","r"],["MulZeroOneClass","toMulOneClass"],["NonAssocSemiring","toMulZeroOneClass"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Module","LocalizedModule","Basic",0,"LocalizedModule","smul_eq_iff_of_mem","_simp_1_2"],"typeFallback":"forall {b : Prop} {a : Prop}, Eq.{1} Prop (Exists.{0} a (fun (_h : a) => b)) (And a b)","typeFull":"∀ {b a : Prop}, (∃ (_ : a), b) = (a ∧ b)","typeReadable":"∀ {b a : Prop}, (∃ (_ : a), b) = (a ∧ b)","typeReferences":[["Exists"],["And"],["Eq"]],"valueReferences":[["exists_prop"],["Exists"],["And"],["propext"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","isTorsionFree_of_forall_isRegular"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} {A : Type.{u_3}} {N : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.6 : CommRing.{u_1} R] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.9 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.12 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.6)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.9] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.16 : CommRing.{u_3} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.19 : AddCommMonoid.{u_4} N] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.22 : Module.{u_3, u_4} A N (CommSemiring.toSemiring.{u_3} A (CommRing.toCommSemiring.{u_3} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.16)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.19] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.26 : Algebra.{u_1, u_3} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.6) (CommSemiring.toSemiring.{u_3} A (CommRing.toCommSemiring.{u_3} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.16))] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.30 : Module.{u_1, u_4} R N (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.6)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.19] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.34 : IsScalarTower.{u_1, u_3, u_4} R A N (Algebra.toSMul.{u_1, u_3} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.6) (CommSemiring.toSemiring.{u_3} A (CommRing.toCommSemiring.{u_3} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.16)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.26) (SMulZeroClass.toSMul.{u_3, u_4} A N (AddZero.toZero.{u_4} N (AddZeroClass.toAddZero.{u_4} N (AddMonoid.toAddZeroClass.{u_4} N (AddCommMonoid.toAddMonoid.{u_4} N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.19)))) (DistribSMul.toSMulZeroClass.{u_3, u_4} A N (AddMonoid.toAddZeroClass.{u_4} N (AddCommMonoid.toAddMonoid.{u_4} N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.19)) (DistribMulAction.toDistribSMul.{u_3, u_4} A N (MonoidWithZero.toMonoid.{u_3} A (Semiring.toMonoidWithZero.{u_3} A (CommSemiring.toSemiring.{u_3} A (CommRing.toCommSemiring.{u_3} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.16)))) (AddCommMonoid.toAddMonoid.{u_4} N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.19) (Module.toDistribMulAction.{u_3, u_4} A N (CommSemiring.toSemiring.{u_3} A (CommRing.toCommSemiring.{u_3} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.16)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.22)))) (SMulZeroClass.toSMul.{u_1, u_4} R N (AddZero.toZero.{u_4} N (AddZeroClass.toAddZero.{u_4} N (AddMonoid.toAddZeroClass.{u_4} N (AddCommMonoid.toAddMonoid.{u_4} N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.19)))) (DistribSMul.toSMulZeroClass.{u_1, u_4} R N (AddMonoid.toAddZeroClass.{u_4} N (AddCommMonoid.toAddMonoid.{u_4} N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.19)) (DistribMulAction.toDistribSMul.{u_1, u_4} R N (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.6)))) (AddCommMonoid.toAddMonoid.{u_4} N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.19) (Module.toDistribMulAction.{u_1, u_4} R N (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.6)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.30))))] (f : LinearMap.{u_1, u_1, u_2, u_4} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.6)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.6)) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.6)))) M N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.30) (S : Submonoid.{u_1} R 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(CommRing.toNonUnitalCommRing.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.6)))))) s)) -> (forall [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.76 : Module.IsTorsionFree.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.6)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.12] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.80 : IsLocalization.{u_1, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.6) S A (CommRing.toCommSemiring.{u_3} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.16) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1166806679._hygCtx._hyg.26] 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[inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] [inst_3 : CommRing A] [inst_4 : AddCommMonoid N] [inst_5 : Module A N] [inst_6 : Algebra R A]\n  [inst_7 : Module R N] [IsScalarTower R A N] (f : M →ₗ[R] N) (S : Submonoid R),\n  (∀ s ∈ S, s ≠ 0 → IsRegular s) →\n    ∀ [Module.IsTorsionFree R M] [IsLocalization S A] [IsLocalizedModule S f], Module.IsTorsionFree A N","typeReadable":"∀ {R : Type u_1} {M : Type u_2} {A : Type u_3} {N : Type u_4} [inst : CommRing R] [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] [inst_3 : CommRing A] [inst_4 : AddCommMonoid N] [inst_5 : Module A N] [inst_6 : Algebra R A]\n  [inst_7 : Module R N] [IsScalarTower R A N] (f : M →ₗ[R] N) (S : Submonoid R),\n  (∀ s ∈ S, s ≠ 0 → IsRegular s) →\n    ∀ [Module.IsTorsionFree R M] [IsLocalization S A] [IsLocalizedModule S f], Module.IsTorsionFree A 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S : 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.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.3))))) {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.11] 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.40 S f) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3046389206._hygCtx._hyg.71 f_1 e_f) a_1._@._internal._hyg.0))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f f_1 : M →ₗ[R] M') (e_f : f = f_1)\n  [inst_5 : IsLocalizedModule S f] (a a_1 : LocalizedModule S M),\n  a = a_1 → IsLocalizedModule.fromLocalizedModule' S f a = IsLocalizedModule.fromLocalizedModule' S f_1 a_1","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2973489541._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2973489541._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2973489541._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2973489541._hygCtx._hyg.40 S f] {N : Type.{u_6}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2973489541._hygCtx._hyg.76 : AddCommMonoid.{u_6} N] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2973489541._hygCtx._hyg.79 : Module.{u_1, u_6} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2973489541._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2973489541._hygCtx._hyg.76] {g : LinearMap.{u_1, u_1, u_3, u_6} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2973489541._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R 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AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M') [IsLocalizedModule S f]\n  {N : Type u_6} [inst_6 : AddCommMonoid N] [inst_7 : Module R N] {g : M' →ₗ[R] N},\n  (∀ {x y : M}, g (f x) = g (f y) → f x = f y) → Function.Injective ⇑g","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M') [IsLocalizedModule S f]\n  {N : Type u_6} [inst_6 : AddCommMonoid N] [inst_7 : Module R N] {g : M' →ₗ[R] N},\n  (∀ {x y : M}, g (f x) = g (f y) → f x = f y) → Function.Injective 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.40 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.14] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.44 : Module.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.17] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.3) (RingHom.id.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.40 S f], IsLocalizedModule.{u_1, u_2, u_4} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.3 M M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.44 S (LinearMap.comp.{u_1, u_1, u_1, u_2, u_3, u_4} R R R M M' M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.44 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.3))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.3))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.3))) (RingHomCompTriple.ids.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.3)))) (LinearEquiv.toLinearMap.{u_1, u_1, u_3, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.3))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.3))) (RingHomInvPair.ids.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.3)) (RingHomInvPair.ids.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.3)) M' M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2410417570._hygCtx._hyg.44 e) f)","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} {M'' : Type u_4}\n  [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M'] [inst_3 : AddCommMonoid M''] [inst_4 : Module R M]\n  [inst_5 : Module R M'] [inst_6 : Module R M''] (f : M →ₗ[R] M') (e : M' ≃ₗ[R] M'') [hf : IsLocalizedModule S f],\n  IsLocalizedModule S (↑e ∘ₗ f)","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} {M'' : Type u_4}\n  [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M'] [inst_3 : AddCommMonoid M''] [inst_4 : Module R M]\n  [inst_5 : Module R M'] [inst_6 : Module R M''] (f : M →ₗ[R] M') (e : M' ≃ₗ[R] M'') [hf : IsLocalizedModule S f],\n  IsLocalizedModule S (↑e ∘ₗ f)","typeReferences":[["IsLocalizedModule"],["RingHomCompTriple","ids"],["Module"],["CommSemiring"],["CommSemiring","toSemiring"],["LinearMap","comp"],["LinearEquiv","toLinearMap"],["MulZeroOneClass","toMulOneClass"],["LinearMap"],["LinearEquiv"],["Submonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["RingHomInvPair","ids"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["smulCommClass_self"],["RingHom"],["Eq","trans"],["RingHomCompTriple","ids"],["Submonoid","smul"],["Membership","mem"],["Iff","mp"],["Exists","intro"],["SMulZeroClass","toSMul"],["map_smul"],["Algebra","id"],["Subtype","val"],["Module","End","isUnit_iff"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["LinearMap","ext"],["Eq","symm"],["DistribMulAction","toMulAction"],["EquivLike","comp_bijective"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["Exists"],["EquivLike","bijective_comp"],["SemilinearMapClass","toMulActionSemiHomClass"],["Module","End","instSemiring"],["SetLike","instMembership"],["Submonoid","instSetLike"],["IsLocalizedModule","surj"],["DistribMulAction","toDistribSMul"],["Function","Bijective"],["Prod","snd"],["LinearMap"],["AddZeroClass","toAddZero"],["CommSemiring","toCommMonoid"],["Prod"],["Exists","casesOn"],["Submonoid"],["LinearEquiv","instSemilinearEquivClass"],["Eq","refl"],["HSMul","hSMul"],["RingHomInvPair","ids"],["id"],["EquivLike","toEmbeddingLike"],["Eq","mpr"],["LinearMap","coe_comp"],["AddZero","toZero"],["algebraMap"],["AddMonoid","toAddZeroClass"],["Subtype"],["LinearEquiv","congr_arg"],["LinearMap","instFunLike"],["CommMonoid","toMonoid"],["Eq","mp"],["IsLocalizedModule","exists_of_eq"],["RingHom","instFunLike"],["LinearMap","comp"],["LinearEquiv","coe_coe"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["LinearEquiv"],["DFunLike","coe"],["Prod","fst"],["congrArg"],["IsLocalizedModule","map_units"],["SemilinearEquivClass","instSemilinearMapClass"],["Module","End","instAlgebra"],["EquivLike","toFunLike"],["Algebra","toSMul"],["MonoidWithZero","toMonoid"],["instHSMul"],["LinearEquiv","instEquivLike"],["Eq"],["propext"],["Module","End"],["True"],["IsScalarTower","left"],["Submonoid","smul_def"],["CommSemiring","toSemiring"],["Function","comp_apply"],["Semiring","toMonoidWithZero"],["LinearEquiv","toLinearMap"],["IsUnit"],["Function","comp"],["_private","Mathlib","Algebra","Module","LocalizedModule","Basic",0,"IsLocalizedModule","of_linearEquiv","_simp_1"],["LinearEquiv","symm"],["eq_self"],["Module","toDistribMulAction"],["of_eq_true"],["LinearEquiv","apply_symm_apply"],["IsLocalizedModule","mk"],["Module","End","instMonoid"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","linearEquiv","eq_1"],"typeFallback":"forall 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IsLocalizedModule S f]\n  [inst_8 : IsLocalizedModule S g],\n  IsLocalizedModule.linearEquiv S f g = (IsLocalizedModule.iso S f).symm.trans (IsLocalizedModule.iso S 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1516545263._hygCtx._hyg.3)))) S)) s₁ s₂))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (m₁ m₂ : M) (s₁ s₂ : ↥S),\n  IsLocalizedModule.mk' f m₁ s₁ + IsLocalizedModule.mk' f m₂ s₂ = IsLocalizedModule.mk' f (s₂ • m₁ + s₁ • m₂) (s₁ * s₂)","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (m₁ m₂ : M) (s₁ s₂ : ↥S),\n  IsLocalizedModule.mk' f m₁ s₁ + IsLocalizedModule.mk' f m₂ s₂ = IsLocalizedModule.mk' f (s₂ • m₁ + s₁ • m₂) (s₁ * s₂)","typeReferences":[["IsLocalizedModule"],["Subtype"],["Module"],["Membership","mem"],["CommSemiring"],["Submonoid","smul"],["HMul","hMul"],["SMulZeroClass","toSMul"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["IsLocalizedModule","mk'"],["SetLike","instMembership"],["Submonoid","instSetLike"],["instHAdd"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["LinearMap"],["Submonoid"],["HAdd","hAdd"],["Module","toDistribMulAction"],["Submonoid","mul"],["AddCommMonoid","toAddCommSemigroup"],["HSMul","hSMul"],["instHMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["IsLocalizedModule","fromLocalizedModule"],["Submonoid","smul"],["Membership","mem"],["HMul","hMul"],["SMulZeroClass","toSMul"],["LocalizedModule","mk"],["LinearMap","semilinearMapClass"],["Semiring","toNonAssocSemiring"],["LocalizedModule","mkLinearMap","_proof_3"],["RingHom","id"],["Eq","symm"],["Semiring","toModule"],["NonAssocSemiring","toMulZeroOneClass"],["IsLocalizedModule","mk'"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["LocalizedModule"],["Submonoid","instSetLike"],["DistribMulAction","toDistribSMul"],["AddZeroClass","toAddZero"],["LinearMap"],["CommSemiring","toCommMonoid"],["map_add"],["Submonoid"],["Eq","refl"],["HSMul","hSMul"],["id"],["instHMul"],["LocalizedModule","algebra'","_proof_4"],["Eq","mpr"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Subtype"],["LinearMap","instFunLike"],["OreLocalization","oreSetComm"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["OreLocalization","instAdd"],["DFunLike","coe"],["congrArg"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Eq"],["OreLocalization","instAddCommMonoidOreLocalization"],["instHAdd"],["CommSemiring","toSemiring"],["OreLocalization","instModuleOfIsScalarTower"],["Semiring","toMonoidWithZero"],["SemilinearMapClass","toAddHomClass"],["HAdd","hAdd"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["Submonoid","mul"],["LocalizedModule","mk_add_mk"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","instMonoid","_proof_11"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 : CommSemiring.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 : Semiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29] {S : 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.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3))))}, autoParam.{0} (forall (n : Nat) (x : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32 S)) (LocalizedModule.instMonoid._proof_2.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32 S)) (inferInstance.{max (succ u_1) (succ u_2)} (One.{max u_2 u_1} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 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(CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32))) (AddMonoidWithOne.toOne.{u_2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29)))))) n x) x)) Monoid.npow_succ._autoParam","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A] {S : Submonoid R},\n  autoParam (∀ (n : ℕ) (x : LocalizedModule S A), npowRecAuto (n + 1) x = npowRecAuto n x * x)\n    Monoid.npow_succ._autoParam","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A] {S : Submonoid R},\n  autoParam (∀ (n : ℕ) (x : LocalizedModule S A), npowRecAuto (n + 1) x = npowRecAuto n x * x)\n    Monoid.npow_succ._autoParam","typeReferences":[["instAddNat"],["Mul","mk"],["OreLocalization","oreSetComm"],["OreLocalization","instOne"],["CommSemiring"],["HMul","hMul"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["Semigroup","toMul"],["Semiring","toNonAssocSemiring"],["Semigroup","mk"],["instOfNatNat"],["Algebra","toModule"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["One"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["LocalizedModule","instMonoid","_proof_2"],["NonAssocSemiring","toAddCommMonoidWithOne"],["NonAssocSemiring","toMulZeroOneClass"],["LocalizedModule"],["instHAdd"],["LocalizedModule","mul"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["OfNat","ofNat"],["CommSemiring","toCommMonoid"],["Algebra"],["HAdd","hAdd"],["Submonoid"],["Module","toDistribMulAction"],["Nat"],["npowRecAuto"],["Monoid","npow_succ","_autoParam"],["AddMonoidWithOne","toOne"],["instHMul"],["inferInstance"],["LocalizedModule","instMonoid","_proof_7"],["autoParam"],["Semiring"]],"valueReferences":[["LocalizedModule","instMonoid","_proof_6"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","smul'_mul"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2070844397._hygCtx._hyg.3 : CommSemiring.{u} R] {S : Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2070844397._hygCtx._hyg.3))))} {T : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2070844397._hygCtx._hyg.17 : CommSemiring.{u_1} T] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2070844397._hygCtx._hyg.20 : Algebra.{u, u_1} R T inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2070844397._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} T inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2070844397._hygCtx._hyg.17)] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2070844397._hygCtx._hyg.24 : IsLocalization.{u, u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2070844397._hygCtx._hyg.3 S T inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2070844397._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2070844397._hygCtx._hyg.20] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2070844397._hygCtx._hyg.29 : Semiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2070844397._hygCtx._hyg.32 : Algebra.{u, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2070844397._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2070844397._hygCtx._hyg.29] (x : T) (p₁ : LocalizedModule.{u, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2070844397._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2070844397._hygCtx._hyg.29))) (Algebra.toModule.{u, u_2} R A 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S : 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.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.3))))) {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.40 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.14] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.36 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.36 R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.36 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.11) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.36))) (IsScalarTower.right.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.3))) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2864189145._hygCtx._hyg.40","typeFull":"{R : Type u_1} →\n  [inst : CommSemiring R] →\n    (S : Submonoid R) →\n      {M : Type u_2} →\n        {M' : Type u_3} →\n          [inst_1 : AddCommMonoid M] →\n            [inst_2 : AddCommMonoid M'] →\n              [inst_3 : Module R M] →\n                [inst_4 : Module R M'] → (f : M →ₗ[R] M') → [IsLocalizedModule S f] → LocalizedModule S M ≃ₗ[R] M'","typeReadable":"{R : Type u_1} →\n  [inst : CommSemiring R] →\n    (S : Submonoid R) →\n      {M : Type u_2} →\n        {M' : Type u_3} →\n          [inst_1 : AddCommMonoid M] →\n            [inst_2 : AddCommMonoid M'] →\n              [inst_3 : Module R M] →\n                [inst_4 : Module R M'] → (f : M →ₗ[R] M') → [IsLocalizedModule S f] → LocalizedModule S M ≃ₗ[R] M'","typeReferences":[["IsLocalizedModule"],["Module"],["OreLocalization","oreSetComm"],["CommSemiring"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["LinearEquiv"],["Algebra","id"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["IsScalarTower","right"],["Semiring","toModule"],["NonAssocSemiring","toMulZeroOneClass"],["LocalizedModule"],["OreLocalization","instAddCommMonoidOreLocalization"],["IsScalarTower","left"],["CommSemiring","toSemiring"],["OreLocalization","instModuleOfIsScalarTower"],["Semiring","toMonoidWithZero"],["LinearMap"],["CommSemiring","toCommMonoid"],["Submonoid"],["Module","toDistribMulAction"],["RingHomInvPair","ids"]],"valueReferences":[["LinearMap","instFunLike"],["IsLocalizedModule","iso","_proof_1"],["OreLocalization","oreSetComm"],["IsLocalizedModule","fromLocalizedModule"],["IsLocalizedModule","iso","_proof_2"],["DFunLike","coe"],["Semiring","toNonAssocSemiring"],["LocalizedModule","mkLinearMap","_proof_3"],["RingHom","id"],["MonoidWithZero","toMonoid"],["LinearEquiv","mk"],["Semiring","toModule"],["OreLocalization","instAddCommMonoidOreLocalization"],["LocalizedModule"],["IsLocalizedModule","fromLocalizedModule","bij"],["CommSemiring","toSemiring"],["OreLocalization","instModuleOfIsScalarTower"],["Semiring","toMonoidWithZero"],["LinearMap"],["CommSemiring","toCommMonoid"],["Module","toDistribMulAction"],["Equiv","invFun"],["Equiv","ofBijective"],["IsLocalizedModule","iso","_proof_3"],["LocalizedModule","algebra'","_proof_4"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","algebraMap_mk'"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2780064200._hygCtx._hyg.3 : CommSemiring.{u} R] {S : Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2780064200._hygCtx._hyg.3))))} (T : Type.{u_1}) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2780064200._hygCtx._hyg.17 : CommSemiring.{u_1} T] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2780064200._hygCtx._hyg.20 : Algebra.{u, u_1} R T inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2780064200._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} T inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2780064200._hygCtx._hyg.17)] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2780064200._hygCtx._hyg.24 : IsLocalization.{u, u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2780064200._hygCtx._hyg.3 S T inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2780064200._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2780064200._hygCtx._hyg.20] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2780064200._hygCtx._hyg.29 : Semiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2780064200._hygCtx._hyg.32 : Algebra.{u, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2780064200._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2780064200._hygCtx._hyg.29] (a : R) (s : Subtype.{succ u} R (fun (x : R) => Membership.mem.{u, u} R (Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2780064200._hygCtx._hyg.3))))) (SetLike.instMembership.{u, u} (Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R 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: CommSemiring R] {S : Submonoid R} (T : Type u_1) [inst_1 : CommSemiring T] [inst_2 : Algebra R T]\n  [inst_3 : IsLocalization S T] {A : Type u_2} [inst_4 : Semiring A] [inst_5 : Algebra R A] (a : R) (s : ↥S),\n  (algebraMap T (LocalizedModule S A)) (IsLocalization.mk' T a s) = LocalizedModule.mk ((algebraMap R A) a) s","typeReadable":"∀ {R : Type u} [inst : CommSemiring R] {S : Submonoid R} (T : Type u_1) [inst_1 : CommSemiring T] [inst_2 : Algebra R T]\n  [inst_3 : IsLocalization S T] {A : Type u_2} [inst_4 : Semiring A] [inst_5 : Algebra R A] (a : R) (s : ↥S),\n  (algebraMap T (LocalizedModule S A)) (IsLocalization.mk' T a s) = LocalizedModule.mk ((algebraMap R A) a) s","typeReferences":[["RingHom"],["Subtype"],["Membership","mem"],["CommSemiring"],["RingHom","instFunLike"],["MulZeroOneClass","toMulOneClass"],["LocalizedModule","instSemiring"],["LocalizedModule","mk"],["DFunLike","coe"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["LocalizedModule","algebraOfIsLocalization"],["SetLike","instMembership"],["LocalizedModule"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["IsLocalization","mk'"],["IsLocalization"],["Algebra"],["Submonoid"],["Semiring"],["algebraMap"]],"valueReferences":[["RingHom"],["Membership","mem"],["HMul","hMul"],["SMulZeroClass","toSMul"],["LocalizedModule","instSemiring"],["LocalizedModule","mk"],["LocalizedModule","mk'_smul_mk"],["Semiring","toNonAssocSemiring"],["NonAssocSemiring","toAddCommMonoidWithOne"],["NonAssocSemiring","toMulZeroOneClass"],["LocalizedModule","smulOfIsLocalization"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["MulOne","toOne"],["LocalizedModule"],["Submonoid","instSetLike"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["AddZeroClass","toAddZero"],["Submonoid"],["Eq","refl"],["AddMonoidWithOne","toOne"],["HSMul","hSMul"],["id"],["instHMul"],["Eq","mpr"],["AddZero","toZero"],["algebraMap"],["AddMonoid","toAddZeroClass"],["MulOneClass","toMulOne"],["Subtype"],["RingHom","instFunLike"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["congrArg"],["Submonoid","toMulOneClass"],["MulOne","toMul"],["Algebra","toModule"],["Algebra","toSMul"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["LocalizedModule","algebraOfIsLocalization"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["IsLocalization","mk'"],["Semiring","toMonoidWithZero"],["mul_one"],["OfNat","ofNat"],["Algebra","algebraMap_eq_smul_one"],["Submonoid","one"],["Module","toDistribMulAction"],["One","toOfNat1"],["Submonoid","mul"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","instSemiring","_proof_16"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 : CommSemiring.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 : Semiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29] {S : 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.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3))))}, autoParam.{0} (forall (n : Nat), Eq.{succ (max u_1 u_2)} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (Monoid.toMulOneClass.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 S) (LocalizedModule.instSemiring._proof_10.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 S) (LocalizedModule.instSemiring._proof_11.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 S)) (LocalizedModule.instSemiring._proof_12.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 S)))))))) (NatCast.natCast.{max u_1 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a","typeReferences":[["NonUnitalNonAssocSemiring","toMul"],["LocalizedModule"],["CommSemiring"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["HMul","hMul"],["MulZeroOneClass","toMulOneClass"],["LocalizedModule","instSemiring"],["Algebra"],["Submonoid"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["inferInstance"],["instHMul"],["Eq"],["Semiring","toNonUnitalSemiring"],["NonAssocSemiring","toMulZeroOneClass"],["Semiring"]],"valueReferences":[["Eq","trans"],["NonUnitalNonAssocSemiring","toMul"],["Prod","mk"],["Membership","mem"],["Submonoid","smul"],["Exists","intro"],["HMul","hMul"],["SMulZeroClass","toSMul"],["Quot","ind"],["LocalizedModule","instSemiring"],["LocalizedModule","mk"],["NonUnitalCommSemiring","toNonUnitalNonAssocCommSemiring"],["Semiring","toNonAssocSemiring"],["mul_comm"],["DistribMulAction","toMulAction"],["Semiring","toNonUnitalSemiring"],["Quot","mk"],["NonAssocSemiring","toMulZeroOneClass"],["NonUnitalCommSemiring","toCommSemigroup"],["DistribSMul","toSMulZeroClass"],["LocalizedModule","mk_eq"],["SetLike","instMembership"],["Exists"],["CommMagma","toMul"],["LocalizedModule"],["NonUnitalNonAssocSemiring","toDistrib"],["Submonoid","instSetLike"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Submonoid","mulAction"],["Prod","snd"],["AddZeroClass","toAddZero"],["CommSemiring","toCommMonoid"],["Prod"],["Submonoid"],["Setoid","r"],["Iff","mpr"],["one_smul"],["HSMul","hSMul"],["instHMul"],["Submonoid","instSubmonoidClass"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Subtype"],["NonUnitalNonAssocCommSemiring","toCommMagma"],["OreLocalization","oreSetComm"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["congrArg"],["Prod","fst"],["OreLocalization","oreEqv"],["Prod","casesOn"],["CommSemiring","toNonUnitalCommSemiring"],["Algebra","toModule"],["congr"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["Submonoid","toMonoid"],["CommSemigroup","toCommMagma"],["True"],["MulMemClass","toCommSemigroup"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Distrib","toMul"],["Semiring","toMonoidWithZero"],["Submonoid","one"],["OfNat","ofNat"],["eq_self"],["Module","toDistribMulAction"],["of_eq_true"],["One","toOfNat1"],["Submonoid","mul"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["inferInstance"],["SubmonoidClass","toMulMemClass"],["Semiring"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","instMonoid","_proof_2"],"typeFallback":"forall 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (LocalizedModule.mul.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32 S))) b c))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A] {S : Submonoid R}\n  (a b c : LocalizedModule S A), a * b * c = a * (b * c)","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A] {S : Submonoid R}\n  (a b c : LocalizedModule S A), a * b * c = a * (b * 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 : CommSemiring.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 : Semiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29] {S : 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.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3))))} (a : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A 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IsLocalizedModule.mk' f m₁ s₁ * IsLocalizedModule.mk' f m₂ s₂ = IsLocalizedModule.mk' f (m₁ * m₂) (s₁ * s₂)","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_6} {M' : Type u_7}\n  [inst_1 : NonUnitalNonAssocSemiring M] [inst_2 : Semiring M'] [inst_3 : Module R M] [inst_4 : Algebra R M']\n  (f : M →ₗ[R] M'),\n  (∀ (m₁ m₂ : M), f (m₁ * m₂) = f m₁ * f m₂) →\n    ∀ [inst_5 : IsLocalizedModule S f] (m₁ m₂ : M) (s₁ s₂ : ↥S),\n      IsLocalizedModule.mk' f m₁ s₁ * IsLocalizedModule.mk' f m₂ s₂ = IsLocalizedModule.mk' f (m₁ * m₂) (s₁ * 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S : 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.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.3))))) {M : Type.{u_2}} {M' : Type.{u_3}} {M'' : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.17 : AddCommMonoid.{u_4} M''] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.40 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.14] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.44 : Module.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.17] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.3) (RingHom.id.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.44) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.71 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.40 S f] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.75 : IsLocalizedModule.{u_1, u_2, u_4} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.3 M M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.44 S g], LinearEquiv.{u_1, u_1, u_3, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2994524416._hygCtx._hyg.3))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R 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           [inst_4 : Module R M] →\n                    [inst_5 : Module R M'] →\n                      [inst_6 : Module R M''] →\n                        (f : M →ₗ[R] M') →\n                          (g : M →ₗ[R] M'') → [IsLocalizedModule S f] → [IsLocalizedModule S g] → M' ≃ₗ[R] M''","typeReadable":"{R : Type u_1} →\n  [inst : CommSemiring R] →\n    (S : Submonoid R) →\n      {M : Type u_2} →\n        {M' : Type u_3} →\n          {M'' : Type u_4} →\n            [inst_1 : AddCommMonoid M] →\n              [inst_2 : AddCommMonoid M'] →\n                [inst_3 : AddCommMonoid M''] →\n                  [inst_4 : Module R M] →\n                    [inst_5 : Module R M'] →\n                      [inst_6 : Module R M''] →\n                        (f : M →ₗ[R] M') →\n                          (g : M →ₗ[R] M'') → [IsLocalizedModule S f] → [IsLocalizedModule S g] → M' ≃ₗ[R] M''","typeReferences":[["IsLocalizedModule"],["Module"],["CommSemiring"],["CommSemiring","toSemiring"],["MulZeroOneClass","toMulOneClass"],["LinearMap"],["LinearEquiv"],["Submonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["RingHomInvPair","ids"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["IsLocalizedModule","lift","_proof_1"],["IsLocalizedModule","iso","_proof_1"],["LocalizedModule"],["OreLocalization","instAddCommMonoidOreLocalization"],["OreLocalization","oreSetComm"],["CommSemiring","toSemiring"],["OreLocalization","instModuleOfIsScalarTower"],["Semiring","toMonoidWithZero"],["LinearEquiv","symm"],["CommSemiring","toCommMonoid"],["LinearEquiv","trans"],["Module","toDistribMulAction"],["Semiring","toNonAssocSemiring"],["LocalizedModule","mkLinearMap","_proof_3"],["RingHom","id"],["IsLocalizedModule","iso"],["MonoidWithZero","toMonoid"],["LocalizedModule","algebra'","_proof_4"],["Semiring","toModule"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","mk'_eq_zero'"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3035130408._hygCtx._hyg.3 : CommSemiring.{u_1} R] {S : 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.Algebra.Module.LocalizedModule.Basic.3035130408._hygCtx._hyg.3))))} {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3035130408._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3035130408._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3035130408._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3035130408._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3035130408._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3035130408._hygCtx._hyg.40 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3035130408._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3035130408._hygCtx._hyg.14] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3035130408._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3035130408._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3035130408._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3035130408._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3035130408._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3035130408._hygCtx._hyg.36 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3)))) (LinearMap.module.{u_1, u_1, u_1, u_2, u_2} R R R M M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.36 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.36 (smulCommClass_self.{u_1, u_2} R M (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_2} R M (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.11) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.36)))) (LinearMap.module.{u_1, u_1, u_1, u_3, u_3} R R R M' M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.40 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.40 (smulCommClass_self.{u_1, u_3} R M' (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_3} R M' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.14) (Module.toDistribMulAction.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.40))))) (LinearMap.{u_1, u_1, u_2, u_2} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3))) M M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.36) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : LinearMap.{u_1, u_1, u_2, u_2} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3))) M M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.36) => LinearMap.{u_1, u_1, u_3, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3))) M' M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.40) (LinearMap.instFunLike.{u_1, u_1, u_2, u_3} R R (LinearMap.{u_1, u_1, u_2, u_2} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3))) M M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.36) (LinearMap.{u_1, u_1, u_3, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3))) M' M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.40) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) (LinearMap.addCommMonoid.{u_1, u_1, u_2, u_2} R R M M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.36 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3)))) (LinearMap.addCommMonoid.{u_1, u_1, u_3, u_3} R R M' M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.40 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3)))) (LinearMap.module.{u_1, u_1, u_1, u_2, u_2} R R R M M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.36 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.36 (smulCommClass_self.{u_1, u_2} R M (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_2} R M (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.11) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.36)))) (LinearMap.module.{u_1, u_1, u_1, u_3, u_3} R R R M' M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.40 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.40 (smulCommClass_self.{u_1, u_3} R M' (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_3} R M' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.14) (Module.toDistribMulAction.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.40)))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3)))) (IsLocalizedModule.map.{u_1, u_2, u_3, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3 S M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.40 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.71 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.40 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.71) (LinearMap.id.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.36)) (LinearMap.id.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3525663135._hygCtx._hyg.40)","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f], (IsLocalizedModule.map S f f) LinearMap.id = LinearMap.id","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f], (IsLocalizedModule.map S f f) LinearMap.id = LinearMap.id","typeReferences":[["smulCommClass_self"],["IsLocalizedModule"],["IsLocalizedModule","map"],["LinearMap","instFunLike"],["CommMonoid","toMonoid"],["Module"],["LinearMap","module"],["CommSemiring"],["CommSemiring","toSemiring"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["LinearMap","id"],["DFunLike","coe"],["CommSemiring","toCommMonoid"],["LinearMap","addCommMonoid"],["Submonoid"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["DistribMulAction","toMulAction"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["smulCommClass_self"],["Subtype"],["Eq","trans"],["CommMonoid","toMonoid"],["LinearMap","instFunLike"],["Prod","mk"],["Membership","mem"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["LinearMap","id"],["congrArg"],["Prod","casesOn"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["LinearMap","ext"],["DistribMulAction","toMulAction"],["congrFun'"],["Eq","ndrec"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["IsLocalizedModule","mk'"],["Function","uncurry"],["IsLocalizedModule","map"],["SetLike","instMembership"],["True"],["Submonoid","instSetLike"],["LinearMap","module"],["CommSemiring","toSemiring"],["LinearMap"],["IsLocalizedModule","map_mk'"],["CommSemiring","toCommMonoid"],["LinearMap","addCommMonoid"],["Exists","casesOn"],["Prod"],["Submonoid"],["eq_self"],["Module","toDistribMulAction"],["of_eq_true"],["IsLocalizedModule","mk'_surjective"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","lift_comp_iso"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S : 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.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.3))))) {M : Type.{u_2}} {M' : Type.{u_3}} {M'' : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.17 : AddCommMonoid.{u_4} M''] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.40 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.14] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.44 : Module.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.17] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.3) (RingHom.id.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.44) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.71 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.40 S f] (h : forall (x : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.44) (Module.End.instMonoid.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.44) (DFunLike.coe.{max (succ u_1) (succ u_4), succ u_1, succ u_4} (RingHom.{u_1, u_4} R (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.44) (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.3))) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.44 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.3))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.3))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.3))) (RingHomCompTriple.ids.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.36)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.14 (OreLocalization.instModuleOfIsScalarTower.{u_1, u_2, u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.36 R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.36 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.3)) 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1561590962._hygCtx._hyg.44 g h)","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} {M'' : Type u_4}\n  [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M'] [inst_3 : AddCommMonoid M''] [inst_4 : Module R M]\n  [inst_5 : Module R M'] [inst_6 : Module R M''] (f : M →ₗ[R] M') (g : M →ₗ[R] M'') [inst_7 : IsLocalizedModule S f]\n  (h : ∀ (x : ↥S), IsUnit ((algebraMap R (Module.End R M'')) ↑x)),\n  IsLocalizedModule.lift S f g h ∘ₗ ↑(IsLocalizedModule.iso S f) = LocalizedModule.lift S g h","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} {M'' : Type u_4}\n  [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M'] [inst_3 : AddCommMonoid M''] [inst_4 : Module R M]\n  [inst_5 : Module R M'] [inst_6 : Module R M''] (f : M →ₗ[R] M') (g : M →ₗ[R] M'') [inst_7 : IsLocalizedModule S f]\n  (h : ∀ (x : ↥S), IsUnit ((algebraMap R (Module.End R M'')) ↑x)),\n  IsLocalizedModule.lift S f g h ∘ₗ ↑(IsLocalizedModule.iso S f) = LocalizedModule.lift S g 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3 : CommSemiring.{u_1} R] {S : 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.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3))))} {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.40 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[inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.71 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.40 S f] (r : R) (m : M) (s : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R 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R M' (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.14) (Module.toDistribMulAction.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.40))))) r (IsLocalizedModule.mk'.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3 S M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.40 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.71 m s)) (IsLocalizedModule.mk'.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3 S M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.40 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.71 (HSMul.hSMul.{u_1, u_2, u_2} R M M (instHSMul.{u_1, u_2} R M (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.11) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.761693692._hygCtx._hyg.36))))) r m) s)","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (r : R) (m : M) (s : ↥S),\n  r • IsLocalizedModule.mk' f m s = IsLocalizedModule.mk' f (r • m) s","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (r : R) (m : M) (s : ↥S),\n  r • IsLocalizedModule.mk' f m s = IsLocalizedModule.mk' f (r • m) s","typeReferences":[["IsLocalizedModule"],["Subtype"],["Module"],["Membership","mem"],["CommSemiring"],["MulZeroOneClass","toMulOneClass"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["IsLocalizedModule","mk'"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["LinearMap"],["AddZeroClass","toAddZero"],["Submonoid"],["Module","toDistribMulAction"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["IsLocalizedModule","mk'_smul"],["Module","toDistribMulAction"],["HSMul","hSMul"],["MonoidWithZero","toMonoid"],["Eq","symm"],["instHSMul"],["AddZero","toZero"],["IsLocalizedModule","mk'"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","numeratorRingHom","_proof_4"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3 : CommSemiring.{u_1} R] {S : 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.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3))))} {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29 : Semiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.32 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29] (x : A) (y : A), Eq.{succ (max u_1 u_2)} (OreLocalization.{u_1, u_2} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R 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(OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3) S) A (DistribMulAction.toMulAction.{u_1, u_2} R A (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29)))) (Module.toDistribMulAction.{u_1, u_2} R A (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A 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(NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.32)))) (AddZeroClass.toAddZero.{max u_1 u_2} (OreLocalization.{u_1, u_2} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3) S) A (DistribMulAction.toMulAction.{u_1, u_2} R A (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29)))) (Module.toDistribMulAction.{u_1, u_2} R A (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.32)))) (AddMonoid.toAddZeroClass.{max u_1 u_2} (OreLocalization.{u_1, u_2} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3) S) A (DistribMulAction.toMulAction.{u_1, u_2} R A (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29)))) (Module.toDistribMulAction.{u_1, u_2} R A (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.32)))) (AddMonoidWithOne.toAddMonoid.{max u_1 u_2} (OreLocalization.{u_1, u_2} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3) S) A (DistribMulAction.toMulAction.{u_1, u_2} R A (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29)))) (Module.toDistribMulAction.{u_1, u_2} R A (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.32)))) (AddCommMonoidWithOne.toAddMonoidWithOne.{max u_1 u_2} (OreLocalization.{u_1, u_2} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3) S) A (DistribMulAction.toMulAction.{u_1, u_2} R A (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29)))) (Module.toDistribMulAction.{u_1, u_2} R A (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.32)))) (NonAssocSemiring.toAddCommMonoidWithOne.{max u_1 u_2} (OreLocalization.{u_1, u_2} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3) S) A (DistribMulAction.toMulAction.{u_1, u_2} R A (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29)))) (Module.toDistribMulAction.{u_1, u_2} R A (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.32)))) (Semiring.toNonAssocSemiring.{max u_1 u_2} (OreLocalization.{u_1, u_2} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3) S) A (DistribMulAction.toMulAction.{u_1, u_2} R A (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29)))) (Module.toDistribMulAction.{u_1, u_2} R A (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.32)))) (LocalizedModule.instSemiring.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.32 S))))))))) (LocalizedModule.mk.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.32) x (OfNat.ofNat.{u_1} (Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3))))) R (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.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3)))))) S x)) 1 (One.toOfNat1.{u_1} (Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3))))) R (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.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3)))))) S x)) (Submonoid.one.{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.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3)))) S)))) (LocalizedModule.mk.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.32) y (OfNat.ofNat.{u_1} (Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R 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(x + y) 1 = LocalizedModule.mk x 1 + LocalizedModule.mk y 1","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A]\n  (x y : A), LocalizedModule.mk (x + y) 1 = LocalizedModule.mk x 1 + LocalizedModule.mk y 1","typeReferences":[["Subtype"],["OreLocalization","oreSetComm"],["Membership","mem"],["CommSemiring"],["MulZeroOneClass","toMulOneClass"],["LocalizedModule","instSemiring"],["AddCommMonoid","toAddMonoid"],["AddMonoidWithOne","toAddMonoid"],["LocalizedModule","mk"],["OreLocalization"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["NonAssocSemiring","toMulZeroOneClass"],["SetLike","instMembership"],["Submonoid","instSetLike"],["instHAdd"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Semiring","toMonoidWithZero"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["Submonoid","one"],["OfNat","ofNat"],["CommSemiring","toCommMonoid"],["Algebra"],["Submonoid"],["HAdd","hAdd"],["Module","toDistribMulAction"],["One","toOfNat1"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Subtype"],["Eq","trans"],["OreLocalization","oreSetComm"],["Membership","mem"],["MulZeroOneClass","toMulOneClass"],["LocalizedModule","instSemiring"],["AddCommMonoid","toAddMonoid"],["LocalizedModule","mk"],["AddMonoidWithOne","toAddMonoid"],["congrArg"],["Semiring","toNonAssocSemiring"],["OreLocalization"],["Algebra","toModule"],["forall_congr"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["NonAssocSemiring","toMulZeroOneClass"],["AddSemigroup","toAdd"],["SetLike","instMembership"],["True"],["Submonoid","instSetLike"],["instHAdd"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["Submonoid","one"],["CommSemiring","toCommMonoid"],["OfNat","ofNat"],["Submonoid"],["HAdd","hAdd"],["implies_true"],["eq_self"],["Module","toDistribMulAction"],["One","toOfNat1"],["of_eq_true"],["AddMonoid","toAddSemigroup"],["OreLocalization","oreDiv"],["OreLocalization","add_oreDiv"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","lift","congr_simp"],"typeFallback":"forall 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[inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 : AddCommMonoid.{u_4} M''] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.40 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.14] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44) (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3)) (Semiring.toNonAssocSemiring.{u_4} (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44) (Module.End.instSemiring.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44))) (algebraMap.{u_1, u_4} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44))) (Algebra.toSMul.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_4} R M'' (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_4} R M'' (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_4} M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17) 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(MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3)))))) S x) x))), Eq.{max (succ u_3) (succ u_4)} (LinearMap.{u_1, u_1, u_3, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3))) M' M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44) (IsLocalizedModule.lift.{u_1, u_2, u_3, u_4} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3 S M M' M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.71 g h) (IsLocalizedModule.lift.{u_1, u_2, u_3, u_4} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3 S_1 M M' M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44 f_1 (Eq.ndrec.{0, succ u_1} (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.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3))))) S (fun (S : 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.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3))))) => IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.40 S f_1) (Eq.ndrec.{0, max (succ u_2) (succ u_3)} (LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.40) f (fun (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.40) => IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.40 S f) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.71 f_1 e_f) S_1 e_S) g_1 (Eq.ndrec.{0, succ u_1} (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.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3))))) S (fun (S : 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.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3))))) => forall (x : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3))))) R (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.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3)))))) S x)), IsUnit.{u_4} (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44) (Module.End.instMonoid.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44) (DFunLike.coe.{max (succ u_1) (succ u_4), succ u_1, succ u_4} (RingHom.{u_1, u_4} R (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44) (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3)) (Semiring.toNonAssocSemiring.{u_4} (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44) (Module.End.instSemiring.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44))) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44) (RingHom.instFunLike.{u_1, u_4} R (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44) (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3)) (Semiring.toNonAssocSemiring.{u_4} (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44) (Module.End.instSemiring.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44))) (algebraMap.{u_1, u_4} R (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3 (Module.End.instSemiring.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44) (Module.End.instAlgebra.{u_1, u_1, u_4} R R M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44 (smulCommClass_self.{u_1, u_4} R M'' (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_4} R M'' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_4} M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17) (Module.toDistribMulAction.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44))) (Algebra.toSMul.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_4} R M'' (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_4} R M'' (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_4} M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17) (Module.toDistribMulAction.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44))))) (Subtype.val.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3))))) R (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.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3)))))) S x) x))) h S_1 e_S)))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S S_1 : Submonoid R) (e_S : S = S_1) {M : Type u_2} {M' : Type u_3}\n  {M'' : Type u_4} [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M'] [inst_3 : AddCommMonoid M'']\n  [inst_4 : Module R M] [inst_5 : Module R M'] [inst_6 : Module R M''] (f f_1 : M →ₗ[R] M') (e_f : f = f_1)\n  [inst_7 : IsLocalizedModule S f] (g g_1 : M →ₗ[R] M''),\n  g = g_1 →\n    ∀ (h : ∀ (x : ↥S), IsUnit ((algebraMap R (Module.End R M'')) ↑x)),\n      IsLocalizedModule.lift S f g h = IsLocalizedModule.lift S_1 f_1 g_1 ⋯","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S S_1 : Submonoid R) (e_S : S = S_1) {M : Type u_2} {M' : Type u_3}\n  {M'' : Type u_4} [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M'] [inst_3 : AddCommMonoid M'']\n  [inst_4 : Module R M] [inst_5 : Module R M'] [inst_6 : Module R M''] (f f_1 : M →ₗ[R] M') (e_f : f = f_1)\n  [inst_7 : IsLocalizedModule S f] (g g_1 : M →ₗ[R] M''),\n  g = g_1 →\n    ∀ (h : ∀ (x : ↥S), IsUnit ((algebraMap R (Module.End R M'')) ↑x)),\n      IsLocalizedModule.lift S f g h = IsLocalizedModule.lift S_1 f_1 g_1 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx._hyg.161)))) (Algebra.toSMul.{u_6, u_7} R S inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx._hyg.161 (Ring.toSemiring.{u_7} S inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx._hyg.164) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx._hyg.170) M)) m x) (OfNat.ofNat.{u_7} S 0 (Zero.toOfNat0.{u_7} S (MulZeroClass.toZero.{u_7} S (NonUnitalNonAssocSemiring.toMulZeroClass.{u_7} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_7} S (NonAssocRing.toNonUnitalNonAssocRing.{u_7} S (Ring.toNonAssocRing.{u_7} S inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx._hyg.164)))))))))) -> (IsLocalizedModule.{u_6, u_7, u_8} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx._hyg.161 S S' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} S 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx._hyg.174) M (AlgHom.toLinearMap.{u_6, u_7, u_8} R S S' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx._hyg.161 (Ring.toSemiring.{u_7} S inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx._hyg.164) (Ring.toSemiring.{u_8} S' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx._hyg.167) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx._hyg.170 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx._hyg.174 f))","typeFull":"∀ {R : Type u_6} {S : Type u_7} {S' : Type u_8} [inst : CommSemiring R] [inst_1 : Ring S] [inst_2 : Ring S']\n  [inst_3 : Algebra R S] [inst_4 : Algebra R S'] (M : Submonoid R) (f : S →ₐ[R] S'),\n  (∀ x ∈ M, IsUnit ((algebraMap R S') x)) →\n    (∀ (y : S'), ∃ x, x.2 • y = f x.1) → (∀ (x : S), f x = 0 → ∃ m, m • x = 0) → IsLocalizedModule M f.toLinearMap","typeReadable":"∀ {R : Type u_6} {S : Type u_7} {S' : Type u_8} [inst : CommSemiring R] [inst_1 : Ring S] [inst_2 : Ring S']\n  [inst_3 : Algebra R S] [inst_4 : Algebra R S'] (M : Submonoid R) (f : S →ₐ[R] S'),\n  (∀ x ∈ M, IsUnit ((algebraMap R S') x)) →\n    (∀ (y : S'), ∃ x, x.2 • y = f x.1) → (∀ (x : S), f x = 0 → ∃ m, m • x = 0) → IsLocalizedModule M 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Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M') [IsLocalizedModule S f]\n  (s : ↥S), Function.Injective fun m => s • m","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M') [IsLocalizedModule S f]\n  (s : ↥S), Function.Injective fun m => s • 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x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx.68.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.118) -> Prop) (x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx.68.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.118 : Prod.{u_1, u_2} M (Subtype.{succ u_2} R (fun (x : R) => Membership.mem.{u_2, u_2} R (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3))))) (SetLike.instMembership.{u_2, u_2} (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.3))))) R (Submonoid.instSetLike.{u_2} R 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x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx.68.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.118 x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3743058621._hygCtx._hyg.70)","typeFull":"∀ {R : Type u_2} [inst : CommSemiring R] (S : Submonoid R) (M : Type u_1) [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] (m2 : M) (s2 : ↥S) (motive : (x : M × ↥S) → LocalizedModule.r S M (m2, s2) x → Prop)\n  (x : M × ↥S) (x_1 : LocalizedModule.r S M (m2, s2) x),\n  (∀ (m3 : M) (s3 : ↥S) (x : LocalizedModule.r S M (m2, s2) (m3, s3)), motive (m3, s3) x) → motive x x_1","typeReadable":"∀ {R : Type u_2} [inst : CommSemiring R] (S : Submonoid R) (M : Type u_1) [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] (m2 : M) (s2 : ↥S) (motive : (x : M × ↥S) → LocalizedModule.r S M (m2, s2) x → Prop)\n  (x : M × ↥S) (x_1 : LocalizedModule.r S M (m2, s2) x),\n  (∀ (m3 : M) (s3 : ↥S) (x : LocalizedModule.r S M (m2, s2) (m3, s3)), motive (m3, s3) x) → motive x x_1","typeReferences":[["Subtype"],["SetLike","instMembership"],["Prod","mk"],["Module"],["Submonoid","instSetLike"],["CommSemiring"],["CommSemiring","toSemiring"],["Membership","mem"],["MulZeroOneClass","toMulOneClass"],["Submonoid"],["Prod"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["LocalizedModule","r"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["Submonoid"],["Prod","casesOn"],["SetLike","instMembership"],["Semiring","toNonAssocSemiring"],["Subtype"],["Prod","mk"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["Membership","mem"],["LocalizedModule","r"],["MulZeroOneClass","toMulOneClass"],["NonAssocSemiring","toMulZeroOneClass"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","surj"],"typeFallback":"forall {R : Type.{u_1}} {inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72 : CommSemiring.{u_1} R} {M : Type.{u_2}} {M' : Type.{u_3}} {inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 : AddCommMonoid.{u_2} M} {inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 : AddCommMonoid.{u_3} M'} {inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80} {inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83} (S : Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 : CommSemiring.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 : Semiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29] {S : 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.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3))))} (a : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.32)))) (LocalizedModule.instSemiring.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.32 S)) (LocalizedModule.numeratorRingHom.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3 S A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.32) (algebraMap.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1134821990._hygCtx._hyg.32)) r))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A]\n  (r : R) (x : LocalizedModule S A),\n  (LocalizedModule.numeratorRingHom.comp (algebraMap R A)) r * x =\n    x * (LocalizedModule.numeratorRingHom.comp (algebraMap R A)) r","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A]\n  (r : R) (x : LocalizedModule S A),\n  (LocalizedModule.numeratorRingHom.comp (algebraMap R A)) r * x =\n    x * (LocalizedModule.numeratorRingHom.comp (algebraMap R A)) r","typeReferences":[["RingHom"],["OreLocalization","oreSetComm"],["CommSemiring"],["HMul","hMul"],["RingHom","instFunLike"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["LocalizedModule","instSemiring"],["DFunLike","coe"],["Semiring","toNonAssocSemiring"],["OreLocalization"],["Algebra","toModule"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Eq"],["RingHom","comp"],["NonAssocSemiring","toMulZeroOneClass"],["LocalizedModule","numeratorRingHom"],["LocalizedModule"],["NonUnitalNonAssocSemiring","toDistrib"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Distrib","toMul"],["Semiring","toMonoidWithZero"],["CommSemiring","toCommMonoid"],["Algebra"],["Submonoid"],["Module","toDistribMulAction"],["instHMul"],["algebraMap"],["Semiring"]],"valueReferences":[["RingHom"],["Membership","mem"],["HMul","hMul"],["LocalizedModule","instSemiring"],["LocalizedModule","mk"],["OreLocalization"],["Semiring","toNonAssocSemiring"],["mul_comm"],["DistribMulAction","toMulAction"],["NonAssocSemiring","toMulZeroOneClass"],["NonUnitalCommSemiring","toCommSemigroup"],["LocalizedModule","instMonoid"],["LocalizedModule","numeratorRingHom"],["SetLike","instMembership"],["LocalizedModule"],["NonUnitalNonAssocSemiring","toDistrib"],["CommMagma","toMul"],["Submonoid","instSetLike"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["LocalizedModule","induction_on"],["CommSemiring","toCommMonoid"],["Submonoid"],["Eq","refl"],["id"],["instHMul"],["Submonoid","instSubmonoidClass"],["Eq","mpr"],["LocalizedModule","mk_mul_mk"],["algebraMap"],["MulOneClass","toMulOne"],["Subtype"],["OreLocalization","oreSetComm"],["RingHom","instFunLike"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["congrArg"],["MulOne","toMul"],["CommSemiring","toNonUnitalCommSemiring"],["Algebra","toModule"],["MonoidWithZero","toMonoid"],["Monoid","toMulOneClass"],["RingHom","comp"],["Eq"],["CommSemigroup","toCommMagma"],["MulMemClass","toCommSemigroup"],["Algebra","commutes"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Distrib","toMul"],["Semiring","toMonoidWithZero"],["OfNat","ofNat"],["Submonoid","one"],["Module","toDistribMulAction"],["One","toOfNat1"],["Submonoid","mul"],["SubmonoidClass","toMulMemClass"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","lift'_add"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S : 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.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.3))))) {M : Type.{u_2}} {M'' : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.17 : AddCommMonoid.{u_4} M''] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.44 : Module.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.17] (g : LinearMap.{u_1, u_1, u_2, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.3))) M M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.44) (h : forall (x : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.3))))) R (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.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.3)))))) S x)), IsUnit.{u_4} (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.44) (Module.End.instMonoid.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.44) (DFunLike.coe.{max (succ u_1) (succ u_4), succ u_1, succ u_4} (RingHom.{u_1, u_4} R (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.44) (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.3)) (Semiring.toNonAssocSemiring.{u_4} (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.44) (Module.End.instSemiring.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.44))) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.44) (RingHom.instFunLike.{u_1, u_4} R (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.44) (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.3)) (Semiring.toNonAssocSemiring.{u_4} (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.44) (Module.End.instSemiring.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.44))) (algebraMap.{u_1, u_4} R (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.44) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3470330668._hygCtx._hyg.3 (Module.End.instSemiring.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R 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{R : Type.{u}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3 : CommSemiring.{u} R] {S : Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3))))} {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29 : Semiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.32 : Algebra.{u, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29], RingHom.{u_2, max u_2 u} A (OreLocalization.{u, u_2} R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3) S) A (DistribMulAction.toMulAction.{u, u_2} R A (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29)))) (Module.toDistribMulAction.{u, u_2} R A (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29))) (Algebra.toModule.{u, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.32)))) (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29) (Semiring.toNonAssocSemiring.{max u u_2} (OreLocalization.{u, u_2} R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3) S) A (DistribMulAction.toMulAction.{u, u_2} R A (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29)))) (Module.toDistribMulAction.{u, u_2} R A (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29))) (Algebra.toModule.{u, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.32)))) (LocalizedModule.instSemiring.{u, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.32 S))","typeFull":"{R : Type u} →\n  [inst : CommSemiring R] →\n    {S : Submonoid R} → {A : Type u_2} → [inst_1 : Semiring A] → [inst_2 : Algebra R A] → A →+* OreLocalization S A","typeReadable":"{R : Type u} →\n  [inst : CommSemiring R] →\n    {S : Submonoid R} → {A : Type u_2} → [inst_1 : Semiring A] → [inst_2 : Algebra R A] → A →+* OreLocalization S A","typeReferences":[["RingHom"],["OreLocalization","oreSetComm"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["MulZeroOneClass","toMulOneClass"],["LocalizedModule","instSemiring"],["AddCommMonoid","toAddMonoid"],["CommSemiring","toCommMonoid"],["Algebra"],["Submonoid"],["Module","toDistribMulAction"],["Semiring","toNonAssocSemiring"],["OreLocalization"],["Algebra","toModule"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["NonAssocSemiring","toMulZeroOneClass"],["Semiring"]],"valueReferences":[["MulOneClass","toMulOne"],["MonoidHom","mk"],["Subtype"],["OreLocalization","oreSetComm"],["Membership","mem"],["LocalizedModule","numeratorRingHom","_proof_1"],["MulZeroOneClass","toMulOneClass"],["LocalizedModule","numeratorRingHom","_proof_3"],["AddCommMonoid","toAddMonoid"],["LocalizedModule","instSemiring"],["LocalizedModule","mk"],["Semiring","toNonAssocSemiring"],["OreLocalization"],["Algebra","toModule"],["LocalizedModule","numeratorRingHom","_proof_4"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["NonAssocSemiring","toMulZeroOneClass"],["SetLike","instMembership"],["MulOne","toOne"],["Submonoid","instSetLike"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["Submonoid","one"],["CommSemiring","toCommMonoid"],["OfNat","ofNat"],["Submonoid"],["Module","toDistribMulAction"],["One","toOfNat1"],["OneHom","mk"],["RingHom","mk"],["LocalizedModule","numeratorRingHom","_proof_2"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","smul_inj"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.3 : CommSemiring.{u_1} R] {S : 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.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.3))))} {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.40 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.14] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.40) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.71 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.40 S f] (s : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R 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(CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.3))))) R (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.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.3)))))) S x)) M' M' (instHSMul.{u_1, u_3} (Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.3))))) R (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.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.3)))))) S x)) M' (Submonoid.smul.{u_1, u_3} R M' (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.3)))) (SMulZeroClass.toSMul.{u_1, u_3} R M' (AddZero.toZero.{u_3} M' (AddZeroClass.toAddZero.{u_3} M' (AddMonoid.toAddZeroClass.{u_3} M' (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.14)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R M' (AddMonoid.toAddZeroClass.{u_3} M' (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.14)) (DistribMulAction.toDistribSMul.{u_1, u_3} R M' (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.14) (Module.toDistribMulAction.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.145617272._hygCtx._hyg.40)))) S)) s m₂)) (Eq.{succ u_3} M' m₁ m₂)","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M') [IsLocalizedModule S f]\n  (s : ↥S) (m₁ m₂ : M'), s • m₁ = s • m₂ ↔ m₁ = m₂","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M') [IsLocalizedModule S f]\n  (s : ↥S) (m₁ m₂ : M'), s • m₁ = s • m₂ ↔ m₁ = m₂","typeReferences":[["IsLocalizedModule"],["Subtype"],["Module"],["Membership","mem"],["CommSemiring"],["Submonoid","smul"],["MulZeroOneClass","toMulOneClass"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["LinearMap"],["AddZeroClass","toAddZero"],["Submonoid"],["Module","toDistribMulAction"],["Iff"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Subtype"],["SetLike","instMembership"],["IsLocalizedModule","smul_injective"],["Submonoid","instSetLike"],["Membership","mem"],["CommSemiring","toSemiring"],["Submonoid","smul"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submonoid"],["Function","Injective","eq_iff"],["Module","toDistribMulAction"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["instHSMul"],["AddZero","toZero"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Module","LocalizedModule","Basic",0,"IsLocalizedModule","isTorsionFree","_simp_1_1"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.GroupWithZero.Regular.985349491._hygCtx._hyg.3 : MulZeroClass.{u_1} R] [inst._@.Mathlib.Algebra.GroupWithZero.Regular.985349491._hygCtx._hyg.6 : IsCancelMulZero.{u_1} R (MulZeroClass.toMul.{u_1} R inst._@.Mathlib.Algebra.GroupWithZero.Regular.985349491._hygCtx._hyg.3) (MulZeroClass.toZero.{u_1} R inst._@.Mathlib.Algebra.GroupWithZero.Regular.985349491._hygCtx._hyg.3)] {a : R} [inst._@.Mathlib.Algebra.GroupWithZero.Regular.985349491._hygCtx._hyg.10 : Nontrivial.{u_1} R], Eq.{1} Prop (IsRegular.{u_1} R (MulZeroClass.toMul.{u_1} R inst._@.Mathlib.Algebra.GroupWithZero.Regular.985349491._hygCtx._hyg.3) a) (Ne.{succ u_1} R a (OfNat.ofNat.{u_1} R 0 (Zero.toOfNat0.{u_1} R (MulZeroClass.toZero.{u_1} R inst._@.Mathlib.Algebra.GroupWithZero.Regular.985349491._hygCtx._hyg.3))))","typeFull":"∀ {R : Type u_1} [inst : MulZeroClass R] [IsCancelMulZero R] {a : R} [Nontrivial R], IsRegular a = (a ≠ 0)","typeReadable":"∀ {R : Type u_1} [inst : MulZeroClass R] [IsCancelMulZero R] {a : R} [Nontrivial R], IsRegular a = (a ≠ 0)","typeReferences":[["MulZeroClass","toZero"],["MulZeroClass"],["Nontrivial"],["MulZeroClass","toMul"],["Zero","toOfNat0"],["Ne"],["IsRegular"],["Eq"],["IsCancelMulZero"],["OfNat","ofNat"]],"valueReferences":[["MulZeroClass","toZero"],["MulZeroClass","toMul"],["Zero","toOfNat0"],["Ne"],["isRegular_iff_ne_zero"],["IsRegular"],["OfNat","ofNat"],["propext"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","smulOfIsLocalization","_proof_2"],"typeFallback":"forall {R : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.824820056._hygCtx._hyg.3 : CommSemiring.{u_2} R] {S : Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.824820056._hygCtx._hyg.3))))} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.824820056._hygCtx._hyg.9 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.824820056._hygCtx._hyg.12 : Module.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.824820056._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.824820056._hygCtx._hyg.9] (T : Type.{u_3}) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.824820056._hygCtx._hyg.17 : CommSemiring.{u_3} T] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.824820056._hygCtx._hyg.20 : Algebra.{u_2, u_3} R T inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.824820056._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_3} T inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.824820056._hygCtx._hyg.17)] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.824820056._hygCtx._hyg.24 : IsLocalization.{u_2, u_3} R 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{R : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3 : CommSemiring.{u_2} R] (S : Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3))))) {M : Type.{u_3}} {M'' : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.11 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 : AddCommMonoid.{u_1} M''] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.36 : Module.{u_2, u_3} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.11] 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44) (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3)) (Semiring.toNonAssocSemiring.{u_1} (Module.End.{u_2, u_1} R M'' (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44) (Module.End.instSemiring.{u_2, u_1} R M'' (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44))) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => Module.End.{u_2, u_1} R M'' 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44) (Module.End.instSemiring.{u_2, u_1} R M'' (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44))) (algebraMap.{u_2, u_1} R (Module.End.{u_2, u_1} R M'' (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3 (Module.End.instSemiring.{u_2, u_1} R M'' (CommSemiring.toSemiring.{u_2} R 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[inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.40 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.14] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.36 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(CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.83 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3)))) (LinearMap.addCommMonoid.{u_1, u_1, u_3, u_7} R R M' N' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.87 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3)))) (LinearMap.module.{u_1, u_1, u_1, u_2, u_6} R R R M N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.36 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.87 g inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3846601649._hygCtx._hyg.100) h))) True)","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] {N : Type u_6} {N' : Type u_7} [inst_6 : AddCommMonoid N] [inst_7 : AddCommMonoid N']\n  [inst_8 : Module R N] [inst_9 : Module R N'] (g : N →ₗ[R] N') [inst_10 : IsLocalizedModule S g] (h : M →ₗ[R] N),\n  Function.Surjective ⇑h → Function.Surjective ⇑((IsLocalizedModule.map S f g) h) = True","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] {N : Type u_6} {N' : Type u_7} [inst_6 : AddCommMonoid N] [inst_7 : AddCommMonoid N']\n  [inst_8 : Module R N] [inst_9 : Module R N'] (g : N →ₗ[R] N') [inst_10 : IsLocalizedModule S g] (h : M →ₗ[R] N),\n  Function.Surjective ⇑h → Function.Surjective ⇑((IsLocalizedModule.map S f g) h) = True","typeReferences":[["IsLocalizedModule"],["smulCommClass_self"],["LinearMap","instFunLike"],["CommMonoid","toMonoid"],["Module"],["CommSemiring"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["DistribMulAction","toMulAction"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["IsLocalizedModule","map"],["True"],["CommSemiring","toSemiring"],["LinearMap","module"],["LinearMap"],["Function","Surjective"],["CommSemiring","toCommMonoid"],["Submonoid"],["LinearMap","addCommMonoid"],["Module","toDistribMulAction"]],"valueReferences":[["smulCommClass_self"],["IsLocalizedModule","map"],["IsLocalizedModule","map_surjective"],["CommMonoid","toMonoid"],["LinearMap","instFunLike"],["CommSemiring","toSemiring"],["LinearMap","module"],["eq_true"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["Function","Surjective"],["DFunLike","coe"],["CommSemiring","toCommMonoid"],["LinearMap","addCommMonoid"],["Module","toDistribMulAction"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["DistribMulAction","toMulAction"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","liftOfLE","_proof_1"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S₁ : 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.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3))))) (S₂ : 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.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3))))), (LE.le.{u_1} (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.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3))))) (Preorder.toLE.{u_1} (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R 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S₂","typeReferences":[["PartialOrder","toPreorder"],["Subtype"],["SetLike","instMembership"],["Submonoid","instSetLike"],["CommSemiring"],["CommSemiring","toSemiring"],["Membership","mem"],["MulZeroOneClass","toMulOneClass"],["Subtype","val"],["Submonoid"],["Submonoid","instPartialOrder"],["Semiring","toNonAssocSemiring"],["LE","le"],["Preorder","toLE"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["Submonoid"],["SetLike","instMembership"],["Semiring","toNonAssocSemiring"],["Submonoid","instSetLike"],["Subtype","property"],["CommSemiring","toSemiring"],["Membership","mem"],["MulZeroOneClass","toMulOneClass"],["Subtype","val"],["NonAssocSemiring","toMulZeroOneClass"]]},{"isProp":true,"kind":"definition","name":["LocalizedModule","divBy","match_3"],"typeFallback":"forall {R : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3 : CommSemiring.{u_2} R] {S : Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R 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x_1","typeReferences":[["LocalizedModule","r","setoid"],["Subtype"],["SetLike","instMembership"],["Prod","mk"],["Module"],["Submonoid","instSetLike"],["CommSemiring"],["CommSemiring","toSemiring"],["Membership","mem"],["MulZeroOneClass","toMulOneClass"],["Submonoid"],["Prod"],["AddCommMonoid"],["HasEquiv","Equiv"],["Semiring","toNonAssocSemiring"],["instHasEquivOfSetoid"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["LocalizedModule","r","setoid"],["Subtype"],["SetLike","instMembership"],["Prod","mk"],["Submonoid","instSetLike"],["Membership","mem"],["CommSemiring","toSemiring"],["MulZeroOneClass","toMulOneClass"],["Prod","casesOn"],["Submonoid"],["Prod"],["Semiring","toNonAssocSemiring"],["HasEquiv","Equiv"],["instHasEquivOfSetoid"],["NonAssocSemiring","toMulZeroOneClass"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","lift","eq_1"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S : 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.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3))))) {M : Type.{u_2}} {M' : Type.{u_3}} {M'' : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 : AddCommMonoid.{u_4} M''] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.40 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.14] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44 : Module.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.40) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.71 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.40 S f] (g : LinearMap.{u_1, u_1, u_2, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3))) M M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44) (h : forall (x : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R 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u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44) (DFunLike.coe.{max (succ u_1) (succ u_4), succ u_1, succ u_4} (RingHom.{u_1, u_4} R (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44) (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3)) (Semiring.toNonAssocSemiring.{u_4} (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44 (LocalizedModule.lift'._proof_2.{u_1, u_4} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3 M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44) (Algebra.toSMul.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3)) (LocalizedModule.mkLinearMap._proof_3.{u_1, u_4} R 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(LinearEquiv.toLinearMap.{u_1, u_1, u_3, max u_1 u_2} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3))) (IsLocalizedModule.iso._proof_1.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) (IsLocalizedModule.iso._proof_1.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) M' (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.36) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.14 (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_2} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.11 (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.36)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.40 (OreLocalization.instModuleOfIsScalarTower.{u_1, u_2, u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.36 R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.36 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3)) 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(OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_2} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.11 (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.36)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.14 (OreLocalization.instModuleOfIsScalarTower.{u_1, u_2, u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.36 R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.36 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3)) (LocalizedModule.mkLinearMap._proof_3.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3 M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.36) (LocalizedModule.algebra'._proof_4.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.40 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3))) (IsLocalizedModule.iso._proof_1.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) (IsLocalizedModule.iso._proof_1.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) (IsLocalizedModule.iso.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3 S M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.40 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.71))))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} {M'' : Type u_4}\n  [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M'] [inst_3 : AddCommMonoid M''] [inst_4 : Module R M]\n  [inst_5 : Module R M'] [inst_6 : Module R M''] (f : M →ₗ[R] M') [inst_7 : IsLocalizedModule S f] (g : M →ₗ[R] M'')\n  (h : ∀ (x : ↥S), IsUnit ((algebraMap R (Module.End R M'')) ↑x)),\n  IsLocalizedModule.lift S f g h = LocalizedModule.lift S g h ∘ₗ ↑(IsLocalizedModule.iso S f).symm","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} {M'' : Type u_4}\n  [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M'] [inst_3 : AddCommMonoid M''] [inst_4 : Module R M]\n  [inst_5 : Module R M'] [inst_6 : Module R M''] (f : M →ₗ[R] M') [inst_7 : IsLocalizedModule S f] (g : M →ₗ[R] M'')\n  (h : ∀ (x : ↥S), IsUnit ((algebraMap R (Module.End R M'')) ↑x)),\n  IsLocalizedModule.lift S f g h = LocalizedModule.lift S g h ∘ₗ ↑(IsLocalizedModule.iso S f).symm","typeReferences":[["IsLocalizedModule","lift","_proof_1"],["IsLocalizedModule"],["RingHom"],["Subtype"],["IsLocalizedModule","iso","_proof_1"],["OreLocalization","oreSetComm"],["Module"],["Membership","mem"],["CommSemiring"],["LinearMap","comp"],["RingHom","instFunLike"],["MulZeroOneClass","toMulOneClass"],["Subtype","val"],["Algebra","id"],["DFunLike","coe"],["LocalizedModule","lift"],["AddCommMonoid"],["Module","End","instAlgebra"],["LocalizedModule","mkLinearMap","_proof_3"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["Algebra","toSMul"],["Semiring","toModule"],["IsLocalizedModule","lift"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["Module","End"],["SetLike","instMembership"],["Module","End","instSemiring"],["OreLocalization","instAddCommMonoidOreLocalization"],["LocalizedModule"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["OreLocalization","instModuleOfIsScalarTower"],["Semiring","toMonoidWithZero"],["LinearEquiv","toLinearMap"],["IsUnit"],["LinearMap"],["LinearEquiv","symm"],["CommSemiring","toCommMonoid"],["Submonoid"],["Module","toDistribMulAction"],["IsLocalizedModule","iso"],["LocalizedModule","lift'","_proof_2"],["LocalizedModule","algebra'","_proof_4"],["Module","End","instMonoid"],["algebraMap"]],"valueReferences":[["Semiring","toNonAssocSemiring"],["RingHom","id"],["Eq","refl"],["CommSemiring","toSemiring"],["IsLocalizedModule","lift"],["LinearMap"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","lift'","_proof_4"],"typeFallback":"forall {R : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3 : CommSemiring.{u_2} R] (S : Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3))))) {M : Type.{u_1}} {M'' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.11 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.17 : AddCommMonoid.{u_3} M''] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.36 : Module.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.44 : Module.{u_2, u_3} R M'' (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.17] (g : LinearMap.{u_2, u_2, u_1, u_3} R R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3) (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3))) M M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.44) (h : forall (x : Subtype.{succ u_2} R (fun (x : R) => Membership.mem.{u_2, u_2} R (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3))))) (SetLike.instMembership.{u_2, u_2} (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3))))) R (Submonoid.instSetLike.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3)))))) S x)), IsUnit.{u_3} (Module.End.{u_2, u_3} R M'' (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.44) (Module.End.instMonoid.{u_2, u_3} R M'' (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.44) (DFunLike.coe.{max (succ u_2) (succ u_3), succ u_2, succ u_3} (RingHom.{u_2, u_3} R (Module.End.{u_2, u_3} R M'' (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.44) (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3)) (Semiring.toNonAssocSemiring.{u_3} (Module.End.{u_2, u_3} R M'' (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.44) (Module.End.instSemiring.{u_2, u_3} R M'' (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.44))) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => Module.End.{u_2, u_3} R M'' 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.44) (Module.End.instSemiring.{u_2, u_3} R M'' (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.44))) (algebraMap.{u_2, u_3} R (Module.End.{u_2, u_3} R M'' (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.44) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3329919914._hygCtx._hyg.3 (Module.End.instSemiring.{u_2, u_3} R M'' (CommSemiring.toSemiring.{u_2} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.156091895._hygCtx._hyg.40 S f) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.156091895._hygCtx._hyg.71 f_1 e_f) m_1 s_1)))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f f_1 : M →ₗ[R] M') (e_f : f = f_1)\n  [inst_5 : IsLocalizedModule S f] (m m_1 : M),\n  m = m_1 → ∀ (s s_1 : ↥S), s = s_1 → IsLocalizedModule.mk' f m s = IsLocalizedModule.mk' f_1 m_1 s_1","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f f_1 : M →ₗ[R] M') (e_f : f = f_1)\n  [inst_5 : IsLocalizedModule S f] (m m_1 : M),\n  m = m_1 → ∀ (s s_1 : ↥S), s = s_1 → IsLocalizedModule.mk' f m s = IsLocalizedModule.mk' f_1 m_1 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[inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.3 : CommSemiring.{u} R] {S : Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.3))))} {T : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.17 : CommSemiring.{u_1} T] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.20 : Algebra.{u, u_1} R T inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} T inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.17)] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.24 : IsLocalization.{u, u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.3 S T inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.20] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.29 : Semiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.32 : Algebra.{u, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.29] (x : T) (p₁ : LocalizedModule.{u, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.29))) (Algebra.toModule.{u, u_2} R A 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(LocalizedModule.smulOfIsLocalization.{u, u_2, u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.29))) (Algebra.toModule.{u, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.32) T inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.24)) x p₂)) (HSMul.hSMul.{u_1, max u u_2, max u u_2} T (LocalizedModule.{u, u_2} R 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(NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.29))) (Algebra.toModule.{u, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.32)) (LocalizedModule.{u, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.29))) (Algebra.toModule.{u, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.3 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.32)) (LocalizedModule.instSemiring.{u, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2288305372._hygCtx._hyg.32 S)))))) p₁ p₂))","typeFull":"∀ {R : Type u} [inst : CommSemiring R] {S : Submonoid R} {T : Type u_1} [inst_1 : CommSemiring T] [inst_2 : Algebra R T]\n  [inst_3 : IsLocalization S T] {A : Type u_2} [inst_4 : Semiring A] [inst_5 : Algebra R A] (x : T)\n  (p₁ p₂ : LocalizedModule S A), p₁ * x • p₂ = x • (p₁ * p₂)","typeReadable":"∀ {R : Type u} [inst : CommSemiring R] {S : Submonoid R} {T : Type u_1} [inst_1 : CommSemiring T] [inst_2 : Algebra R T]\n  [inst_3 : IsLocalization S T] {A : Type u_2} [inst_4 : Semiring A] [inst_5 : Algebra R A] (x : T)\n  (p₁ p₂ : LocalizedModule S A), p₁ * x • p₂ = x • (p₁ * 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1165189301._hygCtx._hyg.12))) (IsScalarTower.right.{u, u} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1165189301._hygCtx._hyg.3 (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1165189301._hygCtx._hyg.3) (Algebra.id.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1165189301._hygCtx._hyg.3))) (OreLocalization.instModuleOfIsScalarTower.{u, v, u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1165189301._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1165189301._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1165189301._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1165189301._hygCtx._hyg.12 R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1165189301._hygCtx._hyg.3) 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1165189301._hygCtx._hyg.12) (LocalizedModule.{u, v} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1165189301._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1165189301._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1165189301._hygCtx._hyg.12) (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1165189301._hygCtx._hyg.3) (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1165189301._hygCtx._hyg.3) (OreLocalization.instAddCommMonoidOreLocalization.{u, v} R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1165189301._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1165189301._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1165189301._hygCtx._hyg.9 (Module.toDistribMulAction.{u, v} R M (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1165189301._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1165189301._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1165189301._hygCtx._hyg.12)) (OreLocalization.instAddCommMonoidOreLocalization.{u, v} R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1165189301._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1165189301._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1165189301._hygCtx._hyg.9 (Module.toDistribMulAction.{u, v} R M (CommSemiring.toSemiring.{u} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.32)) (LocalizedModule.instSemiring.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 A (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.32 S))))))))) SubNegMonoid.zsmul_zero'._autoParam","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Ring A] [inst_2 : Algebra R A] {S : Submonoid R},\n  autoParam (∀ (a : LocalizedModule S A), SubNegMonoid.zsmul 0 a = 0) SubNegMonoid.zsmul_zero'._autoParam","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Ring A] [inst_2 : Algebra R A] {S : Submonoid R},\n  autoParam (∀ (a : LocalizedModule S A), SubNegMonoid.zsmul 0 a = 0) SubNegMonoid.zsmul_zero'._autoParam","typeReferences":[["AddMonoid","toZero"],["Ring","toNonAssocRing"],["OreLocalization","oreSetComm"],["AddCommGroup","toAddGroup"],["SubNegMonoid","zsmul_zero'","_autoParam"],["CommSemiring"],["MulZeroOneClass","toMulOneClass"],["LocalizedModule","instSemiring"],["AddCommMonoid","toAddMonoid"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["MonoidWithZero","toMonoid"],["Zero","toOfNat0"],["AddGroup","toSubNegMonoid"],["Eq"],["Semiring","toNonUnitalSemiring"],["NonAssocSemiring","toMulZeroOneClass"],["NonAssocRing","toNonUnitalNonAssocRing"],["LocalizedModule"],["SubNegMonoid","zsmul"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["OfNat","ofNat"],["CommSemiring","toCommMonoid"],["Int"],["Ring","toSemiring"],["Algebra"],["Ring","toAddCommGroup"],["Submonoid"],["Module","toDistribMulAction"],["instOfNat"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["inferInstance"],["autoParam"],["OreLocalization","instAddCommGroup"],["Ring"]],"valueReferences":[["Ring","toNonAssocRing"],["LocalizedModule"],["OreLocalization","oreSetComm"],["AddCommGroup","toAddGroup"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["SubNegMonoid","zsmul_zero'"],["CommSemiring","toCommMonoid"],["Ring","toSemiring"],["Ring","toAddCommGroup"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Module","toDistribMulAction"],["Algebra","toModule"],["MonoidWithZero","toMonoid"],["inferInstance"],["AddGroup","toSubNegMonoid"],["OreLocalization","instAddCommGroup"],["NonAssocRing","toNonUnitalNonAssocRing"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","liftOn₂","_proof_1"],"typeFallback":"forall {R : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1486043885._hygCtx._hyg.3 : CommSemiring.{u_2} R] {S : Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R (NonAssocSemiring.toMulZeroOneClass.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1486043885._hygCtx._hyg.3))))} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1486043885._hygCtx._hyg.9 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1486043885._hygCtx._hyg.12 : Module.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1486043885._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1486043885._hygCtx._hyg.9] {α : Type.{u_3}} (f : (Prod.{u_1, u_2} M (Subtype.{succ u_2} R (fun (x : R) => Membership.mem.{u_2, u_2} R (Submonoid.{u_2} R (MulZeroOneClass.toMulOneClass.{u_2} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3618933012._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3618933012._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3618933012._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3618933012._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3618933012._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3618933012._hygCtx._hyg.40 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3618933012._hygCtx._hyg.3)))) f m))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (m : M) (s : ↥S),\n  (IsLocalizedModule.iso S f) 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3 M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.36 M₁ M₂ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.87 S₁_1 S₂_1 (Eq.ndrec.{0, succ u_1} (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.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3))))) S₁ (fun (S₁ : Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R 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(Submonoid.instPartialOrder.{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.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3))))))) S₁ S₂_1) (Eq.ndrec.{0, succ u_1} (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.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3))))) S₂ (fun (S₂ : 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.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3))))) => LE.le.{u_1} (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.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3))))) (Preorder.toLE.{u_1} (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.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3))))) (PartialOrder.toPreorder.{u_1} (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.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3))))) (Submonoid.instPartialOrder.{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.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3))))))) S₁ S₂) h S₂_1 e_S₂) S₁_1 e_S₁) f₁_1 f₂_1 (Eq.ndrec.{0, succ u_1} (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.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3))))) S₁ (fun (S₁ : 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.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3))))) => IsLocalizedModule.{u_1, u_2, u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3 M M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.83 S₁ f₁_1) (Eq.ndrec.{0, max (succ u_2) (succ u_6)} (LinearMap.{u_1, u_1, u_2, u_6} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3))) M M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.83) f₁ (fun (f₁ : LinearMap.{u_1, u_1, u_2, u_6} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3))) M M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.83) => IsLocalizedModule.{u_1, u_2, u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3 M M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.83 S₁ f₁) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.118 f₁_1 e_f₁) S₁_1 e_S₁) (Eq.ndrec.{0, succ u_1} (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.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3))))) S₂ (fun (S₂ : 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.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3))))) => IsLocalizedModule.{u_1, u_2, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3 M M₂ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.87 S₂ f���_1) (Eq.ndrec.{0, max (succ u_2) (succ u_7)} (LinearMap.{u_1, u_1, u_2, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3))) M M₂ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.87) f₂ (fun (f₂ : LinearMap.{u_1, u_1, u_2, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3))) M M₂ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.87) => IsLocalizedModule.{u_1, u_2, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3 M M₂ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.87 S₂ f₂) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.122 f₂_1 e_f₂) S₂_1 e_S₂))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {M : Type u_2} [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {M₁ : Type u_6}\n  {M₂ : Type u_7} [inst_3 : AddCommMonoid M₁] [inst_4 : AddCommMonoid M₂] [inst_5 : Module R M₁] [inst_6 : Module R M₂]\n  (S₁ S₁_1 : Submonoid R) (e_S₁ : S₁ = S₁_1) (S₂ S₂_1 : Submonoid R) (e_S₂ : S₂ = S₂_1) (h : S₁ ≤ S₂)\n  (f₁ f₁_1 : M →ₗ[R] M₁) (e_f₁ : f₁ = f₁_1) (f₂ f₂_1 : M →ₗ[R] M₂) (e_f₂ : f₂ = f₂_1) [inst_7 : IsLocalizedModule S₁ f₁]\n  [inst_8 : IsLocalizedModule S₂ f₂],\n  IsLocalizedModule.liftOfLE S₁ S₂ h f₁ f₂ = IsLocalizedModule.liftOfLE S₁_1 S₂_1 ⋯ f₁_1 f₂_1","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {M : Type u_2} [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {M₁ : Type u_6}\n  {M₂ : Type u_7} [inst_3 : AddCommMonoid M₁] [inst_4 : AddCommMonoid M₂] [inst_5 : Module R M₁] [inst_6 : Module R M₂]\n  (S₁ S₁_1 : Submonoid R) (e_S₁ : S₁ = S₁_1) (S₂ S₂_1 : Submonoid R) (e_S₂ : S₂ = S₂_1) (h : S₁ ≤ S₂)\n  (f₁ f₁_1 : M →ₗ[R] M₁) (e_f₁ : f₁ = f₁_1) (f₂ f₂_1 : M →ₗ[R] M₂) (e_f₂ : f₂ = f₂_1) [inst_7 : IsLocalizedModule S₁ f₁]\n  [inst_8 : IsLocalizedModule S₂ f₂],\n  IsLocalizedModule.liftOfLE S₁ S₂ h f₁ f₂ = IsLocalizedModule.liftOfLE S₁_1 S₂_1 ⋯ f₁_1 f₂_1","typeReferences":[["IsLocalizedModule"],["PartialOrder","toPreorder"],["Module"],["CommSemiring"],["CommSemiring","toSemiring"],["MulZeroOneClass","toMulOneClass"],["LinearMap"],["Submonoid"],["IsLocalizedModule","liftOfLE"],["Submonoid","instPartialOrder"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["LE","le"],["Eq","ndrec"],["Eq"],["Preorder","toLE"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["IsLocalizedModule"],["PartialOrder","toPreorder"],["CommSemiring","toSemiring"],["MulZeroOneClass","toMulOneClass"],["LinearMap"],["Submonoid"],["IsLocalizedModule","liftOfLE"],["Submonoid","instPartialOrder"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Eq","refl"],["LE","le"],["Eq","ndrec"],["Eq","rec"],["Eq"],["Preorder","toLE"],["NonAssocSemiring","toMulZeroOneClass"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Module","LocalizedModule","Basic",0,"LocalizedModule","smul_add_aux"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.3 : CommSemiring.{u} R] {S : Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.3))))} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.9 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.12 : Module.{u, v} R M (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.9] {T : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.17 : CommSemiring.{u_1} T] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.20 : Algebra.{u, u_1} R T inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} T inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.17)] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.24 : IsLocalization.{u, u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.3 S T inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.20] (x : T) (p : LocalizedModule.{u, v} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.12) (q : LocalizedModule.{u, v} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.12), Eq.{max (succ u) (succ v)} (LocalizedModule.{u, v} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.12) (HSMul.hSMul.{u_1, max u v, max u v} T (LocalizedModule.{u, v} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.12) (LocalizedModule.{u, v} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4136493946._hygCtx._hyg.9 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CommSemiring T] [inst_4 : Algebra R T] [inst_5 : IsLocalization S T] (x : T)\n  (p q : LocalizedModule S M), x • (p + q) = x • p + x • q","typeReadable":"∀ {R : Type u} [inst : CommSemiring R] {S : Submonoid R} {M : Type v} [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  {T : Type u_1} [inst_3 : CommSemiring T] [inst_4 : Algebra R T] [inst_5 : IsLocalization S T] (x : T)\n  (p q : LocalizedModule S M), x • (p + q) = x • p + x • 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 S f] {N : Type.{u_5}} {N' : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 : AddCommMonoid.{u_5} N] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 : AddCommMonoid.{u_2} N'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 : Module.{u_3, u_5} R N (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87 : Module.{u_3, u_2} R N' (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80] (g : LinearMap.{u_3, u_3, u_5, u_2} R R (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) N N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3)))))))) (IsLocalizedModule.lift.{u_3, u_4, u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3 S M M' N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.71 (LinearMap.comp.{u_3, u_3, u_3, u_4, u_5, u_2} R R R M N N' (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) (RingHomCompTriple.ids.{u_3, u_3} R R (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R (CommSemiring.toSemiring.{u_3} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.71 (LinearMap.comp.{u_3, u_3, u_3, u_4, u_5, u_2} R R R M N N' (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) (RingHomCompTriple.ids.{u_3, u_3} R R (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R (CommSemiring.toSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3)))) g h₂) (IsLocalizedModule.map_units.{u_3, u_5, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3 N N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87 S g inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.100)))","typeFull":"∀ {R : Type u_3} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_4} {M' : Type u_1} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] {N : Type u_5} {N' : Type u_2} [inst_6 : AddCommMonoid N] [inst_7 : AddCommMonoid N']\n  [inst_8 : Module R N] [inst_9 : Module R N'] (g : N →ₗ[R] N') [inst_10 : IsLocalizedModule S g] (h₁ h₂ : M →ₗ[R] N),\n  IsLocalizedModule.lift S f (g ∘ₗ (h₁ + h₂)) ⋯ =\n    IsLocalizedModule.lift S f (g ∘ₗ h₁) ⋯ + IsLocalizedModule.lift S f (g ∘ₗ h₂) ⋯","typeReadable":"∀ {R : Type u_3} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_4} {M' : Type u_1} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] {N : Type u_5} {N' : Type u_2} [inst_6 : AddCommMonoid N] [inst_7 : AddCommMonoid N']\n  [inst_8 : Module R N] [inst_9 : Module R N'] (g : N →ₗ[R] N') [inst_10 : IsLocalizedModule S g] (h₁ h₂ : M →ₗ[R] N),\n  IsLocalizedModule.lift S f (g ∘ₗ (h₁ + h₂)) ⋯ =\n    IsLocalizedModule.lift S f (g ∘ₗ h₁) ⋯ + IsLocalizedModule.lift S f (g ∘ₗ h₂) ⋯","typeReferences":[["IsLocalizedModule"],["instHAdd"],["RingHomCompTriple","ids"],["Module"],["CommSemiring"],["CommSemiring","toSemiring"],["LinearMap","comp"],["MulZeroOneClass","toMulOneClass"],["LinearMap"],["LinearMap","addCommMonoid"],["HAdd","hAdd"],["Submonoid"],["IsLocalizedModule","map_units"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["AddCommMonoid","toAddCommSemigroup"],["RingHom","id"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Eq"],["IsLocalizedModule","lift"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["Eq","trans"],["RingHomCompTriple","ids"],["LinearMap","comp"],["MulZeroOneClass","toMulOneClass"],["IsLocalizedModule","lift_comp"],["congrArg"],["IsLocalizedModule","map_units"],["Semiring","toNonAssocSemiring"],["LinearMap","comp","congr_simp"],["RingHom","id"],["congr"],["LinearMap","instAdd"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["LinearMap","comp_add"],["IsLocalizedModule","lift"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["True"],["instHAdd"],["CommSemiring","toSemiring"],["LinearMap"],["IsLocalizedModule","ext"],["Submonoid"],["HAdd","hAdd"],["LinearMap","addCommMonoid"],["eq_self"],["AddCommMonoid","toAddCommSemigroup"],["of_eq_true"],["Eq","refl"],["IsLocalizedModule","lift","congr_simp"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","liftOn₂_mk"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2372657690._hygCtx._hyg.3 : CommSemiring.{u} R] {S : Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2372657690._hygCtx._hyg.3))))} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2372657690._hygCtx._hyg.9 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2372657690._hygCtx._hyg.12 : Module.{u, v} R M (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2372657690._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2372657690._hygCtx._hyg.9] {α : Type.{u_2}} (f : (Prod.{v, u} M (Subtype.{succ u} R (fun (x : R) => Membership.mem.{u, u} R (Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2372657690._hygCtx._hyg.3))))) (SetLike.instMembership.{u, u} (Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2372657690._hygCtx._hyg.3))))) R (Submonoid.instSetLike.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2372657690._hygCtx._hyg.3)))))) S x))) -> α) (wd : forall (p : Prod.{v, u} M (Subtype.{succ u} R (fun (x : R) => Membership.mem.{u, u} R (Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2372657690._hygCtx._hyg.3))))) (SetLike.instMembership.{u, u} (Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2372657690._hygCtx._hyg.3))))) R (Submonoid.instSetLike.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2372657690._hygCtx._hyg.3)))))) S x))) (q : Prod.{v, u} M (Subtype.{succ u} R (fun (x : R) => Membership.mem.{u, u} R (Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2372657690._hygCtx._hyg.3))))) (SetLike.instMembership.{u, u} (Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2372657690._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2372657690._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2372657690._hygCtx._hyg.12 m' s') f wd) (f (Prod.mk.{v, u} M (Subtype.{succ u} R (fun (x : R) => Membership.mem.{u, u} R (Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2372657690._hygCtx._hyg.3))))) (SetLike.instMembership.{u, u} (Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2372657690._hygCtx._hyg.3))))) R (Submonoid.instSetLike.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2372657690._hygCtx._hyg.3)))))) S x)) m' s'))","typeFull":"∀ {R : Type u} [inst : CommSemiring R] {S : Submonoid R} {M : Type v} [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  {α : Type u_2} (f : M × ↥S → M × ↥S → α) (wd : ∀ (p q p' q' : M × ↥S), p ≈ p' → q ≈ q' → f p q = f p' q') (m m' : M)\n  (s s' : ↥S), (LocalizedModule.mk m s).liftOn₂ (LocalizedModule.mk m' s') f wd = f (m, s) (m', s')","typeReadable":"∀ {R : Type u} [inst : CommSemiring R] {S : Submonoid R} {M : Type v} [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  {α : Type u_2} (f : M × ↥S → M × ↥S → α) (wd : ∀ (p q p' q' : M × ↥S), p ≈ p' → q ≈ q' → f p q = f p' q') (m m' : M)\n  (s s' : ↥S), (LocalizedModule.mk m s).liftOn₂ (LocalizedModule.mk m' s') f wd = f (m, s) (m', s')","typeReferences":[["LocalizedModule","liftOn₂"],["LocalizedModule","r","setoid"],["Subtype"],["SetLike","instMembership"],["Prod","mk"],["Module"],["Submonoid","instSetLike"],["CommSemiring"],["CommSemiring","toSemiring"],["Membership","mem"],["MulZeroOneClass","toMulOneClass"],["LocalizedModule","mk"],["Submonoid"],["Prod"],["AddCommMonoid"],["HasEquiv","Equiv"],["Semiring","toNonAssocSemiring"],["instHasEquivOfSetoid"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["LocalizedModule","r","setoid"],["Subtype"],["Prod","mk"],["Membership","mem"],["Quotient","mk"],["MulZeroOneClass","toMulOneClass"],["LocalizedModule","mk"],["Quotient","liftOn₂"],["HasEquiv","Equiv"],["Semiring","toNonAssocSemiring"],["instHasEquivOfSetoid"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["eq_of_heq"],["Eq","symm"],["Eq"],["Eq","ndrec"],["NonAssocSemiring","toMulZeroOneClass"],["LocalizedModule","liftOn₂"],["SetLike","instMembership"],["HEq","refl"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["Eq","casesOn"],["Submonoid"],["Prod"],["Quotient","liftOn₂_mk"],["Eq","refl"],["HEq"],["Eq","mpr"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","isScalarTower_module"],"typeFallback":"forall {R : Type.{u_6}} {A : Type.{u_7}} {M : Type.{u_8}} {M' : Type.{u_9}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.162 : CommSemiring.{u_6} R] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.165 : CommSemiring.{u_7} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.168 : Algebra.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.162 (CommSemiring.toSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.165)] (S : Submonoid.{u_6} R (MulZeroOneClass.toMulOneClass.{u_6} R (NonAssocSemiring.toMulZeroOneClass.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R (CommSemiring.toSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.162))))) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.174 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.177 : Module.{u_6, u_8} R M (CommSemiring.toSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.162) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.174] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.181 : AddCommMonoid.{u_9} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.184 : Module.{u_6, u_9} R M' (CommSemiring.toSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.162) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.181] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.188 : IsLocalization.{u_6, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.162 S A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.165 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.168] (f : LinearMap.{u_6, u_6, u_8, u_9} R R (CommSemiring.toSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.162) (CommSemiring.toSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.162) (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R (CommSemiring.toSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.162))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.174 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.181 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.177 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.184) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.201 : IsLocalizedModule.{u_6, u_8, u_9} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.162 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.174 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.181 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.177 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.184 S f], IsScalarTower.{u_6, u_7, u_9} R A M' (Algebra.toSMul.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.162 (CommSemiring.toSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.165) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.168) (SMulZeroClass.toSMul.{u_7, u_9} A M' (AddZero.toZero.{u_9} M' (AddZeroClass.toAddZero.{u_9} M' (AddMonoid.toAddZeroClass.{u_9} M' (AddCommMonoid.toAddMonoid.{u_9} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.181)))) (DistribSMul.toSMulZeroClass.{u_7, u_9} A M' (AddMonoid.toAddZeroClass.{u_9} M' (AddCommMonoid.toAddMonoid.{u_9} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.181)) (DistribMulAction.toDistribSMul.{u_7, u_9} A M' (MonoidWithZero.toMonoid.{u_7} A (Semiring.toMonoidWithZero.{u_7} A (CommSemiring.toSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.165))) (AddCommMonoid.toAddMonoid.{u_9} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.181) (Module.toDistribMulAction.{u_7, u_9} A M' (CommSemiring.toSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.165) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.181 (IsLocalizedModule.module.{u_6, u_7, u_8, u_9} R A M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.162 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.165 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.168 S inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.174 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.177 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.181 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.184 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.188 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.201))))) (SMulZeroClass.toSMul.{u_6, u_9} R M' (AddZero.toZero.{u_9} M' (AddZeroClass.toAddZero.{u_9} M' (AddMonoid.toAddZeroClass.{u_9} M' (AddCommMonoid.toAddMonoid.{u_9} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.181)))) (DistribSMul.toSMulZeroClass.{u_6, u_9} R M' (AddMonoid.toAddZeroClass.{u_9} M' (AddCommMonoid.toAddMonoid.{u_9} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.181)) (DistribMulAction.toDistribSMul.{u_6, u_9} R M' (MonoidWithZero.toMonoid.{u_6} R (Semiring.toMonoidWithZero.{u_6} R (CommSemiring.toSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.162))) (AddCommMonoid.toAddMonoid.{u_9} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.181) (Module.toDistribMulAction.{u_6, u_9} R M' (CommSemiring.toSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.162) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.181 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1243286638._hygCtx._hyg.184))))","typeFull":"∀ {R : Type u_6} {A : Type u_7} {M : Type u_8} {M' : Type u_9} [inst : CommSemiring R] [inst_1 : CommSemiring A]\n  [inst_2 : Algebra R A] (S : Submonoid R) [inst_3 : AddCommMonoid M] [inst_4 : Module R M] [inst_5 : AddCommMonoid M']\n  [inst_6 : Module R M'] [inst_7 : IsLocalization S A] (f : M →ₗ[R] M') [inst_8 : IsLocalizedModule S f],\n  IsScalarTower R A M'","typeReadable":"∀ {R : Type u_6} {A : Type u_7} {M : Type u_8} {M' : Type u_9} [inst : CommSemiring R] [inst_1 : CommSemiring A]\n  [inst_2 : Algebra R A] (S : Submonoid R) [inst_3 : AddCommMonoid M] [inst_4 : Module R M] [inst_5 : AddCommMonoid M']\n  [inst_6 : Module R M'] [inst_7 : IsLocalization S A] (f : M →ₗ[R] M') [inst_8 : IsLocalizedModule S f],\n  IsScalarTower R A M'","typeReferences":[["IsLocalizedModule"],["IsLocalizedModule","module"],["Module"],["IsScalarTower"],["CommSemiring"],["MulZeroOneClass","toMulOneClass"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Algebra","toSMul"],["MonoidWithZero","toMonoid"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["LinearMap"],["AddZeroClass","toAddZero"],["IsLocalization"],["Algebra"],["Submonoid"],["Module","toDistribMulAction"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["IsScalarTower","left"],["LocalizedModule"],["OreLocalization","instAddCommMonoidOreLocalization"],["OreLocalization","oreSetComm"],["CommSemiring","toSemiring"],["OreLocalization","instModuleOfIsScalarTower"],["LinearEquiv","isScalarTower"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"],["LocalizedModule","moduleOfIsLocalization"],["LinearEquiv","symm"],["Algebra","id"],["CommSemiring","toCommMonoid"],["Module","toDistribMulAction"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Algebra","toModule"],["IsLocalizedModule","iso"],["LocalizedModule","instIsScalarTower"],["MonoidWithZero","toMonoid"],["RingHomInvPair","ids"],["DistribMulAction","toMulAction"],["IsScalarTower","right"],["Semiring","toModule"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","instMonoid","_proof_7"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 : CommSemiring.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 : Semiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29] {S : 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.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3))))} (a : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (b : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (c : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)), Eq.{succ (max u_1 u_2)} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (HMul.hMul.{max u_1 u_2, max u_1 u_2, max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (instHMul.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (Mul.mk.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (fun (a._@._internal._hyg.0 : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (a_1._@._internal._hyg.0 : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) => LocalizedModule.mul.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32 S a._@._internal._hyg.0 a_1._@._internal._hyg.0))) (HMul.hMul.{max u_1 u_2, max u_1 u_2, max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (instHMul.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (Mul.mk.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (fun (a._@._internal._hyg.0 : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (a_1._@._internal._hyg.0 : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) => LocalizedModule.mul.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32 S a._@._internal._hyg.0 a_1._@._internal._hyg.0))) a b) c) (HMul.hMul.{max u_1 u_2, max u_1 u_2, max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (instHMul.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (Mul.mk.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (fun (a._@._internal._hyg.0 : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (a_1._@._internal._hyg.0 : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) => LocalizedModule.mul.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32 S a._@._internal._hyg.0 a_1._@._internal._hyg.0))) a (HMul.hMul.{max u_1 u_2, max u_1 u_2, max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (instHMul.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (Mul.mk.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (fun (a._@._internal._hyg.0 : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) (a_1._@._internal._hyg.0 : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32)) => LocalizedModule.mul.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3319175497._hygCtx._hyg.32 S a._@._internal._hyg.0 a_1._@._internal._hyg.0))) b c))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A] {S : Submonoid R}\n  (a b c : LocalizedModule S A), a * b * c = a * (b * c)","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A] {S : Submonoid R}\n  (a b c : LocalizedModule S A), a * b * c = a * (b * c)","typeReferences":[["Mul","mk"],["LocalizedModule"],["LocalizedModule","mul"],["CommSemiring"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["HMul","hMul"],["MulZeroOneClass","toMulOneClass"],["Algebra"],["Submonoid"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["instHMul"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["Semiring"]],"valueReferences":[["LocalizedModule","instMonoid","_proof_2"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","fromLocalizedModule","congr_simp"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S : 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.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3))))) {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.40 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.14] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1821731144._hygCtx._hyg.40) (f_1 : 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(CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2851436865._hygCtx._hyg.3))))) R (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.Algebra.Module.LocalizedModule.Basic.2851436865._hygCtx._hyg.3)))))) S x)) (Submonoid.one.{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.Algebra.Module.LocalizedModule.Basic.2851436865._hygCtx._hyg.3)))) S)))))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) (M : Type u_2) [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] (x y : M), LocalizedModule.mk (x + y) 1 = LocalizedModule.mk x 1 + LocalizedModule.mk y 1","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) (M : Type u_2) [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] (x y : M), LocalizedModule.mk (x + y) 1 = LocalizedModule.mk x 1 + LocalizedModule.mk y 1","typeReferences":[["Subtype"],["OreLocalization","oreSetComm"],["Module"],["CommSemiring"],["Membership","mem"],["MulZeroOneClass","toMulOneClass"],["LocalizedModule","mk"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["SetLike","instMembership"],["LocalizedModule"],["OreLocalization","instAddCommMonoidOreLocalization"],["instHAdd"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["CommSemiring","toCommMonoid"],["OfNat","ofNat"],["Submonoid","one"],["HAdd","hAdd"],["Submonoid"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["One","toOfNat1"]],"valueReferences":[["Subtype"],["Eq","trans"],["OreLocalization","oreSetComm"],["Membership","mem"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["LocalizedModule","mk"],["congrArg"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["AddSemigroup","toAdd"],["SetLike","instMembership"],["True"],["LocalizedModule"],["OreLocalization","instAddCommMonoidOreLocalization"],["instHAdd"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["CommSemiring","toCommMonoid"],["OfNat","ofNat"],["Submonoid","one"],["Submonoid"],["HAdd","hAdd"],["eq_self"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["of_eq_true"],["One","toOfNat1"],["AddMonoid","toAddSemigroup"],["OreLocalization","oreDiv"],["OreLocalization","add_oreDiv"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","linearEquiv_symm_apply"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S : 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.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3))))) {M : Type.{u_2}} {M' : Type.{u_3}} {M'' : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.17 : AddCommMonoid.{u_4} M''] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.40 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.14] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.44 : Module.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.17] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.40) (g : LinearMap.{u_1, u_1, u_2, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3))) M M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.44) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.71 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.40 S f] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.75 : IsLocalizedModule.{u_1, u_2, u_4} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3 M M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.44 S g] (x : M), Eq.{succ u_3} M' (DFunLike.coe.{max (succ u_3) (succ u_4), succ u_4, succ u_3} (LinearEquiv.{u_1, u_1, u_4, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R 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(RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3))) (RingHomInvPair.ids.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3)) (RingHomInvPair.ids.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3)) (IsLocalizedModule.linearEquiv.{u_1, u_2, u_3, u_4} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3 S M M' M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.44 f g inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.71 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.75)) (DFunLike.coe.{max (succ u_2) (succ u_4), succ u_2, succ u_4} (LinearMap.{u_1, u_1, u_2, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3))) M M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.44) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M'') (LinearMap.instFunLike.{u_1, u_1, u_2, u_4} R R M M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.44 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M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M') (LinearMap.instFunLike.{u_1, u_1, u_2, u_3} R R M M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.40 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3960095835._hygCtx._hyg.3)))) f x)","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} {M'' : Type u_4}\n  [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M'] [inst_3 : AddCommMonoid M''] [inst_4 : Module R M]\n  [inst_5 : Module R M'] [inst_6 : Module R M''] (f : M →ₗ[R] M') (g : M →ₗ[R] M'') [inst_7 : IsLocalizedModule S f]\n  [inst_8 : IsLocalizedModule S g] (x : M), (IsLocalizedModule.linearEquiv S f g).symm (g x) = f x","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} {M'' : Type u_4}\n  [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M'] [inst_3 : AddCommMonoid M''] [inst_4 : Module R M]\n  [inst_5 : Module R M'] [inst_6 : Module R M''] (f : M →ₗ[R] M') (g : M →ₗ[R] M'') [inst_7 : IsLocalizedModule S f]\n  [inst_8 : IsLocalizedModule S g] (x : M), (IsLocalizedModule.linearEquiv S f g).symm (g x) = f 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 : CommSemiring.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29 : Ring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.32 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29)] {S : 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.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3))))}, autoParam.{0} (forall (n : Nat) (a : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A 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{R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (m : M'),\n  (IsLocalizedModule.iso S f).symm m = LocalizedModule.mk ⋯.choose.1 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.3 : CommSemiring.{u_1} R] {S : 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.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.3))))} {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] (A : Type.{u_5}) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.21 : CommSemiring.{u_5} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.24 : Algebra.{u_1, u_5} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_5} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.21)] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.28 : Module.{u_5, u_3} A M' (CommSemiring.toSemiring.{u_5} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.21) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.14] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.32 : IsLocalization.{u_1, u_5} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.3 S A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.40 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.14] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.48 : IsScalarTower.{u_1, u_5, u_3} R A M' (Algebra.toSMul.{u_1, u_5} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_5} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.21) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.24) (SMulZeroClass.toSMul.{u_5, u_3} A M' (AddZero.toZero.{u_3} M' (AddZeroClass.toAddZero.{u_3} M' (AddMonoid.toAddZeroClass.{u_3} M' (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.14)))) (DistribSMul.toSMulZeroClass.{u_5, u_3} A M' (AddMonoid.toAddZeroClass.{u_3} M' (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.14)) (DistribMulAction.toDistribSMul.{u_5, u_3} A M' (MonoidWithZero.toMonoid.{u_5} A (Semiring.toMonoidWithZero.{u_5} A (CommSemiring.toSemiring.{u_5} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.21))) (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.14) (Module.toDistribMulAction.{u_5, u_3} A M' (CommSemiring.toSemiring.{u_5} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.21) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.28)))) (SMulZeroClass.toSMul.{u_1, u_3} R M' (AddZero.toZero.{u_3} M' (AddZeroClass.toAddZero.{u_3} M' (AddMonoid.toAddZeroClass.{u_3} M' (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.14)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R M' (AddMonoid.toAddZeroClass.{u_3} M' (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.14)) (DistribMulAction.toDistribSMul.{u_1, u_3} R M' (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.14) (Module.toDistribMulAction.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.40))))] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.40) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.71 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.40 S f] (x : R) (m : M) (s : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.3 S M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.40 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.71 m t)) (IsLocalizedModule.mk'.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.3 S M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.40 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.71 (HSMul.hSMul.{u_1, u_2, u_2} R M M (instHSMul.{u_1, u_2} R M (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.11) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.3)))))) S x)) (instHMul.{u_1} (Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.3))))) R (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.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.3)))))) S x)) (Submonoid.mul.{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.Algebra.Module.LocalizedModule.Basic.1482069452._hygCtx._hyg.3)))) S)) s t))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] (A : Type u_5) [inst_3 : CommSemiring A] [inst_4 : Algebra R A] [inst_5 : Module A M']\n  [inst_6 : IsLocalization S A] [inst_7 : Module R M] [inst_8 : Module R M'] [IsScalarTower R A M'] (f : M →ₗ[R] M')\n  [inst_10 : IsLocalizedModule S f] (x : R) (m : M) (s t : ↥S),\n  IsLocalization.mk' A x s • IsLocalizedModule.mk' f m t = IsLocalizedModule.mk' f (x • m) (s * t)","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] (A : Type u_5) [inst_3 : CommSemiring A] [inst_4 : Algebra R A] [inst_5 : Module A M']\n  [inst_6 : IsLocalization S A] [inst_7 : Module R M] [inst_8 : Module R M'] [IsScalarTower R A M'] (f : M →ₗ[R] M')\n  [inst_10 : IsLocalizedModule S f] (x : R) (m : M) (s t : ↥S),\n  IsLocalization.mk' A x s • IsLocalizedModule.mk' f m t = IsLocalizedModule.mk' f (x • m) (s * 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S : 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.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3))))) {M : Type.{u_2}} {M' : Type.{u_3}} {M'' : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.17 : AddCommMonoid.{u_4} M''] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.40 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.14] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.44 : Module.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.17] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.40) (e : LinearEquiv.{u_1, u_1, u_4, u_2} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3))) (RingHomInvPair.ids.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3)) (RingHomInvPair.ids.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3)) M'' M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.44 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.36) [hf : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.40 S f], IsLocalizedModule.{u_1, u_4, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3 M'' M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.44 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.40 S (LinearMap.comp.{u_1, u_1, u_1, u_4, u_2, u_3} R R R M'' M M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.44 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.40 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3))) (RingHomCompTriple.ids.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3)))) f (LinearEquiv.toLinearMap.{u_1, u_1, u_4, u_2} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3))) (RingHomInvPair.ids.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3)) (RingHomInvPair.ids.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.3)) M'' M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.44 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2653861037._hygCtx._hyg.36 e))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} {M'' : Type u_4}\n  [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M'] [inst_3 : AddCommMonoid M''] [inst_4 : Module R M]\n  [inst_5 : Module R M'] [inst_6 : Module R M''] (f : M →ₗ[R] M') (e : M'' ≃ₗ[R] M) [hf : IsLocalizedModule S f],\n  IsLocalizedModule S (f ∘ₗ ↑e)","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} {M'' : Type u_4}\n  [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M'] [inst_3 : AddCommMonoid M''] [inst_4 : Module R M]\n  [inst_5 : Module R M'] [inst_6 : Module R M''] (f : M →ₗ[R] M') (e : M'' ≃ₗ[R] M) [hf : IsLocalizedModule S f],\n  IsLocalizedModule S (f ∘ₗ ↑e)","typeReferences":[["IsLocalizedModule"],["RingHomCompTriple","ids"],["Module"],["CommSemiring"],["CommSemiring","toSemiring"],["LinearMap","comp"],["LinearEquiv","toLinearMap"],["MulZeroOneClass","toMulOneClass"],["LinearMap"],["LinearEquiv"],["Submonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["RingHomInvPair","ids"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["LinearEquiv","symm_apply_apply"],["Eq","trans"],["RingHomCompTriple","ids"],["Prod","mk"],["Submonoid","smul"],["Membership","mem"],["Exists","intro"],["SMulZeroClass","toSMul"],["map_smul"],["Subtype","val"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["SemilinearMapClass","toMulActionSemiHomClass"],["SetLike","instMembership"],["Exists"],["Submonoid","instSetLike"],["IsLocalizedModule","surj"],["DistribMulAction","toDistribSMul"],["Prod","snd"],["AddZeroClass","toAddZero"],["LinearMap"],["Submonoid"],["Exists","casesOn"],["Prod"],["LinearEquiv","instSemilinearEquivClass"],["HSMul","hSMul"],["RingHomInvPair","ids"],["id"],["Eq","mpr"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Subtype"],["LinearMap","instFunLike"],["Eq","mp"],["IsLocalizedModule","exists_of_eq"],["LinearMap","comp"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["LinearEquiv"],["DFunLike","coe"],["Prod","fst"],["congrArg"],["Prod","casesOn"],["SemilinearEquivClass","instSemilinearMapClass"],["IsLocalizedModule","map_units"],["congr"],["MonoidWithZero","toMonoid"],["EquivLike","toFunLike"],["instHSMul"],["Eq"],["LinearEquiv","instEquivLike"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["LinearEquiv","toLinearMap"],["LinearEquiv","symm"],["Module","toDistribMulAction"],["LinearEquiv","apply_symm_apply"],["IsLocalizedModule","mk"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","divBy_apply"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3 : CommSemiring.{u} R] {S : Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))))} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12 : Module.{u, v} R M (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9] (s : Subtype.{succ u} R (fun (x : R) => Membership.mem.{u, u} R (Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))))) (SetLike.instMembership.{u, u} (Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))))) R (Submonoid.instSetLike.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3)))))) S x)) (p : LocalizedModule.{u, v} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12), Eq.{succ (max u v)} (LocalizedModule.{u, v} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12) (DFunLike.coe.{succ (max v u), succ (max v u), succ (max v u)} (LinearMap.{u, u, max v u, max v u} R R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))) (LocalizedModule.{u, v} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12) (LocalizedModule.{u, v} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12) (OreLocalization.instAddCommMonoidOreLocalization.{u, v} R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 (Module.toDistribMulAction.{u, v} R M (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12)) (OreLocalization.instAddCommMonoidOreLocalization.{u, v} R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 (Module.toDistribMulAction.{u, v} R M (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12)) (OreLocalization.instModuleOfIsScalarTower.{u, v, u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12 R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12 (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3)) (IsScalarTower.left.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9) (Module.toDistribMulAction.{u, v} R M (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12))) (IsScalarTower.right.{u, u} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3 (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) (Algebra.id.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))) (OreLocalization.instModuleOfIsScalarTower.{u, v, u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12 R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12 (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3)) (IsScalarTower.left.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9) (Module.toDistribMulAction.{u, v} R M (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12))) (IsScalarTower.right.{u, u} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3 (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) (Algebra.id.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3)))) (LocalizedModule.{u, v} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : LocalizedModule.{u, v} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12) => LocalizedModule.{u, v} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12) (LinearMap.instFunLike.{u, u, max v u, max v u} R R (LocalizedModule.{u, v} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12) (LocalizedModule.{u, v} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12) (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) (OreLocalization.instAddCommMonoidOreLocalization.{u, v} R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 (Module.toDistribMulAction.{u, v} R M (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12)) (OreLocalization.instAddCommMonoidOreLocalization.{u, v} R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 (Module.toDistribMulAction.{u, v} R M (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12)) (OreLocalization.instModuleOfIsScalarTower.{u, v, u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12 R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12 (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3)) (IsScalarTower.left.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9) (Module.toDistribMulAction.{u, v} R M (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12))) (IsScalarTower.right.{u, u} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3 (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) (Algebra.id.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))) (OreLocalization.instModuleOfIsScalarTower.{u, v, u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u} R (CommSemiring.toCommMonoid.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12 R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12 (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3)) (IsScalarTower.left.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9) (Module.toDistribMulAction.{u, v} R M (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12))) (IsScalarTower.right.{u, u} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3 (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3) (Algebra.id.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3))) (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3)))) (LocalizedModule.divBy.{u, v} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.12 s) p) (LocalizedModule.liftOn.{u, v, max u v} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4202080348._hygCtx._hyg.3 S M 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[inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f],\n  IsLocalizedModule.fromLocalizedModule S f =\n    { toFun := IsLocalizedModule.fromLocalizedModule' S f, map_add' := ⋯, map_smul' := ⋯ 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[inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.71 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.40 S f] {N : Type.{u_6}} {N' : Type.{u_7}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.77 : AddCommMonoid.{u_6} N] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.80 : AddCommMonoid.{u_7} N'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.83 : Module.{u_1, u_6} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.3) 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.87 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.3)))) (LinearMap.module.{u_1, u_1, u_1, u_2, u_6} R R R M N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.36 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.83)))) (LinearMap.module.{u_1, u_1, u_1, u_3, u_7} R R R M' N' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.87 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.87 (smulCommClass_self.{u_1, u_7} R N' (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_7} R N' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_7} N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.80) (Module.toDistribMulAction.{u_1, u_7} R N' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.87)))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.3)))) (IsLocalizedModule.map.{u_1, u_2, u_3, u_6, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.3 S M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.40 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.71 N N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.87 g inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1141048161._hygCtx._hyg.100) h))) True)","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] {N : Type u_6} {N' : Type u_7} [inst_6 : AddCommMonoid N] [inst_7 : AddCommMonoid N']\n  [inst_8 : Module R N] [inst_9 : Module R N'] (g : N →ₗ[R] N') [inst_10 : IsLocalizedModule S g] (h : M →ₗ[R] N),\n  Function.Injective ⇑h → Function.Injective ⇑((IsLocalizedModule.map S f g) h) = True","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] {N : Type u_6} {N' : Type u_7} [inst_6 : AddCommMonoid N] [inst_7 : AddCommMonoid N']\n  [inst_8 : Module R N] [inst_9 : Module R N'] (g : N →ₗ[R] N') [inst_10 : IsLocalizedModule S g] (h : M →ₗ[R] N),\n  Function.Injective ⇑h → Function.Injective ⇑((IsLocalizedModule.map S f g) h) = True","typeReferences":[["IsLocalizedModule"],["smulCommClass_self"],["LinearMap","instFunLike"],["CommMonoid","toMonoid"],["Module"],["CommSemiring"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["DistribMulAction","toMulAction"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["IsLocalizedModule","map"],["True"],["CommSemiring","toSemiring"],["LinearMap","module"],["LinearMap"],["CommSemiring","toCommMonoid"],["Submonoid"],["LinearMap","addCommMonoid"],["Module","toDistribMulAction"],["Function","Injective"]],"valueReferences":[["smulCommClass_self"],["IsLocalizedModule","map"],["CommMonoid","toMonoid"],["LinearMap","instFunLike"],["IsLocalizedModule","map_injective"],["CommSemiring","toSemiring"],["LinearMap","module"],["eq_true"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["DFunLike","coe"],["CommSemiring","toCommMonoid"],["LinearMap","addCommMonoid"],["Module","toDistribMulAction"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["DistribMulAction","toMulAction"],["Function","Injective"]]},{"isProp":true,"kind":"definition","name":["IsLocalizedModule","mk","_flat_ctor"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72 : CommSemiring.{u_1} R] {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83] {S : 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.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))))} {f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109}, (forall (x : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))))) (SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))))) R (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.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)))))) S x)), IsUnit.{u_3} (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (Module.End.instMonoid.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (DFunLike.coe.{max (succ u_1) (succ u_3), succ u_1, succ u_3} (RingHom.{u_1, u_3} R (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)) (Semiring.toNonAssocSemiring.{u_3} (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (Module.End.instSemiring.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109))) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (RingHom.instFunLike.{u_1, u_3} R (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)) (Semiring.toNonAssocSemiring.{u_3} (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (Module.End.instSemiring.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109))) (algebraMap.{u_1, u_3} R (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72 (Module.End.instSemiring.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (Module.End.instAlgebra.{u_1, u_1, u_3} R R M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109 (smulCommClass_self.{u_1, u_3} R M' (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) (DistribMulAction.toMulAction.{u_1, u_3} R M' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)) (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83) (Module.toDistribMulAction.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109))) (Algebra.toSMul.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)) (IsScalarTower.left.{u_1, u_3} R M' (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R 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M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M') (LinearMap.instFunLike.{u_1, u_1, u_2, u_3} R R M M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 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(AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105)))) S)) c x₂)))) -> (IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109 S f)","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] {S : Submonoid R} {f : M →ₗ[R] M'},\n  (∀ (x : ↥S), IsUnit ((algebraMap R (Module.End R M')) ↑x)) →\n    (∀ (y : M'), ∃ x, x.2 • y = f x.1) → (∀ {x₁ x₂ : M}, f x₁ = f x₂ → ∃ c, c • x₁ = c • x₂) → IsLocalizedModule S f","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] {S : Submonoid R} {f : M →ₗ[R] M'},\n  (∀ (x : ↥S), IsUnit ((algebraMap R 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S : 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.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.3))))) {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.40 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.14] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.40) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.71 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.40 S f] (m : M) (s : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (Submonoid.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.40 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.3)))) (Units.val.{u_3} (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.40) (Module.End.instMonoid.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R 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(CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.40 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1811580152._hygCtx._hyg.3)))) f m))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (m : M) (s : ↥S),\n  IsLocalizedModule.fromLocalizedModule' S f (LocalizedModule.mk m s) = ↑⋯.unit⁻¹ (f m)","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (m : M) (s : ↥S),\n  IsLocalizedModule.fromLocalizedModule' S f (LocalizedModule.mk m s) = ↑⋯.unit⁻¹ (f m)","typeReferences":[["IsUnit","unit"],["smulCommClass_self"],["RingHom"],["IsLocalizedModule"],["IsLocalizedModule","fromLocalizedModule'"],["Subtype"],["CommMonoid","toMonoid"],["LinearMap","instFunLike"],["Module"],["Membership","mem"],["CommSemiring"],["Units","instInv"],["RingHom","instFunLike"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["LocalizedModule","mk"],["Subtype","val"],["Algebra","id"],["DFunLike","coe"],["IsLocalizedModule","map_units"],["Module","End","instAlgebra"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["Algebra","toSMul"],["DistribMulAction","toMulAction"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["Units"],["Module","End"],["Inv","inv"],["Module","End","instSemiring"],["SetLike","instMembership"],["IsScalarTower","left"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["LinearMap"],["CommSemiring","toCommMonoid"],["Submonoid"],["Module","toDistribMulAction"],["Units","val"],["Module","End","instMonoid"],["algebraMap"]],"valueReferences":[["rfl"],["IsLocalizedModule","fromLocalizedModule'"],["LocalizedModule","mk"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","instSemiring","_proof_18"],"typeFallback":"forall 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29)))) (Module.toDistribMulAction.{u_1, u_2} R A (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.32)))) 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S : 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.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3))))) {M : Type.{u_2}} {M' : Type.{u_3}} {M'' : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 : AddCommMonoid.{u_4} M''] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.40 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.14] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44 : Module.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) (RingHom.id.{u_1} R 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S f], (LinearMap.{u_1, u_1, u_2, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3))) M M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44) -> (forall (x : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44))) (algebraMap.{u_1, u_4} R (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3 (Module.End.instSemiring.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44) (Module.End.instAlgebra.{u_1, u_1, u_4} R R M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44 (smulCommClass_self.{u_1, u_4} R M'' (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_4} R M'' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_4} M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17) (Module.toDistribMulAction.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.44))) (Algebra.toSMul.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_4} R M'' (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412006._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_4} R M'' (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.3 : CommSemiring.{u_1} R] {S : 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.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.3))))} {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.40 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.14] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.40) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.71 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.40 S f] (m : M) (s₁ : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R 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(Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.3)))) S)) s₁ s₂)) (IsLocalizedModule.mk'.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.3 S M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.40 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.736369829._hygCtx._hyg.71 m s₂)","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (m : M) (s₁ s₂ : ↥S),\n  IsLocalizedModule.mk' f (s₁ • m) (s₁ * s₂) = IsLocalizedModule.mk' f m s₂","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (m : M) (s₁ s₂ : ↥S),\n  IsLocalizedModule.mk' f (s₁ • m) (s₁ * s₂) = IsLocalizedModule.mk' f m s₂","typeReferences":[["IsLocalizedModule"],["Subtype"],["Module"],["Membership","mem"],["CommSemiring"],["Submonoid","smul"],["HMul","hMul"],["MulZeroOneClass","toMulOneClass"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["IsLocalizedModule","mk'"],["SetLike","instMembership"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["LinearMap"],["AddZeroClass","toAddZero"],["Submonoid"],["Module","toDistribMulAction"],["Submonoid","mul"],["HSMul","hSMul"],["instHMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Subtype"],["LinearMap","instFunLike"],["IsLocalizedModule","fromLocalizedModule"],["OreLocalization","oreSetComm"],["Membership","mem"],["Submonoid","smul"],["HMul","hMul"],["MulZeroOneClass","toMulOneClass"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["LocalizedModule","mk"],["DFunLike","coe"],["congrArg"],["LocalizedModule","mkLinearMap","_proof_3"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["instHSMul"],["Semiring","toModule"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["IsLocalizedModule","mk'"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["OreLocalization","instAddCommMonoidOreLocalization"],["LocalizedModule"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["OreLocalization","instModuleOfIsScalarTower"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["LinearMap"],["AddZeroClass","toAddZero"],["CommSemiring","toCommMonoid"],["Submonoid"],["Module","toDistribMulAction"],["Submonoid","mul"],["Eq","refl"],["HSMul","hSMul"],["id"],["LocalizedModule","algebra'","_proof_4"],["instHMul"],["Eq","mpr"],["LocalizedModule","mk_cancel_common_left"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","instCommSemiring","_proof_2"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 : CommSemiring.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29 : CommSemiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.32 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29)] {S : 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.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3))))} (a : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.32)) (b : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.32)), Eq.{succ (max u_1 u_2)} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.32)) (HMul.hMul.{max u_1 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.32)) (instHMul.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.32)) (NonUnitalNonAssocSemiring.toMul.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.32)) (Semiring.toNonUnitalSemiring.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.32)) (LocalizedModule.instSemiring.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 A (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.32 S))))) a b) (HMul.hMul.{max u_1 u_2, max u_1 u_2, max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.32)) (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.32)) (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.32)) (instHMul.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.32)) (NonUnitalNonAssocSemiring.toMul.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.32)) (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.32)) (Semiring.toNonUnitalSemiring.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.32)) (LocalizedModule.instSemiring.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.3 A (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1580257103._hygCtx._hyg.32 S))))) b a)","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : CommSemiring A] [inst_2 : Algebra R A]\n  {S : Submonoid R} (a b : LocalizedModule S A), a * b = b * a","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : CommSemiring A] [inst_2 : Algebra R A]\n  {S : Submonoid R} (a b : LocalizedModule S A), a * b = b * a","typeReferences":[["NonUnitalNonAssocSemiring","toMul"],["LocalizedModule"],["CommSemiring"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["HMul","hMul"],["MulZeroOneClass","toMulOneClass"],["LocalizedModule","instSemiring"],["Algebra"],["Submonoid"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["instHMul"],["Eq"],["Semiring","toNonUnitalSemiring"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["LocalizedModule","instCommSemiring","_proof_1"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","instSemiring","_proof_11"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 : CommSemiring.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 : Semiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29] {S : 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.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3))))} (a : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.40) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.71 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.40 S f] {N : Type.{u_6}} {N' : Type.{u_7}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.77 : AddCommMonoid.{u_6} N] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.80 : AddCommMonoid.{u_7} N'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.83 : Module.{u_1, u_6} R N (CommSemiring.toSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.87 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.87 (smulCommClass_self.{u_1, u_7} R N' (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_7} R N' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_7} N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.80) (Module.toDistribMulAction.{u_1, u_7} R N' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.87))))) (LinearMap.{u_1, u_1, u_2, u_6} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3))) M N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.83) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : LinearMap.{u_1, u_1, u_2, u_6} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3))) M N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.83) => LinearMap.{u_1, u_1, u_3, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3))) M' N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.87) (LinearMap.instFunLike.{u_1, u_1, max u_2 u_6, max u_3 u_7} R R (LinearMap.{u_1, u_1, u_2, u_6} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3))) M N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.83) (LinearMap.{u_1, u_1, u_3, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3))) M' N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.87) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) (LinearMap.addCommMonoid.{u_1, u_1, u_2, u_6} R R M N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.83 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3)))) (LinearMap.addCommMonoid.{u_1, u_1, u_3, u_7} R R M' N' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.87 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3)))) (LinearMap.module.{u_1, u_1, u_1, u_2, u_6} R R R M N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.83 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.83 (smulCommClass_self.{u_1, u_6} R N (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_6} R N (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_6} N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.77) (Module.toDistribMulAction.{u_1, u_6} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.83)))) (LinearMap.module.{u_1, u_1, u_1, u_3, u_7} R R R M' N' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.87 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.87 (smulCommClass_self.{u_1, u_7} R N' (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_7} R N' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_7} N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.80) (Module.toDistribMulAction.{u_1, u_7} R N' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.87)))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3)))) (IsLocalizedModule.map.{u_1, u_2, u_3, u_6, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3 S M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.40 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.71 N N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.87 g inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.100) h) f) (LinearMap.comp.{u_1, u_1, u_1, u_2, u_6, u_7} R R R M N N' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.87 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3))) (RingHomCompTriple.ids.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2359094227._hygCtx._hyg.3) 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[inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.116 : Module.{u_1, u_9} R M₀' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.109] (f₀ : LinearMap.{u_1, u_1, u_6, u_9} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) M₀ M₀' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.109 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.116) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.129 : IsLocalizedModule.{u_1, u_6, u_9} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3 M₀ M₀' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.109 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.116 S f₀] {M₁ : Type.{u_7}} {M₁' : Type.{u_8}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 : AddCommMonoid.{u_7} M₁] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.138 : AddCommMonoid.{u_8} M₁'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141 : Module.{u_1, u_7} R M₁ 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.138 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.145) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.158 : IsLocalizedModule.{u_1, u_7, u_8} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3 M₁ M₁' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.138 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.145 S f₁] (g : LinearMap.{u_1, u_1, u_6, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) M₀ M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141), Eq.{max (max (succ u_1) (succ u_6)) (succ u_8)} (LinearMap.{u_1, u_1, max u_1 u_6, u_8} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R 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(CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_6} R M₀ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_6} R M₀ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_6} M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106) (Module.toDistribMulAction.{u_1, u_6} R M₀ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112))) (IsScalarTower.right.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.145) (LinearMap.comp.{u_1, u_1, u_1, max u_1 u_6, u_9, u_8} R R R (LocalizedModule.{u_1, u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3 S M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112) M₀' M₁' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_6} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) S) M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 (Module.toDistribMulAction.{u_1, u_6} R M₀ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.109 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.138 (OreLocalization.instModuleOfIsScalarTower.{u_1, u_6, u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) S) M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112 R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) 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LinearMap.{u_1, u_1, u_6, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) M₀ M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141) => LinearMap.{u_1, u_1, u_9, u_8} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) M₀ M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141) (LinearMap.{u_1, u_1, u_9, u_8} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) M₀' M₁' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.109 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.138 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.116 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.145) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (LinearMap.addCommMonoid.{u_1, u_1, u_6, u_7} R R M₀ M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3)))) (LinearMap.addCommMonoid.{u_1, u_1, u_9, u_8} R R M₀' M₁' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.109 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.138 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.116 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.145 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3)))) (LinearMap.module.{u_1, u_1, u_1, u_6, u_7} R R R M₀ M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141 (smulCommClass_self.{u_1, u_7} R M₁ (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_7} R M₁ (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_7} M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135) (Module.toDistribMulAction.{u_1, u_7} R M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141)))) (LinearMap.module.{u_1, u_1, u_1, u_9, u_8} R R R M₀' M₁' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.109 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.138 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.116 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.145 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.145 (smulCommClass_self.{u_1, u_8} R M₁' (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_8} R M₁' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_8} M₁' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.138) (Module.toDistribMulAction.{u_1, u_8} R M₁' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.138 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.145)))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3)))) (IsLocalizedModule.map.{u_1, u_6, u_9, u_7, u_8} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3 S M₀ M₀' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.109 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.116 f₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.129 M₁ M₁' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.138 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.145 f₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.158) g) (LinearEquiv.toLinearMap.{u_1, u_1, max u_1 u_6, u_9} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) (RingHomInvPair.ids.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3)) (RingHomInvPair.ids.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3)) (LocalizedModule.{u_1, u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3 S M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112) M₀' (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_6} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) S) M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 (Module.toDistribMulAction.{u_1, u_6} R M₀ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.109 (OreLocalization.instModuleOfIsScalarTower.{u_1, u_6, u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) S) M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112 R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_6} R M₀ (MonoidWithZero.toMonoid.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.116 (IsLocalizedModule.iso.{u_1, u_6, u_9} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3 S M₀ M₀' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.109 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.116 f₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.129))) (LinearMap.comp.{u_1, u_1, u_1, max u_1 u_6, max u_1 u_7, u_8} R R R (LocalizedModule.{u_1, u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3 S M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112) (LocalizedModule.{u_1, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3 S M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141) M₁' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_6} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) S) M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 (Module.toDistribMulAction.{u_1, u_6} R M₀ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112)) (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_7} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_7} R M₁ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_7} R M₁ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_7} M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135) (Module.toDistribMulAction.{u_1, u_7} R M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141))) (IsScalarTower.right.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) (smulCommClass_self.{u_1, max u_1 u_7} R (LocalizedModule.{u_1, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3 S M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141) (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, max u_1 u_7} R (LocalizedModule.{u_1, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3 S M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141) (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{max u_1 u_7} (LocalizedModule.{u_1, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3 S M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141) (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_7} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) S) M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 (Module.toDistribMulAction.{u_1, u_7} R M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141))) (Module.toDistribMulAction.{u_1, max u_1 u_7} R (LocalizedModule.{u_1, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3 S M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_7} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) S) M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 (Module.toDistribMulAction.{u_1, u_7} R M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141)) (OreLocalization.instModuleOfIsScalarTower.{u_1, u_7, u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) S) M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141 R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_7} R M₁ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_7} R M₁ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_7} M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135) (Module.toDistribMulAction.{u_1, u_7} R M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141))) (IsScalarTower.right.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3)))))))) (LinearMap.{u_1, u_1, u_6, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) M₀ M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : LinearMap.{u_1, u_1, u_6, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) M₀ M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141) => LinearMap.{u_1, u_1, max u_1 u_6, max u_1 u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) (LocalizedModule.{u_1, u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3 S M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112) (LocalizedModule.{u_1, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3 S M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141) (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_6} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) S) M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 (Module.toDistribMulAction.{u_1, u_6} R M₀ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112)) (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_7} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) S) M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 (Module.toDistribMulAction.{u_1, u_7} R M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141)) (OreLocalization.instModuleOfIsScalarTower.{u_1, u_6, u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) S) M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112 R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_6} R M₀ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_6} R M₀ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_6} M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106) (Module.toDistribMulAction.{u_1, u_6} R M₀ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112))) (IsScalarTower.right.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) (OreLocalization.instModuleOfIsScalarTower.{u_1, u_7, u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) S) M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141 R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_7} R M₁ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_7} R M₁ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_7} M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135) (Module.toDistribMulAction.{u_1, u_7} R M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141))) (IsScalarTower.right.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3)))) (LinearMap.instFunLike.{u_1, u_1, max u_6 u_7, max (max u_1 u_6) u_7} R R (LinearMap.{u_1, u_1, u_6, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) 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(CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_6} R M₀ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_6} R M₀ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_6} M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106) (Module.toDistribMulAction.{u_1, u_6} R M₀ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112))) (IsScalarTower.right.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) (LocalizedModule.mkLinearMap.{u_1, u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3 S M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112) (localizedModuleIsLocalizedModule.{u_1, u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3 S M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.112) M₁ (LocalizedModule.{u_1, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3 S M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_7} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) S) M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 (Module.toDistribMulAction.{u_1, u_7} R M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141 (OreLocalization.instModuleOfIsScalarTower.{u_1, u_7, u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) S) M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141 R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_7} R M₁ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_7} R M₁ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_7} M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135) (Module.toDistribMulAction.{u_1, u_7} R M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141))) (IsScalarTower.right.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3))) (LocalizedModule.mkLinearMap.{u_1, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3 S M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141) (localizedModuleIsLocalizedModule.{u_1, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.3 S M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1957199518._hygCtx._hyg.141)) g))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M₀ : Type u_6} {M₀' : Type u_9} [inst_1 : AddCommMonoid M₀]\n  [inst_2 : AddCommMonoid M₀'] [inst_3 : Module R M₀] [inst_4 : Module R M₀'] (f₀ : M₀ →ₗ[R] M₀')\n  [inst_5 : IsLocalizedModule S f₀] {M₁ : Type u_7} {M₁' : Type u_8} [inst_6 : AddCommMonoid M₁]\n  [inst_7 : AddCommMonoid M₁'] [inst_8 : Module R M₁] [inst_9 : Module R M₁'] (f₁ : M₁ →ₗ[R] M₁')\n  [inst_10 : IsLocalizedModule S f₁] (g : M₀ →ₗ[R] M₁),\n  (IsLocalizedModule.map S f₀ f₁) g ∘ₗ ↑(IsLocalizedModule.iso S f₀) =\n    ↑(IsLocalizedModule.iso S f₁) ∘ₗ\n      (IsLocalizedModule.map S (LocalizedModule.mkLinearMap S M₀) (LocalizedModule.mkLinearMap S M₁)) g","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M₀ : Type u_6} {M₀' : Type u_9} [inst_1 : AddCommMonoid M₀]\n  [inst_2 : AddCommMonoid M₀'] [inst_3 : Module R M₀] [inst_4 : Module R M₀'] (f₀ : M₀ →ₗ[R] M₀')\n  [inst_5 : IsLocalizedModule S f₀] {M₁ : Type u_7} {M₁' : Type u_8} [inst_6 : AddCommMonoid M₁]\n  [inst_7 : AddCommMonoid M₁'] [inst_8 : Module R M₁] [inst_9 : Module R M₁'] (f₁ : M₁ →ₗ[R] M₁')\n  [inst_10 : IsLocalizedModule S f₁] (g : M₀ →ₗ[R] M₁),\n  (IsLocalizedModule.map S f₀ f₁) g ∘ₗ ↑(IsLocalizedModule.iso S f₀) =\n    ↑(IsLocalizedModule.iso S f₁) ∘ₗ\n      (IsLocalizedModule.map S (LocalizedModule.mkLinearMap S M₀) (LocalizedModule.mkLinearMap S M₁)) 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.3 : CommSemiring.{u_1} R] {S : 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.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.3))))} {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.40 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.14] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.40) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.71 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.40 S f] (m₁ : M) (m₂ : M) (s₁ : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} 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(CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.3)))))) S x)) M (Submonoid.smul.{u_1, u_2} R M (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.3)))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.11) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.4007826754._hygCtx._hyg.36)))) S)) s₂ m₁))))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (m₁ m₂ : M) (s₁ s₂ : ↥S),\n  (IsLocalizedModule.mk' f m₁ s₁ = IsLocalizedModule.mk' f m₂ s₂) = ∃ s, s • s₁ • m₂ = s • s₂ • m₁","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n  [inst_5 : IsLocalizedModule S f] (m₁ m₂ : M) (s₁ s₂ : ↥S),\n  (IsLocalizedModule.mk' f m₁ s₁ = IsLocalizedModule.mk' f m₂ s₂) = ∃ s, s • s₁ • m₂ = s • s₂ • m₁","typeReferences":[["IsLocalizedModule"],["Subtype"],["Module"],["Membership","mem"],["CommSemiring"],["Submonoid","smul"],["MulZeroOneClass","toMulOneClass"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["IsLocalizedModule","mk'"],["SetLike","instMembership"],["Exists"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["LinearMap"],["AddZeroClass","toAddZero"],["Submonoid"],["Module","toDistribMulAction"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Subtype"],["Membership","mem"],["Submonoid","smul"],["MulZeroOneClass","toMulOneClass"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["propext"],["IsLocalizedModule","mk'"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["Exists"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["IsLocalizedModule","mk'_eq_mk'_iff"],["AddZeroClass","toAddZero"],["Submonoid"],["Module","toDistribMulAction"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","instSemiring","_proof_7"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 : CommSemiring.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 : Semiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29] {S : 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.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3))))} (n : Nat), Eq.{succ (max u_1 u_2)} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx._hyg.161 (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx._hyg.164) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx._hyg.170) M)) m x) (OfNat.ofNat.{u_2} S 0 (Zero.toOfNat0.{u_2} S (MulZeroClass.toZero.{u_2} S (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} S (NonAssocRing.toNonUnitalNonAssocRing.{u_2} S (Ring.toNonAssocRing.{u_2} S inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx._hyg.164))))))))) -> Prop) (x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx.291.Mathlib.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx._hyg.299 : Exists.{succ u_1} (Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx._hyg.161 (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx._hyg.164) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx._hyg.170) M)) m x) (OfNat.ofNat.{u_2} S 0 (Zero.toOfNat0.{u_2} S (MulZeroClass.toZero.{u_2} S (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} S (NonAssocRing.toNonUnitalNonAssocRing.{u_2} S (Ring.toNonAssocRing.{u_2} S inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx._hyg.164))))))))), (forall (m : R) (hm : Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx._hyg.161))))) (SetLike.instMembership.{u_1, u_1} (Submonoid.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx._hyg.161)))))) M x)) S (Submonoid.smul.{u_1, u_2} R S (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx._hyg.161)))) (Algebra.toSMul.{u_1, u_2} R S inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx._hyg.161 (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx._hyg.164) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1954505467._hygCtx._hyg.170) M)) m x) (OfNat.ofNat.{u_2} S 0 (Zero.toOfNat0.{u_2} S (MulZeroClass.toZero.{u_2} S (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} S (NonAssocRing.toNonUnitalNonAssocRing.{u_2} S (Ring.toNonAssocRing.{u_2} S 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[inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44 : Module.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17] (g : LinearMap.{u_1, u_1, u_2, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3))) M M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.36 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3))) M M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44) g g_1) -> (forall (h : forall (x : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3))))) R (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.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3)))))) S x)), IsUnit.{u_4} (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44) (Module.End.instMonoid.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44) (DFunLike.coe.{max (succ u_1) (succ u_4), succ u_1, succ u_4} (RingHom.{u_1, u_4} R (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44) (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3)) (Semiring.toNonAssocSemiring.{u_4} (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44) (Module.End.instSemiring.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44))) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44) (RingHom.instFunLike.{u_1, u_4} R (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44) (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3)) (Semiring.toNonAssocSemiring.{u_4} (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44) (Module.End.instSemiring.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44))) (algebraMap.{u_1, u_4} R (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3 (Module.End.instSemiring.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44) (Module.End.instAlgebra.{u_1, u_1, u_4} R R M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44 (smulCommClass_self.{u_1, u_4} R M'' (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_4} R M'' (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_4} M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17) (Module.toDistribMulAction.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44))) (Algebra.toSMul.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_4} R M'' (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_4} R M'' (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_4} M'' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17) (Module.toDistribMulAction.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44))))) (Subtype.val.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.36) M'' (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_2} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.11 (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.36)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 (OreLocalization.instModuleOfIsScalarTower.{u_1, u_2, u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.36 R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.36 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1136412005._hygCtx._hyg.44 g_1 h))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M'' : Type u_4} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M''] [inst_3 : Module R M] [inst_4 : Module R M''] (g g_1 : M →ₗ[R] M''),\n  g = g_1 →\n    ∀ (h : ∀ (x : ↥S), IsUnit ((algebraMap R (Module.End R M'')) ↑x)),\n      LocalizedModule.lift S g h = LocalizedModule.lift S g_1 h","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M'' : Type u_4} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M''] [inst_3 : Module R M] [inst_4 : Module R M''] (g g_1 : M →ₗ[R] M''),\n  g = g_1 →\n    ∀ (h : ∀ (x : ↥S), IsUnit ((algebraMap R (Module.End R M'')) ↑x)),\n      LocalizedModule.lift S g h = LocalizedModule.lift S g_1 h","typeReferences":[["RingHom"],["smulCommClass_self"],["Subtype"],["CommMonoid","toMonoid"],["OreLocalization","oreSetComm"],["Module"],["Membership","mem"],["CommSemiring"],["RingHom","instFunLike"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["Subtype","val"],["Algebra","id"],["DFunLike","coe"],["LocalizedModule","lift"],["AddCommMonoid"],["Module","End","instAlgebra"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["Algebra","toSMul"],["DistribMulAction","toMulAction"],["IsScalarTower","right"],["Semiring","toModule"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["Module","End"],["SetLike","instMembership"],["Module","End","instSemiring"],["OreLocalization","instAddCommMonoidOreLocalization"],["LocalizedModule"],["IsScalarTower","left"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["OreLocalization","instModuleOfIsScalarTower"],["Semiring","toMonoidWithZero"],["IsUnit"],["LinearMap"],["CommSemiring","toCommMonoid"],["Submonoid"],["Module","toDistribMulAction"],["Module","End","instMonoid"],["algebraMap"]],"valueReferences":[["RingHom"],["smulCommClass_self"],["Subtype"],["CommMonoid","toMonoid"],["OreLocalization","oreSetComm"],["Membership","mem"],["RingHom","instFunLike"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["Subtype","val"],["Algebra","id"],["DFunLike","coe"],["LocalizedModule","lift"],["Module","End","instAlgebra"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["Algebra","toSMul"],["DistribMulAction","toMulAction"],["IsScalarTower","right"],["Semiring","toModule"],["Eq"],["Eq","rec"],["NonAssocSemiring","toMulZeroOneClass"],["Module","End"],["SetLike","instMembership"],["Module","End","instSemiring"],["OreLocalization","instAddCommMonoidOreLocalization"],["LocalizedModule"],["IsScalarTower","left"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["OreLocalization","instModuleOfIsScalarTower"],["Semiring","toMonoidWithZero"],["IsUnit"],["LinearMap"],["CommSemiring","toCommMonoid"],["Submonoid"],["Module","toDistribMulAction"],["Eq","refl"],["Module","End","instMonoid"],["algebraMap"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","iso_mk_one"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.478966506._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S : 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.Algebra.Module.LocalizedModule.Basic.478966506._hygCtx._hyg.3))))) {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.478966506._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.478966506._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.478966506._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.478966506._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.478966506._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.478966506._hygCtx._hyg.40 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[inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112 : Module.{u_1, u_6} R M₀ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106] {M₁ : Type.{u_7}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 : AddCommMonoid.{u_7} M₁] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141 : Module.{u_1, u_7} R M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135] (g : LinearMap.{u_1, u_1, u_6, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141 R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_7} R M₁ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_7} R M₁ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_7} M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135) (Module.toDistribMulAction.{u_1, u_7} R M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141))) (IsScalarTower.right.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3)))) (LocalizedModule.{u_1, u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 S M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : LocalizedModule.{u_1, u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 S M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112) => LocalizedModule.{u_1, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 S M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141) (LinearMap.instFunLike.{u_1, u_1, max u_1 u_6, max u_1 u_7} R R (LocalizedModule.{u_1, u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 S M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112) (LocalizedModule.{u_1, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 S M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_6} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) S) M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 (Module.toDistribMulAction.{u_1, u_6} R M₀ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112)) (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_7} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) S) M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 (Module.toDistribMulAction.{u_1, u_7} R M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141)) (OreLocalization.instModuleOfIsScalarTower.{u_1, u_6, u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) S) M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112 R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_6} R M₀ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_6} R M₀ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_6} M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106) (Module.toDistribMulAction.{u_1, u_6} R M₀ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112))) (IsScalarTower.right.{u_1, u_1} 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141 (smulCommClass_self.{u_1, u_7} R M₁ (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_7} R M₁ (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_7} M₁ 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_6} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) S) M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 (Module.toDistribMulAction.{u_1, u_6} R M₀ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112)) (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_7} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) S) M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 (Module.toDistribMulAction.{u_1, u_7} R M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141)) 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_7} R M₁ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_7} R M₁ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_7} M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135) (Module.toDistribMulAction.{u_1, u_7} R M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141))) (IsScalarTower.right.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) (smulCommClass_self.{u_1, max u_1 u_7} R (LocalizedModule.{u_1, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 S M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141) (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, max u_1 u_7} R (LocalizedModule.{u_1, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 S M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141) (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{max u_1 u_7} (LocalizedModule.{u_1, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 S M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141) (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_7} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_7} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) S) M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 (Module.toDistribMulAction.{u_1, u_7} R M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141)) 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_7} R M₁ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_7} M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135) (Module.toDistribMulAction.{u_1, u_7} R M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141))) (IsScalarTower.right.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3)))))))) (LinearMap.{u_1, u_1, u_6, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) M₀ M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : LinearMap.{u_1, u_1, u_6, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) M₀ M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141) => LinearMap.{u_1, u_1, max u_1 u_6, max u_1 u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (RingHom.id.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 (Module.toDistribMulAction.{u_1, u_7} R M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141)) (OreLocalization.instModuleOfIsScalarTower.{u_1, u_6, u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) S) M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112 R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_6} R M₀ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_6} R M₀ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_6} M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106) (Module.toDistribMulAction.{u_1, u_6} R M₀ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112))) (IsScalarTower.right.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) (OreLocalization.instModuleOfIsScalarTower.{u_1, u_7, u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) S) M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141 R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_7} R M₁ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_7} R M₁ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_7} M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135) 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 S M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112) (LocalizedModule.{u_1, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 S M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141) (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_6} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) S) M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 (Module.toDistribMulAction.{u_1, u_6} R M₀ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112)) (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_7} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) S) M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 (Module.toDistribMulAction.{u_1, u_7} R M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141)) (OreLocalization.instModuleOfIsScalarTower.{u_1, u_6, u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) S) M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112 R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_6} R M₀ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_6} R M₀ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_6} M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106) (Module.toDistribMulAction.{u_1, u_6} R M₀ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112))) (IsScalarTower.right.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) (OreLocalization.instModuleOfIsScalarTower.{u_1, u_7, u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) S) M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141 R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141 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(IsScalarTower.right.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) (smulCommClass_self.{u_1, max u_1 u_7} R (LocalizedModule.{u_1, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 S M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141) (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, max u_1 u_7} R (LocalizedModule.{u_1, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 S M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141) (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{max u_1 u_7} (LocalizedModule.{u_1, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 S M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141) (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_7} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) S) M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 (Module.toDistribMulAction.{u_1, u_7} R M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141))) (Module.toDistribMulAction.{u_1, max u_1 u_7} R (LocalizedModule.{u_1, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 S M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_7} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) S) M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 (Module.toDistribMulAction.{u_1, u_7} R M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141)) (OreLocalization.instModuleOfIsScalarTower.{u_1, u_7, u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) S) M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141 R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_7} R M₁ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_7} R M₁ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_7} M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135) (Module.toDistribMulAction.{u_1, u_7} R M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141))) (IsScalarTower.right.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))))))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3)))) (IsLocalizedModule.map.{u_1, u_6, max u_1 u_6, u_7, max u_1 u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 S M₀ (LocalizedModule.{u_1, u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 S M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_6} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) S) M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 (Module.toDistribMulAction.{u_1, u_6} R M₀ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112 (OreLocalization.instModuleOfIsScalarTower.{u_1, u_6, u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) S) M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112 R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_6} R M₀ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_6} R M₀ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_6} M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106) (Module.toDistribMulAction.{u_1, u_6} R M₀ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112))) (IsScalarTower.right.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) (LocalizedModule.mkLinearMap.{u_1, u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 S M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112) (localizedModuleIsLocalizedModule.{u_1, u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 S M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112) M₁ (LocalizedModule.{u_1, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 S M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_7} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) S) M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 (Module.toDistribMulAction.{u_1, u_7} R M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141)) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141 (OreLocalization.instModuleOfIsScalarTower.{u_1, u_7, u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) S) M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141 R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_7} R M₁ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_7} R M₁ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_7} M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135) (Module.toDistribMulAction.{u_1, u_7} R M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141))) (IsScalarTower.right.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) (LocalizedModule.mkLinearMap.{u_1, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 S M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141) (localizedModuleIsLocalizedModule.{u_1, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 S M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141)) g) (LocalizedModule.mk.{u_1, u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 S M₀ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112 m s)) (LocalizedModule.mk.{u_1, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3 S M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141 (DFunLike.coe.{max (succ u_6) (succ u_7), succ u_6, succ u_7} (LinearMap.{u_1, u_1, u_6, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3))) M₀ M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141) M₀ (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M₀) => M₁) (LinearMap.instFunLike.{u_1, u_1, u_6, u_7} R R M₀ M₁ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.106 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.135 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.112 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.141 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1091960139._hygCtx._hyg.3)))) g m) s)","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M₀ : Type u_6} [inst_1 : AddCommMonoid M₀]\n  [inst_2 : Module R M₀] {M₁ : Type u_7} [inst_3 : AddCommMonoid M₁] [inst_4 : Module R M₁] (g : M₀ →ₗ[R] M₁) (m : M₀)\n  (s : ↥S),\n  ((IsLocalizedModule.map S (LocalizedModule.mkLinearMap S M₀) (LocalizedModule.mkLinearMap S M₁)) g)\n      (LocalizedModule.mk m s) =\n    LocalizedModule.mk (g m) s","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M₀ : Type u_6} [inst_1 : AddCommMonoid M₀]\n  [inst_2 : Module R M₀] {M₁ : Type u_7} [inst_3 : AddCommMonoid M₁] [inst_4 : Module R M₁] (g : M₀ →ₗ[R] M₁) (m : M₀)\n  (s : ↥S),\n  ((IsLocalizedModule.map S (LocalizedModule.mkLinearMap S M₀) (LocalizedModule.mkLinearMap S M₁)) g)\n      (LocalizedModule.mk m s) =\n    LocalizedModule.mk (g m) 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S : 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.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))))) {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.71 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 S f] {N : Type.{u_6}} {N' : Type.{u_7}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 : AddCommMonoid.{u_6} N] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 : AddCommMonoid.{u_7} N'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 : Module.{u_1, u_6} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87 : Module.{u_1, u_7} R N' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80] (g : LinearMap.{u_1, u_1, u_6, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) N N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.100 : IsLocalizedModule.{u_1, u_6, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3 N N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87 S g], LinearMap.{u_1, u_1, max u_6 u_2, max u_7 u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) (LinearMap.{u_1, u_1, u_2, u_6} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) M N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83) (LinearMap.{u_1, u_1, u_3, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) M' N' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87) (LinearMap.addCommMonoid.{u_1, u_1, u_2, u_6} R R M N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3)))) (LinearMap.addCommMonoid.{u_1, u_1, u_3, u_7} R R M' N' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.87 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3)))) (LinearMap.module.{u_1, u_1, u_1, u_2, u_6} R R R M N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.83 (smulCommClass_self.{u_1, u_6} R N (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3) (DistribMulAction.toMulAction.{u_1, u_6} R N (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_6} N inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3880878766._hygCtx._hyg.77) 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→\n  [inst : CommSemiring R] →\n    (S : Submonoid R) →\n      {M : Type u_2} →\n        {M' : Type u_3} →\n          [inst_1 : AddCommMonoid M] →\n            [inst_2 : AddCommMonoid M'] →\n              [inst_3 : Module R M] →\n                [inst_4 : Module R M'] →\n                  (f : M →ₗ[R] M') →\n                    [IsLocalizedModule S f] →\n                      {N : Type u_6} →\n                        {N' : Type u_7} →\n                          [inst_6 : AddCommMonoid N] →\n                            [inst_7 : AddCommMonoid N'] →\n                              [inst_8 : Module R N] →\n                                [inst_9 : Module R N'] →\n                                  (g : N →ₗ[R] N') → [IsLocalizedModule S g] → (M →ₗ[R] N) →ₗ[R] M' →ₗ[R] N'","typeReferences":[["smulCommClass_self"],["IsLocalizedModule"],["CommMonoid","toMonoid"],["Module"],["CommSemiring"],["CommSemiring","toSemiring"],["LinearMap","module"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["CommSemiring","toCommMonoid"],["Submonoid"],["LinearMap","addCommMonoid"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["DistribMulAction","toMulAction"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["IsLocalizedModule","map","_proof_3"],["IsLocalizedModule","lift","_proof_1"],["LinearMap","mk"],["CommSemiring","toSemiring"],["LinearMap","module"],["AddHom","mk"],["LinearMap","comp"],["LinearMap"],["LinearMap","addCommMonoid"],["IsLocalizedModule","map_units"],["Semiring","toNonAssocSemiring"],["AddCommMonoid","toAddCommSemigroup"],["RingHom","id"],["LocalizedModule","lift'","_proof_2"],["IsLocalizedModule","map","_proof_2"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["IsLocalizedModule","lift"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","mk_add_mk"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2393162376._hygCtx._hyg.3 : CommSemiring.{u} R] {S : Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2393162376._hygCtx._hyg.3))))} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2393162376._hygCtx._hyg.9 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2393162376._hygCtx._hyg.12 : Module.{u, v} R M (CommSemiring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2393162376._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2393162376._hygCtx._hyg.9] {m1 : M} {m2 : M} {s1 : Subtype.{succ u} R (fun (x : R) => Membership.mem.{u, u} R (Submonoid.{u} R (MulZeroOneClass.toMulOneClass.{u} R (NonAssocSemiring.toMulZeroOneClass.{u} R (Semiring.toNonAssocSemiring.{u} R 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v} [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  {m1 m2 : M} {s1 s2 : ↥S},\n  LocalizedModule.mk m1 s1 + LocalizedModule.mk m2 s2 = LocalizedModule.mk (s2 • m1 + s1 • m2) (s1 * s2)","typeReadable":"∀ {R : Type u} [inst : CommSemiring R] {S : Submonoid R} {M : Type v} [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  {m1 m2 : M} {s1 s2 : ↥S},\n  LocalizedModule.mk m1 s1 + LocalizedModule.mk m2 s2 = LocalizedModule.mk (s2 • m1 + s1 • m2) (s1 * 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2480668754._hygCtx._hyg.3 : CommSemiring.{u_1} R] {S : 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.Algebra.Module.LocalizedModule.Basic.2480668754._hygCtx._hyg.3))))} {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2480668754._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2480668754._hygCtx._hyg.14 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2480668754._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2480668754._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2480668754._hygCtx._hyg.11] 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(Module.toDistribMulAction.{u_1, u_2} R A (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32))))))) a) (OfNat.ofNat.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A 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(AddMonoid.toZero.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (AddCommMonoid.toAddMonoid.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)) (OreLocalization.instAddCommMonoidOreLocalization.{u_1, u_2} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3) S) A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Module.toDistribMulAction.{u_1, u_2} R A (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.221733677._hygCtx._hyg.32)))))))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A] {S : Submonoid R}\n  (a : LocalizedModule S A), 0 * a = 0","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A] {S : Submonoid R}\n  (a : LocalizedModule S A), 0 * a = 0","typeReferences":[["MulOneClass","toMulOne"],["AddMonoid","toZero"],["OreLocalization","oreSetComm"],["CommSemiring"],["HMul","hMul"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["MulOne","toMul"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["Zero","toOfNat0"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["LocalizedModule","instMonoid"],["LocalizedModule"],["OreLocalization","instAddCommMonoidOreLocalization"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["CommSemiring","toCommMonoid"],["OfNat","ofNat"],["Algebra"],["Submonoid"],["Module","toDistribMulAction"],["instHMul"],["Semiring"]],"valueReferences":[["LocalizedModule","instSemiring","_proof_4"]]},{"isProp":false,"kind":"inductive","name":["IsLocalizedModule"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72 : CommSemiring.{u_1} R] {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83], (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.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))))) -> (LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) -> Prop","typeFull":"{R : Type u_1} →\n  [inst : CommSemiring R] →\n    {M : Type u_2} →\n      {M' : Type u_3} →\n        [inst_1 : AddCommMonoid M] →\n          [inst_2 : AddCommMonoid M'] →\n            [inst_3 : Module R M] → [inst_4 : Module R M'] → Submonoid R → (M →ₗ[R] M') → Prop","typeReadable":"{R : Type u_1} →\n  [inst : CommSemiring R] →\n    {M : Type u_2} →\n      {M' : Type u_3} →\n        [inst_1 : AddCommMonoid M] →\n          [inst_2 : AddCommMonoid M'] →\n            [inst_3 : Module R M] → [inst_4 : Module R M'] → Submonoid R → (M →ₗ[R] M') → Prop","typeReferences":[["Submonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Module"],["CommSemiring","toSemiring"],["CommSemiring"],["MulZeroOneClass","toMulOneClass"],["LinearMap"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Module","LocalizedModule","Basic",0,"IsLocalizedModule","liftOfLE_mk'","_simp_1_1"],"typeFallback":"forall {F : Type.{u_8}} {M : Type.{u_9}} {X : Type.{u_10}} {Y : Type.{u_11}} [inst._@.Mathlib.GroupTheory.GroupAction.Hom.4062404420._hygCtx._hyg.42 : SMul.{u_9, u_10} M X] [inst._@.Mathlib.GroupTheory.GroupAction.Hom.4062404420._hygCtx._hyg.46 : SMul.{u_9, u_11} M Y] [inst._@.Mathlib.GroupTheory.GroupAction.Hom.4062404420._hygCtx._hyg.50 : FunLike.{succ u_8, succ u_10, succ u_11} F X Y] [inst._@.Mathlib.GroupTheory.GroupAction.Hom.4062404420._hygCtx._hyg.55 : MulActionHomClass.{u_8, u_9, u_10, u_11} F M X Y 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.993575611._hygCtx._hyg.3) S) M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.993575611._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.993575611._hygCtx._hyg.36 R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.993575611._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.993575611._hygCtx._hyg.36 (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.993575611._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.993575611._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1116576693._hygCtx._hyg.81) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1116576693._hygCtx._hyg.84 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1116576693._hygCtx._hyg.88) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1116576693._hygCtx._hyg.101 : IsLocalizedModule.{u_1, u_6, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1116576693._hygCtx._hyg.3 M M' (AddCommGroup.toAddCommMonoid.{u_6} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1116576693._hygCtx._hyg.78) (AddCommGroup.toAddCommMonoid.{u_7} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1116576693._hygCtx._hyg.81) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1116576693._hygCtx._hyg.84 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1116576693._hygCtx._hyg.88 S f] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.1116576693._hygCtx._hyg.105 : AddCommGroup.{u_8} N] 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(SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3))))) R (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.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3)))))) S x)) M (Submonoid.smul.{u_1, u_2} R M (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3)))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.11) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.36)))) S)) c x₁) (HSMul.hSMul.{u_1, u_2, u_2} (Subtype.{succ u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.36)))) S)) c x₂))) -> Prop) (x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx.110.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.118 : Exists.{succ u_1} (Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R 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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.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3)))))) S x)) => Eq.{succ u_2} M (HSMul.hSMul.{u_1, u_2, u_2} (Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3))))) R (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.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3)))))) S x)) M M (instHSMul.{u_1, u_2} (Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3))))) R (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.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3)))))) S x)) M (Submonoid.smul.{u_1, u_2} R M (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3)))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.11) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.36)))) S)) c x₁) (HSMul.hSMul.{u_1, u_2, u_2} (Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R 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(CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3))))) R (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.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3)))))) S x)) M (Submonoid.smul.{u_1, u_2} R M (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3)))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.11) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.36)))) S)) c x₂))), (forall (c : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R 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(AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.11) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.36)))) S)) c x₁) (HSMul.hSMul.{u_1, u_2, u_2} (Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3))))) R (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 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x)) M (Submonoid.smul.{u_1, u_2} R M (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3)))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.11) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.36)))) S)) c x₂)), motive (Exists.intro.{succ u_1} (Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R 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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.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3)))))) S x)) => Eq.{succ u_2} M (HSMul.hSMul.{u_1, u_2, u_2} (Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3))))) R (Submonoid.instSetLike.{u_1} R 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(CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.11) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.36)))) S)) c x₁) (HSMul.hSMul.{u_1, u_2, u_2} (Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R 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(CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3))))) R (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.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3)))))) S x)) M (Submonoid.smul.{u_1, u_2} R M (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3)))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.11) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.36)))) S)) c x₂)) c h)) -> (motive x._@.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx.110.Mathlib.Algebra.Module.LocalizedModule.Basic.787447925._hygCtx._hyg.118)","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] {x₁ x₂ : M} (motive : (∃ c, c • x₁ = c • x₂) → Prop) (x : ∃ c, c • x₁ = c • x₂),\n  (∀ (c : ↥S) (h : c • x₁ = c • x₂), motive ⋯) → motive x","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] {x₁ x₂ : M} (motive : (∃ c, c • x₁ = c • x₂) → Prop) (x : ∃ c, c • x₁ = c • x₂),\n  (∀ (c : ↥S) (h : c • x₁ = c • x₂), motive ⋯) → motive x","typeReferences":[["Subtype"],["Module"],["Membership","mem"],["CommSemiring"],["Submonoid","smul"],["Exists","intro"],["MulZeroOneClass","toMulOneClass"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["Exists"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Submonoid"],["Module","toDistribMulAction"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Subtype"],["SetLike","instMembership"],["Submonoid","instSetLike"],["Submonoid","smul"],["Membership","mem"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Exists","casesOn"],["Submonoid"],["Module","toDistribMulAction"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["instHSMul"],["Eq"],["AddZero","toZero"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","lift'_mk"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2554959060._hygCtx._hyg.3 : CommSemiring.{u_1} R] (S : 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.Algebra.Module.LocalizedModule.Basic.2554959060._hygCtx._hyg.3))))) {M : Type.{u_2}} {M'' : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2554959060._hygCtx._hyg.11 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2554959060._hygCtx._hyg.17 : AddCommMonoid.{u_4} M''] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2554959060._hygCtx._hyg.36 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2554959060._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2554959060._hygCtx._hyg.11] 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2554959060._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2554959060._hygCtx._hyg.44) (Module.End.instSemiring.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2554959060._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2554959060._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2554959060._hygCtx._hyg.44))) (algebraMap.{u_1, u_4} R (Module.End.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2554959060._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2554959060._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2554959060._hygCtx._hyg.44) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2554959060._hygCtx._hyg.3 (Module.End.instSemiring.{u_1, u_4} R M'' (CommSemiring.toSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2554959060._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2554959060._hygCtx._hyg.44))) (Algebra.toSMul.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2554959060._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2554959060._hygCtx._hyg.3) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2554959060._hygCtx._hyg.3)) (IsScalarTower.left.{u_1, u_4} R M'' (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2554959060._hygCtx._hyg.3))) (DistribMulAction.toMulAction.{u_1, u_4} R M'' (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2554959060._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_4} M'' 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(CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2554959060._hygCtx._hyg.3)))) g m))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M'' : Type u_4} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M''] [inst_3 : Module R M] [inst_4 : Module R M''] (g : M →ₗ[R] M'')\n  (h : ∀ (x : ↥S), IsUnit ((algebraMap R (Module.End R M'')) ↑x)) (m : M) (s : ↥S),\n  LocalizedModule.lift' S g h (LocalizedModule.mk m s) = ↑⋯.unit⁻¹ (g m)","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} {M'' : Type u_4} [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M''] [inst_3 : Module R M] [inst_4 : Module R M''] (g : M →ₗ[R] M'')\n  (h : ∀ (x : ↥S), IsUnit ((algebraMap R (Module.End R M'')) ↑x)) (m : M) (s : ↥S),\n  LocalizedModule.lift' S g h (LocalizedModule.mk m s) = ↑⋯.unit⁻¹ (g 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3))) M M₂ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.87) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.118 : IsLocalizedModule.{u_1, u_2, u_6} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3 M M₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.83 S₁ f₁] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.122 : IsLocalizedModule.{u_1, u_2, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3 M M₂ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.87 S₂ f₂], Eq.{max (succ u_6) (succ u_7)} (LinearMap.{u_1, u_1, u_6, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3))) M₁ M₂ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.87) (IsLocalizedModule.liftOfLE.{u_1, u_2, u_6, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3 M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.36 M₁ M₂ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.87 S₁ S₂ h f₁ f₂ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.118 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.122) (IsLocalizedModule.lift.{u_1, u_2, u_6, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3 S₁ M M₁ M₂ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.87 f₁ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.118 f₂ (IsLocalizedModule.liftOfLE._proof_2.{u_1, u_7, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.3 M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.36 M₂ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.87 S₁ S₂ h f₂ inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2178164565._hygCtx._hyg.122))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {M : Type u_2} [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {M₁ : Type u_6}\n  {M₂ : Type u_7} [inst_3 : AddCommMonoid M₁] [inst_4 : AddCommMonoid M₂] [inst_5 : Module R M₁] [inst_6 : Module R M₂]\n  (S₁ S₂ : Submonoid R) (h : S₁ ≤ S₂) (f₁ : M →ₗ[R] M₁) (f₂ : M →ₗ[R] M₂) [inst_7 : IsLocalizedModule S₁ f₁]\n  [inst_8 : IsLocalizedModule S₂ f₂], IsLocalizedModule.liftOfLE S₁ S₂ h f₁ f₂ = IsLocalizedModule.lift S₁ f₁ f₂ ⋯","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {M : Type u_2} [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {M₁ : Type u_6}\n  {M₂ : Type u_7} [inst_3 : AddCommMonoid M₁] [inst_4 : AddCommMonoid M₂] [inst_5 : Module R M₁] [inst_6 : Module R M₂]\n  (S₁ S₂ : Submonoid R) (h : S₁ ≤ S₂) (f₁ : M →ₗ[R] M₁) (f₂ : M →ₗ[R] M₂) [inst_7 : IsLocalizedModule S₁ f₁]\n  [inst_8 : IsLocalizedModule S₂ f₂], IsLocalizedModule.liftOfLE S₁ S₂ h f₁ f₂ = IsLocalizedModule.lift S₁ f₁ f₂ ⋯","typeReferences":[["IsLocalizedModule"],["PartialOrder","toPreorder"],["Module"],["CommSemiring"],["CommSemiring","toSemiring"],["MulZeroOneClass","toMulOneClass"],["IsLocalizedModule","liftOfLE","_proof_2"],["LinearMap"],["Submonoid"],["IsLocalizedModule","liftOfLE"],["Submonoid","instPartialOrder"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["LE","le"],["IsLocalizedModule","lift"],["Eq"],["Preorder","toLE"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["IsLocalizedModule","liftOfLE"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Eq","refl"],["CommSemiring","toSemiring"],["LinearMap"]]},{"isProp":true,"kind":"theorem","name":["LocalizedModule","instRing","_proof_7"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 : CommSemiring.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29 : Ring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.32 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29)] {S : 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.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3))))} (a : LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonAssocRing.toNonUnitalNonAssocRing.{u_2} A (Ring.toNonAssocRing.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.32)), Eq.{succ (max u_1 u_2)} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonAssocRing.toNonUnitalNonAssocRing.{u_2} A (Ring.toNonAssocRing.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.32)) (HAdd.hAdd.{max u_1 u_2, max u_1 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.32)) (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonAssocRing.toNonUnitalNonAssocRing.{u_2} A (Ring.toNonAssocRing.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29) 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(NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonAssocRing.toNonUnitalNonAssocRing.{u_2} A (Ring.toNonAssocRing.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.32)) (AddMonoid.toAddSemigroup.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonAssocRing.toNonUnitalNonAssocRing.{u_2} A (Ring.toNonAssocRing.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.32)) (AddCommMonoid.toAddMonoid.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonAssocRing.toNonUnitalNonAssocRing.{u_2} A (Ring.toNonAssocRing.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.32)) 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(NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonAssocRing.toNonUnitalNonAssocRing.{u_2} A (Ring.toNonAssocRing.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.32)) (Semiring.toNonUnitalSemiring.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonAssocRing.toNonUnitalNonAssocRing.{u_2} A (Ring.toNonAssocRing.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.32)) (LocalizedModule.instSemiring.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 A (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.32 S)))))))) (Neg.neg.{max u_1 u_2} (LocalizedModule.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonAssocRing.toNonUnitalNonAssocRing.{u_2} A (Ring.toNonAssocRing.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.401737074._hygCtx._hyg.29)))) 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.32)))) (Semiring.toNonAssocSemiring.{max u_1 u_2} (OreLocalization.{u_1, u_2} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3))) S (OreLocalization.oreSetComm.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3) S) A (DistribMulAction.toMulAction.{u_1, u_2} R A (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.32 S))))))) (LocalizedModule.mk.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3 S A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.32) x (OfNat.ofNat.{u_1} (Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R 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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.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3)))))) S x)) (Submonoid.one.{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.Algebra.Module.LocalizedModule.Basic.3325664169._hygCtx._hyg.3)))) S)))))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A]\n  (x y : A), LocalizedModule.mk (x * y) 1 = LocalizedModule.mk x 1 * LocalizedModule.mk y 1","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {A : Type u_2} [inst_1 : Semiring A] [inst_2 : Algebra R A]\n  (x y : A), LocalizedModule.mk (x * y) 1 = LocalizedModule.mk x 1 * LocalizedModule.mk y 1","typeReferences":[["MulOneClass","toMulOne"],["Subtype"],["OreLocalization","oreSetComm"],["Membership","mem"],["CommSemiring"],["HMul","hMul"],["MulZeroOneClass","toMulOneClass"],["LocalizedModule","instSemiring"],["AddCommMonoid","toAddMonoid"],["LocalizedModule","mk"],["MulOne","toMul"],["OreLocalization"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["SetLike","instMembership"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Semiring","toMonoidWithZero"],["Submonoid","one"],["OfNat","ofNat"],["CommSemiring","toCommMonoid"],["Algebra"],["Submonoid"],["Module","toDistribMulAction"],["One","toOfNat1"],["instHMul"],["Semiring"]],"valueReferences":[["MulOneClass","toMulOne"],["Subtype"],["Eq","trans"],["OreLocalization","oreSetComm"],["Membership","mem"],["HMul","hMul"],["MulZeroOneClass","toMulOneClass"],["LocalizedModule","instSemiring"],["AddCommMonoid","toAddMonoid"],["LocalizedModule","mk"],["congrArg"],["Submonoid","toMulOneClass"],["MulOne","toMul"],["Semiring","toNonAssocSemiring"],["OreLocalization"],["Algebra","toModule"],["Monoid","toMulOneClass"],["forall_congr"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["LocalizedModule","instMonoid"],["SetLike","instMembership"],["LocalizedModule"],["NonUnitalNonAssocSemiring","toDistrib"],["True"],["Submonoid","instSetLike"],["Distrib","toMul"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["mul_one"],["Submonoid","one"],["CommSemiring","toCommMonoid"],["OfNat","ofNat"],["Submonoid"],["implies_true"],["eq_self"],["Module","toDistribMulAction"],["Submonoid","mul"],["One","toOfNat1"],["of_eq_true"],["instHMul"],["LocalizedModule","mk_mul_mk"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","mk'_mul_mk'"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.3 : CommSemiring.{u_1} R] {S : 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.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.3))))} {M : Type.{u_6}} {M' : Type.{u_7}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.77 : Semiring.{u_6} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.80 : Semiring.{u_7} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.83 : Algebra.{u_1, u_6} R M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.77] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.87 : Algebra.{u_1, u_7} R M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.80] (f : AlgHom.{u_1, u_6, u_7} R M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.87) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.97 : IsLocalizedModule.{u_1, u_6, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.3 M M' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_6} M (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_6} M (Semiring.toNonAssocSemiring.{u_6} M 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.87 f)] (m₁ : M) (m₂ : M) (s₁ : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.3))))) R (Submonoid.instSetLike.{u_1} R 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.87 f) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.97 m₁ s₁) (IsLocalizedModule.mk'.{u_1, u_6, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.3 S M M' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_6} M (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_6} M (Semiring.toNonAssocSemiring.{u_6} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.77))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} M' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} M' (Semiring.toNonAssocSemiring.{u_7} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.80))) (Algebra.toModule.{u_1, u_6} R M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.83) (Algebra.toModule.{u_1, u_7} R M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.87) (AlgHom.toLinearMap.{u_1, u_6, u_7} R M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.87 f) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.97 m₂ s₂)) (IsLocalizedModule.mk'.{u_1, u_6, u_7} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.3 S M M' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_6} M (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_6} M (Semiring.toNonAssocSemiring.{u_6} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.77))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} M' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} M' (Semiring.toNonAssocSemiring.{u_7} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.80))) (Algebra.toModule.{u_1, u_6} R M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.77 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.83) (Algebra.toModule.{u_1, u_7} R M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.3 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.3))))) R (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.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.3)))))) S x)) (Submonoid.mul.{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.Algebra.Module.LocalizedModule.Basic.2418543810._hygCtx._hyg.3)))) S)) s₁ s₂))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_6} {M' : Type u_7} [inst_1 : Semiring M]\n  [inst_2 : Semiring M'] [inst_3 : Algebra R M] [inst_4 : Algebra R M'] (f : M →ₐ[R] M')\n  [inst_5 : IsLocalizedModule S f.toLinearMap] (m₁ m₂ : M) (s₁ s₂ : ↥S),\n  IsLocalizedModule.mk' f.toLinearMap m₁ s₁ * IsLocalizedModule.mk' f.toLinearMap m₂ s₂ =\n    IsLocalizedModule.mk' f.toLinearMap (m₁ * m₂) (s₁ * s₂)","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_6} {M' : Type u_7} [inst_1 : Semiring M]\n  [inst_2 : Semiring M'] [inst_3 : Algebra R M] [inst_4 : Algebra R M'] (f : M →ₐ[R] M')\n  [inst_5 : IsLocalizedModule S f.toLinearMap] (m₁ m₂ : M) (s₁ s₂ : ↥S),\n  IsLocalizedModule.mk' f.toLinearMap m₁ s₁ * IsLocalizedModule.mk' f.toLinearMap m₂ s₂ =\n    IsLocalizedModule.mk' f.toLinearMap (m₁ * m₂) (s₁ * s₂)","typeReferences":[["IsLocalizedModule"],["Subtype"],["CommSemiring"],["Membership","mem"],["HMul","hMul"],["MulZeroOneClass","toMulOneClass"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["IsLocalizedModule","mk'"],["AlgHom"],["SetLike","instMembership"],["NonUnitalNonAssocSemiring","toDistrib"],["Submonoid","instSetLike"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Distrib","toMul"],["AlgHom","toLinearMap"],["Algebra"],["Submonoid"],["Submonoid","mul"],["instHMul"],["Semiring"]],"valueReferences":[["AlgHom"],["RingHomClass","toNonUnitalRingHomClass"],["NonUnitalNonAssocSemiring","toDistrib"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Distrib","toMul"],["AlgHom","toLinearMap"],["NonUnitalRingHomClass","toMulHomClass"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["IsLocalizedModule","mk'_mul_mk'_of_map_mul"],["map_mul"],["AlgHom","funLike"],["AlgHomClass","toRingHomClass"],["AlgHom","algHomClass"]]},{"isProp":false,"kind":"definition","name":["IsLocalizedModule","recOn"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72 : CommSemiring.{u_1} R] {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83] {S : 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.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))))} {f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109} {motive : (IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109 S f) -> Sort.{u}} (t : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109 S f), (forall (map_units : forall (x : Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))))) (SetLike.instMembership.{u_1, u_1} (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.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))))) R (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.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)))))) S x)), IsUnit.{u_3} (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (Module.End.instMonoid.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (DFunLike.coe.{max (succ u_1) (succ u_3), succ u_1, succ u_3} (RingHom.{u_1, u_3} R (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)) (Semiring.toNonAssocSemiring.{u_3} (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (Module.End.instSemiring.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109))) R (fun 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (Module.End.instSemiring.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109))) (algebraMap.{u_1, u_3} R (Module.End.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72 (Module.End.instSemiring.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109) (Module.End.instAlgebra.{u_1, u_1, u_3} R R M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109 (smulCommClass_self.{u_1, u_3} R M' (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) (DistribMulAction.toMulAction.{u_1, u_3} R M' 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)))))) S x)) M (Submonoid.smul.{u_1, u_2} R M (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105)))) S)) c x₁) (HSMul.hSMul.{u_1, u_2, u_2} (Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (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.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))))) (SetLike.instMembership.{u_1, u_1} (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R 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(CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))))) R (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.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)))))) S x)) M (Submonoid.smul.{u_1, u_2} R M (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72)))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72))) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105)))) S)) c x₂)))), motive (IsLocalizedModule.mk.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.72 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.83 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.105 inst._@.Mathlib.Algebra.Module.LocalizedModule.Basic.2956491502._hygCtx._hyg.109 S f map_units surj exists_of_eq)) -> (motive t)","typeFull":"{R : Type u_1} →\n  [inst : CommSemiring R] →\n    {M : Type u_2} →\n      {M' : Type u_3} →\n        [inst_1 : AddCommMonoid M] →\n          [inst_2 : AddCommMonoid M'] →\n            [inst_3 : Module R M] →\n              [inst_4 : Module R M'] →\n                {S : Submonoid R} →\n                  {f : M →ₗ[R] M'} →\n                    {motive : IsLocalizedModule S f → Sort u} →\n                      (t : IsLocalizedModule S f) →\n                        ((map_units : ∀ (x : ↥S), IsUnit ((algebraMap R (Module.End R M')) ↑x)) →\n                            (surj : ∀ (y : M'), ∃ x, x.2 • y = f x.1) →\n                              (exists_of_eq : ∀ {x₁ x₂ : M}, f x₁ = f x₂ → ∃ c, c • x₁ = c • x₂) → motive ⋯) →\n                          motive t","typeReadable":"{R : Type u_1} →\n  [inst : CommSemiring R] →\n    {M : Type u_2} →\n      {M' : Type u_3} →\n        [inst_1 : AddCommMonoid M] →\n          [inst_2 : AddCommMonoid M'] →\n            [inst_3 : Module R M] →\n              [inst_4 : Module R M'] →\n                {S : Submonoid R} →\n                  {f : M →ₗ[R] M'} →\n                    {motive : IsLocalizedModule S f → Sort u} →\n                      (t : IsLocalizedModule S f) →\n                        ((map_units : ∀ (x : ↥S), IsUnit ((algebraMap R (Module.End R M')) ↑x)) →\n                            (surj : ∀ (y : M'), ∃ x, x.2 • y = f x.1) →\n                              (exists_of_eq : ∀ {x₁ x₂ : M}, f x₁ = f x₂ → ∃ c, c • x₁ = c • x₂) → motive ⋯) →\n                          motive 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