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@@ -222,3 +222,30 @@ data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.NumberTheory.Padics.Height
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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.BigOperators.Ring.Nat.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Group.Action.TypeTags.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Group.Nat.Range.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Group.Nat.TypeTags.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.GroupWithZero.Action.Units.sym.json

@@ -0,0 +1 @@
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inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.91091314._hygCtx._hyg.7 y) b))","typeFull":"∀ {M : Type u_1} {α : Type u_2} [inst : Monoid M] [inst_1 : AddMonoid α] [inst_2 : DistribMulAction M α] (x y : Mˣ)\n  (b : α), (↑x * ↑y) • b = ↑x • ↑y • b","typeReadable":"∀ {M : Type u_1} {α : Type u_2} [inst : Monoid M] [inst_1 : AddMonoid α] [inst_2 : DistribMulAction M α] (x y : Mˣ)\n  (b : α), (↑x * ↑y) • b = ↑x • ↑y • b","typeReferences":[["SemigroupAction","toSMul"],["HMul","hMul"],["AddMonoid"],["Semigroup","toMul"],["Units","val"],["HSMul","hSMul"],["Monoid"],["DistribMulAction","toMulAction"],["instHSMul"],["instHMul"],["Monoid","toSemigroup"],["Eq"],["MulAction","toSemigroupAction"],["DistribMulAction"],["Units"]],"valueReferences":[["Units","val"],["DistribMulAction","toMulAction"],["Monoid","toSemigroup"],["MulAction","toSemigroupAction"],["SemigroupAction","mul_smul"]]},{"isProp":true,"kind":"theorem","name":["Equiv","smulRight_symm_apply"],"typeFallback":"forall {α : Type.{u_4}} {β : Type.{u_5}} [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2414492958._hygCtx._hyg.7 : GroupWithZero.{u_4} α] [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2414492958._hygCtx._hyg.10 : MulAction.{u_4, u_5} α β (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2414492958._hygCtx._hyg.7))] {a : α} (ha : Ne.{succ u_4} α a (OfNat.ofNat.{u_4} α 0 (Zero.toOfNat0.{u_4} α (MulZeroClass.toZero.{u_4} α (MulZeroOneClass.toMulZeroClass.{u_4} α (MonoidWithZero.toMulZeroOneClass.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2414492958._hygCtx._hyg.7))))))) (b : β), Eq.{succ u_5} β (DFunLike.coe.{succ u_5, succ u_5, succ u_5} (Equiv.{succ u_5, succ u_5} β β) β (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : β) => β) (EquivLike.toFunLike.{succ u_5, succ u_5, succ u_5} (Equiv.{succ u_5, succ u_5} β β) β β (Equiv.instEquivLike.{succ 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inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2414492958._hygCtx._hyg.7)))) a) b)","typeFull":"∀ {α : Type u_4} {β : Type u_5} [inst : GroupWithZero α] [inst_1 : MulAction α β] {a : α} (ha : a ≠ 0) (b : β),\n  (Equiv.smulRight ha).symm b = a⁻¹ • b","typeReadable":"∀ {α : Type u_4} {β : Type u_5} [inst : GroupWithZero α] [inst_1 : MulAction α β] {a : α} (ha : a ≠ 0) (b : β),\n  (Equiv.smulRight ha).symm b = a⁻¹ • b","typeReferences":[["Equiv","instEquivLike"],["SemigroupAction","toSMul"],["MonoidWithZero","toMulZeroOneClass"],["DFunLike","coe"],["Equiv"],["GroupWithZero","toMonoidWithZero"],["MonoidWithZero","toMonoid"],["EquivLike","toFunLike"],["Equiv","symm"],["instHSMul"],["Zero","toOfNat0"],["Monoid","toSemigroup"],["Eq"],["Equiv","smulRight"],["Inv","inv"],["InvOneClass","toInv"],["GroupWithZero"],["OfNat","ofNat"],["DivInvOneMonoid","toInvOneClass"],["MulZeroOneClass","toMulZeroClass"],["MulAction"],["MulZeroClass","toZero"],["HSMul","hSMul"],["Ne"],["MulAction","toSemigroupAction"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["Equiv","instEquivLike"],["Eq","refl"],["EquivLike","toFunLike"],["Equiv","symm"],["DFunLike","coe"],["Equiv"],["Equiv","smulRight"]]},{"isProp":true,"kind":"theorem","name":["Commute","smul_right_iff₀"],"typeFallback":"forall {α : Type.{u_4}} {β : Type.{u_5}} [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.7 : GroupWithZero.{u_4} α] [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.10 : MulAction.{u_4, u_5} α β (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.7))] {a : α} [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.15 : Mul.{u_5} β] [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.18 : SMulCommClass.{u_4, u_5, u_5} α β β (SemigroupAction.toSMul.{u_4, u_5} α β (Monoid.toSemigroup.{u_4} α (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_4, u_5} α β (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.10)) (instSMulOfMul.{u_5} β inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.23 : IsScalarTower.{u_4, u_5, u_5} α β β (SemigroupAction.toSMul.{u_4, u_5} α β (Monoid.toSemigroup.{u_4} α (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_4, u_5} α β (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.10)) (instSMulOfMul.{u_5} β inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.15) (SemigroupAction.toSMul.{u_4, u_5} α β (Monoid.toSemigroup.{u_4} α (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_4, u_5} α β (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.10))] {x : β} {y : β}, (Ne.{succ u_4} α a (OfNat.ofNat.{u_4} α 0 (Zero.toOfNat0.{u_4} α (MulZeroClass.toZero.{u_4} α (MulZeroOneClass.toMulZeroClass.{u_4} α (MonoidWithZero.toMulZeroOneClass.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.7))))))) -> (Iff (Commute.{u_5} β inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.15 x (HSMul.hSMul.{u_4, u_5, u_5} α β β (instHSMul.{u_4, u_5} α β (SemigroupAction.toSMul.{u_4, u_5} α β (Monoid.toSemigroup.{u_4} α (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_4, u_5} α β (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.10))) a y)) (Commute.{u_5} β inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.15 x y))","typeFull":"∀ {α : Type u_4} {β : Type u_5} [inst : GroupWithZero α] [inst_1 : MulAction α β] {a : α} [inst_2 : Mul β]\n  [SMulCommClass α β β] [IsScalarTower α β β] {x y : β}, a ≠ 0 → (Commute x (a • y) ↔ Commute x y)","typeReadable":"∀ {α : Type u_4} {β : Type u_5} [inst : GroupWithZero α] [inst_1 : MulAction α β] {a : α} [inst_2 : Mul β]\n  [SMulCommClass α β β] [IsScalarTower α β β] {x y : β}, a ≠ 0 → (Commute x (a • y) ↔ Commute x y)","typeReferences":[["IsScalarTower"],["Mul"],["SemigroupAction","toSMul"],["instSMulOfMul"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["OfNat","ofNat"],["SMulCommClass"],["MulZeroOneClass","toMulZeroClass"],["MulAction"],["Commute"],["MulZeroClass","toZero"],["Iff"],["HSMul","hSMul"],["MonoidWithZero","toMonoid"],["GroupWithZero","toMonoidWithZero"],["instHSMul"],["Zero","toOfNat0"],["Ne"],["Monoid","toSemigroup"],["MulAction","toSemigroupAction"]],"valueReferences":[["Units","smulCommClass_left"],["Units","instGroup"],["SemigroupAction","toSMul"],["instSMulOfMul"],["Units","instIsScalarTower"],["Units","mk0"],["Commute","smul_right_iff"],["Units","instMulAction"],["MonoidWithZero","toMonoid"],["GroupWithZero","toMonoidWithZero"],["Monoid","toSemigroup"],["MulAction","toSemigroupAction"],["Units"]]},{"isProp":true,"kind":"theorem","name":["IsUnit","smul_eq_zero","_simp_1"],"typeFallback":"forall {G : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3584149550._hygCtx._hyg.7 : Monoid.{u_2} G] [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3584149550._hygCtx._hyg.10 : AddMonoid.{u_3} M] [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3584149550._hygCtx._hyg.13 : DistribMulAction.{u_2, u_3} G M inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3584149550._hygCtx._hyg.7 inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3584149550._hygCtx._hyg.10] {u : G} {x : M}, (IsUnit.{u_2} G inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3584149550._hygCtx._hyg.7 u) -> (Eq.{1} Prop (Eq.{succ u_3} M (HSMul.hSMul.{u_2, u_3, u_3} G M M (instHSMul.{u_2, u_3} G M (SMulZeroClass.toSMul.{u_2, u_3} G M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3584149550._hygCtx._hyg.10))) (DistribSMul.toSMulZeroClass.{u_2, u_3} G M (AddMonoid.toAddZeroClass.{u_3} M inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3584149550._hygCtx._hyg.10) (DistribMulAction.toDistribSMul.{u_2, u_3} G M inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3584149550._hygCtx._hyg.7 inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3584149550._hygCtx._hyg.10 inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3584149550._hygCtx._hyg.13)))) u x) (OfNat.ofNat.{u_3} M 0 (Zero.toOfNat0.{u_3} M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3584149550._hygCtx._hyg.10)))))) (Eq.{succ u_3} M x (OfNat.ofNat.{u_3} M 0 (Zero.toOfNat0.{u_3} M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3584149550._hygCtx._hyg.10)))))))","typeFull":"∀ {G : Type u_2} {M : Type u_3} [inst : Monoid G] [inst_1 : AddMonoid M] [inst_2 : DistribMulAction G M] {u : G}\n  {x : M}, IsUnit u → (u • x = 0) = (x = 0)","typeReadable":"∀ {G : Type u_2} {M : Type u_3} [inst : Monoid G] [inst_1 : AddMonoid M] [inst_2 : DistribMulAction G M] {u : G}\n  {x : M}, IsUnit u → (u • x = 0) = (x = 0)","typeReferences":[["DistribMulAction","toDistribSMul"],["AddMonoid"],["IsUnit"],["SMulZeroClass","toSMul"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["HSMul","hSMul"],["Monoid"],["instHSMul"],["Zero","toOfNat0"],["Eq"],["AddZero","toZero"],["DistribMulAction"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["DistribMulAction","toDistribSMul"],["SMulZeroClass","toSMul"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["HSMul","hSMul"],["IsUnit","smul_eq_zero"],["instHSMul"],["Zero","toOfNat0"],["Eq"],["AddZero","toZero"],["propext"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":false,"kind":"definition","name":["Units","instSMulZeroClass"],"typeFallback":"forall {M : Type.{u_3}} {α : Type.{u_4}} [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3486630159._hygCtx._hyg.7 : Monoid.{u_3} M] [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3486630159._hygCtx._hyg.10 : Zero.{u_4} α] [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3486630159._hygCtx._hyg.13 : SMulZeroClass.{u_3, u_4} M α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3486630159._hygCtx._hyg.10], SMulZeroClass.{u_3, u_4} (Units.{u_3} M inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3486630159._hygCtx._hyg.7) α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3486630159._hygCtx._hyg.10","typeFull":"{M : Type u_3} → {α : Type u_4} → [inst : Monoid M] → [inst_1 : Zero α] → [SMulZeroClass M α] → SMulZeroClass Mˣ α","typeReadable":"{M : Type u_3} → {α : Type u_4} → [inst : Monoid M] → [inst_1 : Zero α] → [SMulZeroClass M α] → SMulZeroClass Mˣ α","typeReferences":[["SMulZeroClass"],["Monoid"],["Zero"],["Units"]],"valueReferences":[["SMulZeroClass","mk"],["Units","instSMulZeroClass","_proof_1"],["SMulZeroClass","toSMul"],["Units","instSMul"],["Units"]]},{"isProp":true,"kind":"theorem","name":["Commute","smul_left_iff₀","_simp_1"],"typeFallback":"forall {α : Type.{u_4}} {β : Type.{u_5}} [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.7 : GroupWithZero.{u_4} α] [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.10 : MulAction.{u_4, u_5} α β (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.7))] {a : α} [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.15 : Mul.{u_5} β] [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.18 : SMulCommClass.{u_4, u_5, u_5} α β β (SemigroupAction.toSMul.{u_4, u_5} α β (Monoid.toSemigroup.{u_4} α (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_4, u_5} α β (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.10)) (instSMulOfMul.{u_5} β inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.23 : IsScalarTower.{u_4, u_5, u_5} α β β (SemigroupAction.toSMul.{u_4, u_5} α β (Monoid.toSemigroup.{u_4} α (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_4, u_5} α β (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.10)) (instSMulOfMul.{u_5} β inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.15) (SemigroupAction.toSMul.{u_4, u_5} α β (Monoid.toSemigroup.{u_4} α (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_4, u_5} α β (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.10))] {x : β} {y : β}, (Ne.{succ u_4} α a (OfNat.ofNat.{u_4} α 0 (Zero.toOfNat0.{u_4} α (MulZeroClass.toZero.{u_4} α (MulZeroOneClass.toMulZeroClass.{u_4} α (MonoidWithZero.toMulZeroOneClass.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.7))))))) -> (Eq.{1} Prop (Commute.{u_5} β inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.15 (HSMul.hSMul.{u_4, u_5, u_5} α β β (instHSMul.{u_4, u_5} α β (SemigroupAction.toSMul.{u_4, u_5} α β (Monoid.toSemigroup.{u_4} α (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_4, u_5} α β (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.10))) a x) y) (Commute.{u_5} β inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.15 x y))","typeFull":"∀ {α : Type u_4} {β : Type u_5} [inst : GroupWithZero α] [inst_1 : MulAction α β] {a : α} [inst_2 : Mul β]\n  [SMulCommClass α β β] [IsScalarTower α β β] {x y : β}, a ≠ 0 → Commute (a • x) y = Commute x y","typeReadable":"∀ {α : Type u_4} {β : Type u_5} [inst : GroupWithZero α] [inst_1 : MulAction α β] {a : α} [inst_2 : Mul β]\n  [SMulCommClass α β β] [IsScalarTower α β β] {x y : β}, a ≠ 0 → Commute (a • x) y = Commute x 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α","typeReferences":[["AddZeroClass"],["Monoid"],["DistribSMul"],["Units"]],"valueReferences":[["DistribSMul","mk"],["AddZeroClass","toAddZero"],["Units","instSMulZeroClass"],["AddZero","toZero"],["Units"],["Units","instDistribSMulUnits","_proof_1"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Units","instDistribMulAction","_proof_4"],"typeFallback":"forall {M : Type.{u_1}} {α : Type.{u_2}} [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.91091314._hygCtx._hyg.7 : Monoid.{u_1} M] [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.91091314._hygCtx._hyg.10 : AddMonoid.{u_2} α] [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.91091314._hygCtx._hyg.13 : DistribMulAction.{u_1, u_2} M α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.91091314._hygCtx._hyg.7 inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.91091314._hygCtx._hyg.10] (a : Units.{u_1} M inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.91091314._hygCtx._hyg.7) (x : α) (y : α), 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inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.91091314._hygCtx._hyg.10 inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.91091314._hygCtx._hyg.13))))) a x) (HSMul.hSMul.{u_1, u_2, u_2} (Units.{u_1} M inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.91091314._hygCtx._hyg.7) α α (instHSMul.{u_1, u_2} (Units.{u_1} M inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.91091314._hygCtx._hyg.7) α (SMulZeroClass.toSMul.{u_1, u_2} (Units.{u_1} M inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.91091314._hygCtx._hyg.7) α (AddZero.toZero.{u_2} α (AddZeroClass.toAddZero.{u_2} α (AddMonoid.toAddZeroClass.{u_2} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.91091314._hygCtx._hyg.10))) (DistribSMul.toSMulZeroClass.{u_1, u_2} (Units.{u_1} M inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.91091314._hygCtx._hyg.7) α (AddMonoid.toAddZeroClass.{u_2} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.91091314._hygCtx._hyg.10) (Units.instDistribSMulUnits.{u_1, u_2} M α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.91091314._hygCtx._hyg.7 (AddMonoid.toAddZeroClass.{u_2} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.91091314._hygCtx._hyg.10) (DistribMulAction.toDistribSMul.{u_1, u_2} M α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.91091314._hygCtx._hyg.7 inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.91091314._hygCtx._hyg.10 inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.91091314._hygCtx._hyg.13))))) a y))","typeFull":"∀ {M : Type u_1} {α : Type u_2} [inst : Monoid M] [inst_1 : AddMonoid α] [inst_2 : DistribMulAction M α] (a : Mˣ)\n  (x y : α), a • (x + y) = a • x + a • y","typeReadable":"∀ {M : Type u_1} {α : Type u_2} [inst : Monoid M] [inst_1 : AddMonoid α] [inst_2 : DistribMulAction M α] (a : Mˣ)\n  (x y : α), a • (x + y) = a • x + a • y","typeReferences":[["instHAdd"],["DistribMulAction","toDistribSMul"],["AddMonoid"],["SMulZeroClass","toSMul"],["Units","instDistribSMulUnits"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["HAdd","hAdd"],["HSMul","hSMul"],["Monoid"],["instHSMul"],["Eq"],["AddZero","toZero"],["DistribMulAction"],["Units"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["DistribSMul","smul_add"],["DistribMulAction","toDistribSMul"],["Units","instDistribSMulUnits"],["Units"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["inv_smul_smul₀"],"typeFallback":"forall {α : Type.{u_4}} {β : Type.{u_5}} [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3090395900._hygCtx._hyg.7 : GroupWithZero.{u_4} α] [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3090395900._hygCtx._hyg.10 : MulAction.{u_4, u_5} α β (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α 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(InvOneClass.toInv.{u_4} α (DivInvOneMonoid.toInvOneClass.{u_4} α (DivisionMonoid.toDivInvOneMonoid.{u_4} α (GroupWithZero.toDivisionMonoid.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3090395900._hygCtx._hyg.7)))) a) (HSMul.hSMul.{u_4, u_5, u_5} α β β (instHSMul.{u_4, u_5} α β (SemigroupAction.toSMul.{u_4, u_5} α β (Monoid.toSemigroup.{u_4} α (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3090395900._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_4, u_5} α β (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3090395900._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3090395900._hygCtx._hyg.10))) a x)) x)","typeFull":"∀ {α : Type u_4} {β : Type u_5} [inst : GroupWithZero α] [inst_1 : MulAction α β] {a : α},\n  a ≠ 0 → ∀ (x : β), a⁻¹ • a • x = x","typeReadable":"∀ {α : Type u_4} {β : Type u_5} [inst : GroupWithZero α] [inst_1 : MulAction α β] {a : α},\n  a ≠ 0 → ∀ (x : β), a⁻¹ • a • x = x","typeReferences":[["Inv","inv"],["InvOneClass","toInv"],["SemigroupAction","toSMul"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["OfNat","ofNat"],["DivInvOneMonoid","toInvOneClass"],["MulAction"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["MonoidWithZero","toMonoid"],["GroupWithZero","toMonoidWithZero"],["HSMul","hSMul"],["instHSMul"],["Ne"],["Zero","toOfNat0"],["Monoid","toSemigroup"],["Eq"],["MulAction","toSemigroupAction"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["Units","instGroup"],["Units","instMulAction"],["inv_smul_smul"],["GroupWithZero","toMonoidWithZero"],["MonoidWithZero","toMonoid"],["Units","mk0"],["Units"]]},{"isProp":true,"kind":"theorem","name":["inv_smul_eq_iff₀"],"typeFallback":"forall {α : Type.{u_4}} {β : Type.{u_5}} [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.1355589418._hygCtx._hyg.7 : GroupWithZero.{u_4} α] [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.1355589418._hygCtx._hyg.10 : MulAction.{u_4, u_5} α β (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.1355589418._hygCtx._hyg.7))] {a : α}, (Ne.{succ u_4} α a (OfNat.ofNat.{u_4} α 0 (Zero.toOfNat0.{u_4} α (MulZeroClass.toZero.{u_4} α (MulZeroOneClass.toMulZeroClass.{u_4} α (MonoidWithZero.toMulZeroOneClass.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.1355589418._hygCtx._hyg.7))))))) -> (forall {x : β} {y : β}, Iff (Eq.{succ u_5} β (HSMul.hSMul.{u_4, u_5, u_5} α β β (instHSMul.{u_4, u_5} α β (SemigroupAction.toSMul.{u_4, u_5} α β (Monoid.toSemigroup.{u_4} α (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.1355589418._hygCtx._hyg.7))) 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y","typeReferences":[["Inv","inv"],["InvOneClass","toInv"],["SemigroupAction","toSMul"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["OfNat","ofNat"],["DivInvOneMonoid","toInvOneClass"],["MulAction"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Iff"],["MonoidWithZero","toMonoid"],["GroupWithZero","toMonoidWithZero"],["HSMul","hSMul"],["instHSMul"],["Ne"],["Zero","toOfNat0"],["Monoid","toSemigroup"],["Eq"],["MulAction","toSemigroupAction"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["Units","instGroup"],["Units","instMulAction"],["GroupWithZero","toMonoidWithZero"],["MonoidWithZero","toMonoid"],["inv_smul_eq_iff"],["Units","mk0"],["Units"]]},{"isProp":true,"kind":"theorem","name":["Units","instMulDistribMulAction","_proof_2"],"typeFallback":"forall {M : Type.{u_1}} {α : Type.{u_2}} [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2775186516._hygCtx._hyg.7 : Monoid.{u_1} M] [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2775186516._hygCtx._hyg.10 : Monoid.{u_2} α] [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2775186516._hygCtx._hyg.13 : MulDistribMulAction.{u_1, u_2} M α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2775186516._hygCtx._hyg.7 inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2775186516._hygCtx._hyg.10] (m : Units.{u_1} M inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2775186516._hygCtx._hyg.7), Eq.{succ u_2} α (HSMul.hSMul.{u_1, u_2, u_2} M α α (instHSMul.{u_1, u_2} M α (SemigroupAction.toSMul.{u_1, u_2} M α (Monoid.toSemigroup.{u_1} M inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2775186516._hygCtx._hyg.7) (MulAction.toSemigroupAction.{u_1, u_2} M α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2775186516._hygCtx._hyg.7 (MulDistribMulAction.toMulAction.{u_1, u_2} M α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2775186516._hygCtx._hyg.7 inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2775186516._hygCtx._hyg.10 inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2775186516._hygCtx._hyg.13)))) (Units.val.{u_1} M inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2775186516._hygCtx._hyg.7 m) (OfNat.ofNat.{u_2} α 1 (One.toOfNat1.{u_2} α (MulOne.toOne.{u_2} α (MulOneClass.toMulOne.{u_2} α (Monoid.toMulOneClass.{u_2} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2775186516._hygCtx._hyg.10)))))) (OfNat.ofNat.{u_2} α 1 (One.toOfNat1.{u_2} α (MulOne.toOne.{u_2} α (MulOneClass.toMulOne.{u_2} α (Monoid.toMulOneClass.{u_2} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2775186516._hygCtx._hyg.10)))))","typeFull":"∀ {M : Type u_1} {α : Type u_2} [inst : Monoid M] [inst_1 : Monoid α] [inst_2 : MulDistribMulAction M α] (m : Mˣ),\n  ↑m • 1 = 1","typeReadable":"∀ {M : Type u_1} {α : Type u_2} [inst : Monoid M] [inst_1 : Monoid α] [inst_2 : MulDistribMulAction M α] (m : Mˣ),\n  ↑m • 1 = 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(GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2414492958._hygCtx._hyg.7))))))) -> (Equiv.{succ u_5, succ u_5} β β)","typeFull":"{α : Type u_4} → {β : Type u_5} → [inst : GroupWithZero α] → [MulAction α β] → {a : α} → a ≠ 0 → β ≃ β","typeReadable":"{α : Type u_4} → {β : Type u_5} → [inst : GroupWithZero α] → [MulAction α β] → {a : α} → a ≠ 0 → β ≃ β","typeReferences":[["MulZeroOneClass","toMulZeroClass"],["MulAction"],["MulZeroClass","toZero"],["GroupWithZero","toMonoidWithZero"],["MonoidWithZero","toMonoid"],["MonoidWithZero","toMulZeroOneClass"],["Zero","toOfNat0"],["Ne"],["GroupWithZero"],["Equiv"],["OfNat","ofNat"]],"valueReferences":[["Inv","inv"],["InvOneClass","toInv"],["Equiv","mk"],["SemigroupAction","toSMul"],["DivInvOneMonoid","toInvOneClass"],["inv_smul_smul₀"],["HSMul","hSMul"],["MonoidWithZero","toMonoid"],["GroupWithZero","toMonoidWithZero"],["instHSMul"],["Monoid","toSemigroup"],["MulAction","toSemigroupAction"],["smul_inv_smul₀"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]]},{"isProp":true,"kind":"theorem","name":["IsUnit","smul_eq_zero"],"typeFallback":"forall {G : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3584149550._hygCtx._hyg.7 : Monoid.{u_2} G] [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3584149550._hygCtx._hyg.10 : 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inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3584149550._hygCtx._hyg.10 inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3584149550._hygCtx._hyg.13)))) u x) (OfNat.ofNat.{u_3} M 0 (Zero.toOfNat0.{u_3} M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3584149550._hygCtx._hyg.10)))))) (Eq.{succ u_3} M x (OfNat.ofNat.{u_3} M 0 (Zero.toOfNat0.{u_3} M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3584149550._hygCtx._hyg.10)))))))","typeFull":"∀ {G : Type u_2} {M : Type u_3} [inst : Monoid G] [inst_1 : AddMonoid M] [inst_2 : DistribMulAction G M] {u : G}\n  {x : M}, IsUnit u → (u • x = 0 ↔ x = 0)","typeReadable":"∀ {G : Type u_2} {M : Type u_3} [inst : Monoid G] [inst_1 : AddMonoid M] [inst_2 : DistribMulAction G M] {u : G}\n  {x : M}, IsUnit u → (u • x = 0 ↔ x = 0)","typeReferences":[["DistribMulAction","toDistribSMul"],["AddMonoid"],["IsUnit"],["SMulZeroClass","toSMul"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Iff"],["HSMul","hSMul"],["Monoid"],["instHSMul"],["Zero","toOfNat0"],["Eq"],["AddZero","toZero"],["DistribMulAction"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["IsUnit","unit"],["smul_eq_zero_iff_eq"],["Units","instGroup"],["Units","instDistribMulAction"],["Units"]]},{"isProp":true,"kind":"theorem","name":["Commute","smul_left_iff₀"],"typeFallback":"forall {α : Type.{u_4}} {β : Type.{u_5}} [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.7 : GroupWithZero.{u_4} α] [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.10 : MulAction.{u_4, u_5} α β (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.7))] {a : α} [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.15 : Mul.{u_5} β] [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.18 : SMulCommClass.{u_4, u_5, u_5} α β β (SemigroupAction.toSMul.{u_4, u_5} α β (Monoid.toSemigroup.{u_4} α (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_4, u_5} α β (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.10)) (instSMulOfMul.{u_5} β inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.23 : IsScalarTower.{u_4, u_5, u_5} α β β (SemigroupAction.toSMul.{u_4, u_5} α β (Monoid.toSemigroup.{u_4} α (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_4, u_5} α β (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.10)) (instSMulOfMul.{u_5} β inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.15) (SemigroupAction.toSMul.{u_4, u_5} α β (Monoid.toSemigroup.{u_4} α (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_4, u_5} α β (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.10))] {x : β} {y : β}, (Ne.{succ u_4} α a (OfNat.ofNat.{u_4} α 0 (Zero.toOfNat0.{u_4} α (MulZeroClass.toZero.{u_4} α (MulZeroOneClass.toMulZeroClass.{u_4} α (MonoidWithZero.toMulZeroOneClass.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.7))))))) -> (Iff (Commute.{u_5} β inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.15 (HSMul.hSMul.{u_4, u_5, u_5} α β β (instHSMul.{u_4, u_5} α β (SemigroupAction.toSMul.{u_4, u_5} α β (Monoid.toSemigroup.{u_4} α (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_4, u_5} α β (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.10))) a x) y) (Commute.{u_5} β inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2242874324._hygCtx._hyg.15 x y))","typeFull":"∀ {α : Type u_4} {β : Type u_5} [inst : GroupWithZero α] [inst_1 : MulAction α β] {a : α} [inst_2 : Mul β]\n  [SMulCommClass α β β] [IsScalarTower α β β] {x y : β}, a ≠ 0 → (Commute (a • x) y ↔ Commute x y)","typeReadable":"∀ {α : Type u_4} {β : Type u_5} [inst : GroupWithZero α] [inst_1 : MulAction α β] {a : α} [inst_2 : Mul β]\n  [SMulCommClass α β β] [IsScalarTower α β β] {x y : β}, a ≠ 0 → (Commute (a • x) y ↔ Commute x y)","typeReferences":[["IsScalarTower"],["Mul"],["SemigroupAction","toSMul"],["instSMulOfMul"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["OfNat","ofNat"],["SMulCommClass"],["MulZeroOneClass","toMulZeroClass"],["MulAction"],["Commute"],["MulZeroClass","toZero"],["Iff"],["HSMul","hSMul"],["MonoidWithZero","toMonoid"],["GroupWithZero","toMonoidWithZero"],["instHSMul"],["Zero","toOfNat0"],["Ne"],["Monoid","toSemigroup"],["MulAction","toSemigroupAction"]],"valueReferences":[["Units","smulCommClass_left"],["Units","instGroup"],["Commute","smul_left_iff"],["SemigroupAction","toSMul"],["instSMulOfMul"],["Units","instIsScalarTower"],["Units","mk0"],["Units","instMulAction"],["MonoidWithZero","toMonoid"],["GroupWithZero","toMonoidWithZero"],["Monoid","toSemigroup"],["MulAction","toSemigroupAction"],["Units"]]},{"isProp":true,"kind":"theorem","name":["Equiv","smulRight_apply"],"typeFallback":"forall {α : Type.{u_4}} {β : Type.{u_5}} [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2414492958._hygCtx._hyg.7 : GroupWithZero.{u_4} α] [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2414492958._hygCtx._hyg.10 : MulAction.{u_4, u_5} α β (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2414492958._hygCtx._hyg.7))] {a : α} (ha : Ne.{succ u_4} α a (OfNat.ofNat.{u_4} α 0 (Zero.toOfNat0.{u_4} α (MulZeroClass.toZero.{u_4} α (MulZeroOneClass.toMulZeroClass.{u_4} α (MonoidWithZero.toMulZeroOneClass.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2414492958._hygCtx._hyg.7))))))) (b : β), Eq.{succ u_5} β (DFunLike.coe.{succ u_5, succ u_5, succ u_5} (Equiv.{succ u_5, succ u_5} β β) β (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : β) => β) (EquivLike.toFunLike.{succ u_5, succ u_5, succ u_5} (Equiv.{succ u_5, succ u_5} β β) β β (Equiv.instEquivLike.{succ u_5, succ u_5} β β)) (Equiv.smulRight.{u_4, u_5} α β inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2414492958._hygCtx._hyg.7 inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2414492958._hygCtx._hyg.10 a ha) b) (HSMul.hSMul.{u_4, u_5, u_5} α β β (instHSMul.{u_4, u_5} α β (SemigroupAction.toSMul.{u_4, u_5} α β (Monoid.toSemigroup.{u_4} α (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2414492958._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_4, u_5} α β (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2414492958._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2414492958._hygCtx._hyg.10))) a b)","typeFull":"∀ {α : Type u_4} {β : Type u_5} [inst : GroupWithZero α] [inst_1 : MulAction α β] {a : α} (ha : a ≠ 0) (b : β),\n  (Equiv.smulRight ha) b = a • b","typeReadable":"∀ {α : Type u_4} {β : Type u_5} [inst : GroupWithZero α] [inst_1 : MulAction α β] {a : α} (ha : a ≠ 0) (b : β),\n  (Equiv.smulRight ha) b = a • b","typeReferences":[["Equiv","instEquivLike"],["SemigroupAction","toSMul"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["DFunLike","coe"],["OfNat","ofNat"],["Equiv"],["MulAction"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["HSMul","hSMul"],["EquivLike","toFunLike"],["MonoidWithZero","toMonoid"],["GroupWithZero","toMonoidWithZero"],["instHSMul"],["Ne"],["Zero","toOfNat0"],["Monoid","toSemigroup"],["Eq"],["MulAction","toSemigroupAction"],["Equiv","smulRight"]],"valueReferences":[["Equiv","instEquivLike"],["Eq","refl"],["EquivLike","toFunLike"],["DFunLike","coe"],["Equiv"],["Equiv","smulRight"]]},{"isProp":true,"kind":"theorem","name":["Commute","smul_right_iff₀","_simp_1"],"typeFallback":"forall {α : Type.{u_4}} {β : Type.{u_5}} [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.7 : GroupWithZero.{u_4} α] [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.10 : MulAction.{u_4, u_5} α β (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.7))] {a : α} [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.15 : Mul.{u_5} β] [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.18 : SMulCommClass.{u_4, u_5, u_5} α β β (SemigroupAction.toSMul.{u_4, u_5} α β (Monoid.toSemigroup.{u_4} α (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_4, u_5} α β (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.10)) (instSMulOfMul.{u_5} β inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.23 : IsScalarTower.{u_4, u_5, u_5} α β β (SemigroupAction.toSMul.{u_4, u_5} α β (Monoid.toSemigroup.{u_4} α (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_4, u_5} α β (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.10)) (instSMulOfMul.{u_5} β inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.15) (SemigroupAction.toSMul.{u_4, u_5} α β (Monoid.toSemigroup.{u_4} α (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_4, u_5} α β (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.10))] {x : β} {y : β}, (Ne.{succ u_4} α a (OfNat.ofNat.{u_4} α 0 (Zero.toOfNat0.{u_4} α (MulZeroClass.toZero.{u_4} α (MulZeroOneClass.toMulZeroClass.{u_4} α (MonoidWithZero.toMulZeroOneClass.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.7))))))) -> (Eq.{1} Prop (Commute.{u_5} β inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.2460374789._hygCtx._hyg.15 x (HSMul.hSMul.{u_4, u_5, u_5} α β β (instHSMul.{u_4, u_5} α β (SemigroupAction.toSMul.{u_4, u_5} α β (Monoid.toSemigroup.{u_4} α (MonoidWithZero.toMonoid.{u_4} α (GroupWithZero.toMonoidWithZero.{u_4} α 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y","typeReferences":[["IsScalarTower"],["Mul"],["SemigroupAction","toSMul"],["instSMulOfMul"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["OfNat","ofNat"],["SMulCommClass"],["MulZeroOneClass","toMulZeroClass"],["MulAction"],["Commute"],["MulZeroClass","toZero"],["HSMul","hSMul"],["MonoidWithZero","toMonoid"],["GroupWithZero","toMonoidWithZero"],["instHSMul"],["Zero","toOfNat0"],["Ne"],["Monoid","toSemigroup"],["Eq"],["MulAction","toSemigroupAction"]],"valueReferences":[["Commute","smul_right_iff₀"],["Commute"],["GroupWithZero","toMonoidWithZero"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["SemigroupAction","toSMul"],["instHSMul"],["Monoid","toSemigroup"],["MulAction","toSemigroupAction"],["propext"]]},{"isProp":true,"kind":"theorem","name":["Units","instDistribSMulUnits","_proof_1"],"typeFallback":"forall {M : Type.{u_1}} {α : Type.{u_2}} [inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3043935366._hygCtx._hyg.7 : Monoid.{u_1} M] 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[inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3486630159._hygCtx._hyg.13 : SMulZeroClass.{u_1, u_2} M α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3486630159._hygCtx._hyg.10] (m : Units.{u_1} M inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3486630159._hygCtx._hyg.7), Eq.{succ u_2} α (HSMul.hSMul.{u_1, u_2, u_2} M α α (instHSMul.{u_1, u_2} M α (SMulZeroClass.toSMul.{u_1, u_2} M α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3486630159._hygCtx._hyg.10 inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3486630159._hygCtx._hyg.13)) (Units.val.{u_1} M inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3486630159._hygCtx._hyg.7 m) (OfNat.ofNat.{u_2} α 0 (Zero.toOfNat0.{u_2} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3486630159._hygCtx._hyg.10))) (OfNat.ofNat.{u_2} α 0 (Zero.toOfNat0.{u_2} α inst._@.Mathlib.Algebra.GroupWithZero.Action.Units.3486630159._hygCtx._hyg.10))","typeFull":"∀ {M : Type u_1} {α : Type u_2} [inst : Monoid M] [inst_1 : Zero α] [inst_2 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(x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => A) (RingHom.instFunLike.{u_1, u_2} R A (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.3)) (Semiring.toNonAssocSemiring.{u_2} A (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.10))) (algebraMap.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.13) (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 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inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.13 S s))) (IsLocalizedModule.mk'.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.3 S A Aₛ (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.IsLocalization.1935041079._hygCtx._hyg.10)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} Aₛ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} Aₛ (Semiring.toNonAssocSemiring.{u_3} Aₛ (CommSemiring.toSemiring.{u_3} Aₛ inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.17)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.13) (Algebra.toModule.{u_1, u_3} R Aₛ inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_3} Aₛ inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.17) inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.24) (AlgHom.toLinearMap.{u_1, u_2, u_3} R A Aₛ inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.10) (CommSemiring.toSemiring.{u_3} Aₛ inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.17) inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.24 (IsScalarTower.toAlgHom.{u_1, u_2, u_3} R A Aₛ inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.10 (CommSemiring.toSemiring.{u_3} Aₛ inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.17) inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.28)) (instIsLocalizedModuleToLinearMapToAlgHomOfIsLocalizationAlgebraMapSubmonoid.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.3 S A Aₛ inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.1935041079._hygCtx._hyg.33) x s)","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {A : Type u_2} {Aₛ : Type u_3} [inst_1 : CommSemiring A]\n  [inst_2 : Algebra R A] [inst_3 : CommSemiring Aₛ] [inst_4 : Algebra A Aₛ] [inst_5 : Algebra R Aₛ]\n  [inst_6 : IsScalarTower R A Aₛ] [inst_7 : IsLocalization (Algebra.algebraMapSubmonoid A S) Aₛ] {x : A} {s : ↥S},\n  IsLocalization.mk' Aₛ x ⟨(algebraMap R A) ↑s, ⋯⟩ =\n    IsLocalizedModule.mk' (IsScalarTower.toAlgHom R A Aₛ).toLinearMap x s","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {A : Type u_2} {Aₛ : Type u_3} [inst_1 : CommSemiring A]\n  [inst_2 : Algebra R A] [inst_3 : CommSemiring Aₛ] [inst_4 : Algebra A Aₛ] [inst_5 : Algebra R Aₛ]\n  [inst_6 : IsScalarTower R A Aₛ] [inst_7 : IsLocalization (Algebra.algebraMapSubmonoid A S) Aₛ] {x : A} {s : ↥S},\n  IsLocalization.mk' Aₛ x ⟨(algebraMap R A) ↑s, ⋯⟩ =\n    IsLocalizedModule.mk' (IsScalarTower.toAlgHom R A Aₛ).toLinearMap x s","typeReferences":[["RingHom"],["instIsLocalizedModuleToLinearMapToAlgHomOfIsLocalizationAlgebraMapSubmonoid"],["Subtype"],["IsScalarTower"],["Membership","mem"],["CommSemiring"],["RingHom","instFunLike"],["MulZeroOneClass","toMulOneClass"],["IsScalarTower","toAlgHom"],["DFunLike","coe"],["Subtype","val"],["Algebra","algebraMapSubmonoid"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["Algebra","toSMul"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["IsLocalizedModule","mk'"],["Algebra","mem_algebraMapSubmonoid_of_mem"],["SetLike","instMembership"],["Submonoid","instSetLike"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["IsLocalization","mk'"],["AlgHom","toLinearMap"],["IsLocalization"],["Algebra"],["Submonoid"],["Subtype","mk"],["algebraMap"]],"valueReferences":[["RingHom"],["instIsLocalizedModuleToLinearMapToAlgHomOfIsLocalizationAlgebraMapSubmonoid"],["Membership","mem"],["Submonoid","smul"],["SMulZeroClass","toSMul"],["IsScalarTower","toAlgHom"],["Subtype","val"],["Algebra","algebraMapSubmonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Eq","symm"],["NonAssocSemiring","toMulZeroOneClass"],["IsLocalizedModule","mk'_cancel'"],["IsLocalizedModule","mk'"],["DistribSMul","toSMulZeroClass"],["Algebra","mem_algebraMapSubmonoid_of_mem"],["SetLike","instMembership"],["Submonoid","instSetLike"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["AlgHom","toLinearMap"],["AddZeroClass","toAddZero"],["LinearMap"],["Submonoid"],["HSMul","hSMul"],["id"],["Eq","mpr"],["AddZero","toZero"],["IsLocalizedModule","smul_inj"],["algebraMap"],["AddMonoid","toAddZeroClass"],["algebraMap_smul"],["Subtype"],["LinearMap","instFunLike"],["RingHom","instFunLike"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["congrArg"],["Algebra","toModule"],["IsLocalization","smul_mk'_self"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["propext"],["Submonoid","smul_def"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["IsLocalization","mk'"],["Semiring","toMonoidWithZero"],["Module","toDistribMulAction"],["Subtype","mk"]]},{"isProp":true,"kind":"theorem","name":["instIsLocalizedModuleToLinearMapToAlgHomOfIsLocalizationAlgebraMapSubmonoid"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._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.IsLocalization.3717745716._hygCtx._hyg.3))))) {A : Type.{u_2}} {Aₛ : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.10 : CommSemiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.13 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.10)] [inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.17 : CommSemiring.{u_3} Aₛ] [inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.20 : Algebra.{u_2, u_3} A Aₛ inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.10 (CommSemiring.toSemiring.{u_3} Aₛ inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.17)] [inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.24 : Algebra.{u_1, u_3} R Aₛ inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_3} Aₛ inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.17)] [inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.28 : IsScalarTower.{u_1, u_2, u_3} R A Aₛ (Algebra.toSMul.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.13) (Algebra.toSMul.{u_2, u_3} A Aₛ inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.10 (CommSemiring.toSemiring.{u_3} Aₛ inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.17) inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.20) (Algebra.toSMul.{u_1, u_3} R Aₛ inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_3} Aₛ inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.17) inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.24)] [inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.33 : IsLocalization.{u_2, u_3} A inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.10 (Algebra.algebraMapSubmonoid.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.3 A (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.13 S) Aₛ inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.20], IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.3 A Aₛ (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.IsLocalization.3717745716._hygCtx._hyg.10)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} Aₛ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} Aₛ (Semiring.toNonAssocSemiring.{u_3} Aₛ (CommSemiring.toSemiring.{u_3} Aₛ inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.17)))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.13) (Algebra.toModule.{u_1, u_3} R Aₛ inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_3} Aₛ inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.17) inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.24) S (AlgHom.toLinearMap.{u_1, u_2, u_3} R A Aₛ inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.10) (CommSemiring.toSemiring.{u_3} Aₛ inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.17) inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.24 (IsScalarTower.toAlgHom.{u_1, u_2, u_3} R A Aₛ inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.10 (CommSemiring.toSemiring.{u_3} Aₛ inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.17) inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.3717745716._hygCtx._hyg.28))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {A : Type u_2} {Aₛ : Type u_3} [inst_1 : CommSemiring A]\n  [inst_2 : Algebra R A] [inst_3 : CommSemiring Aₛ] [inst_4 : Algebra A Aₛ] [inst_5 : Algebra R Aₛ]\n  [inst_6 : IsScalarTower R A Aₛ] [IsLocalization (Algebra.algebraMapSubmonoid A S) Aₛ],\n  IsLocalizedModule S (IsScalarTower.toAlgHom R A Aₛ).toLinearMap","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {A : Type u_2} {Aₛ : Type u_3} [inst_1 : CommSemiring A]\n  [inst_2 : Algebra R A] [inst_3 : CommSemiring Aₛ] [inst_4 : Algebra A Aₛ] [inst_5 : Algebra R Aₛ]\n  [inst_6 : IsScalarTower R A Aₛ] [IsLocalization (Algebra.algebraMapSubmonoid A S) Aₛ],\n  IsLocalizedModule S (IsScalarTower.toAlgHom R A Aₛ).toLinearMap","typeReferences":[["IsLocalizedModule"],["IsScalarTower"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["CommSemiring"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["MulZeroOneClass","toMulOneClass"],["AlgHom","toLinearMap"],["IsScalarTower","toAlgHom"],["Algebra"],["IsLocalization"],["Submonoid"],["Algebra","algebraMapSubmonoid"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["Algebra","toSMul"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["Algebra","algebraMapSubmonoid"],["IsLocalizedModule"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["Iff","mpr"],["CommSemiring","toSemiring"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["isLocalizedModule_iff_isLocalization"],["AlgHom","toLinearMap"],["IsScalarTower","toAlgHom"],["IsLocalization"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Module","LocalizedModule","IsLocalization",0,"isLocalizedModule_iff_isLocalization","_simp_1_5"],"typeFallback":"forall (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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(CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.5) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.8))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.11) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.11) (smulCommClass_self.{u_1, u_2} R A (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.5) (DistribMulAction.toMulAction.{u_1, u_2} R A (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.8)))) (Module.toDistribMulAction.{u_1, u_2} R A (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.5) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.8))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.11)))) (Algebra.toSMul.{u_1, u_1} R R inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.5 (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.5) (Algebra.id.{u_1} R inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.5)) (IsScalarTower.left.{u_1, u_2} R A (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.5))) (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.Algebra.Bilinear.1861906991._hygCtx._hyg.5))) (AddCommMonoid.toAddMonoid.{u_2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.8)))) (Module.toDistribMulAction.{u_1, u_2} R A (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.5) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.8))) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.11)))))) (Algebra.lmul.{u_1, u_2} R A inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.11) x)) (IsUnit.{u_2} A (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A inst._@.Mathlib.Algebra.Algebra.Bilinear.1861906991._hygCtx._hyg.8)) x)","typeFull":"∀ {R : Type u_1} {A : Type u_2} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A] {x : A},\n  IsUnit ((Algebra.lmul R A) x) = IsUnit x","typeReadable":"∀ {R : Type u_1} {A : Type u_2} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A] {x : A},\n  IsUnit ((Algebra.lmul R A) x) = IsUnit x","typeReferences":[["smulCommClass_self"],["CommMonoid","toMonoid"],["Algebra","lmul"],["CommSemiring"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["Algebra","id"],["Module","End","instAlgebra"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["MonoidWithZero","toMonoid"],["Algebra","toSMul"],["DistribMulAction","toMulAction"],["AlgHom","funLike"],["Eq"],["AlgHom"],["Module","End"],["Module","End","instSemiring"],["IsScalarTower","left"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["IsUnit"],["CommSemiring","toCommMonoid"],["Algebra"],["Module","toDistribMulAction"],["Module","End","instMonoid"],["Semiring"]],"valueReferences":[["smulCommClass_self"],["CommMonoid","toMonoid"],["Algebra","lmul"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["Algebra","id"],["Module","End","instAlgebra"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["Algebra","toSMul"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["AlgHom","funLike"],["propext"],["AlgHom"],["Module","End"],["Module","End","instSemiring"],["IsScalarTower","left"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["IsUnit"],["CommSemiring","toCommMonoid"],["Module","toDistribMulAction"],["Module","End","instMonoid"],["Algebra","lmul_isUnit_iff"]]},{"isProp":true,"kind":"theorem","name":["IsLocalization","mk'_eq_mk'"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.308450374._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.IsLocalization.308450374._hygCtx._hyg.3))))) (A : Type.{u_2}) [inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.308450374._hygCtx._hyg.10 : CommSemiring.{u_2} A] [inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.308450374._hygCtx._hyg.13 : Algebra.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.308450374._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.308450374._hygCtx._hyg.10)] [inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.308450374._hygCtx._hyg.33 : IsLocalization.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.308450374._hygCtx._hyg.3 S A inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.308450374._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.308450374._hygCtx._hyg.13] (x : R) (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.IsLocalization.308450374._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.IsLocalization.308450374._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.IsLocalization.308450374._hygCtx._hyg.3)))))) S x)), Eq.{succ u_2} A (IsLocalization.mk'.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.308450374._hygCtx._hyg.3 S A inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.308450374._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.308450374._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.308450374._hygCtx._hyg.33 x s) (IsLocalizedModule.mk'.{u_1, u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.308450374._hygCtx._hyg.3 S R A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.308450374._hygCtx._hyg.3)))) (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.IsLocalization.308450374._hygCtx._hyg.10)))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.308450374._hygCtx._hyg.3)) (Algebra.toModule.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.308450374._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.308450374._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.308450374._hygCtx._hyg.13) (Algebra.linearMap.{u_1, u_2} R A inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.308450374._hygCtx._hyg.3 (CommSemiring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.308450374._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.308450374._hygCtx._hyg.13) (instIsLocalizedModuleLinearMapOfIsLocalization.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.308450374._hygCtx._hyg.3 S A inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.308450374._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.308450374._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LocalizedModule.IsLocalization.308450374._hygCtx._hyg.33) x s)","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) (A : Type u_2) [inst_1 : CommSemiring A]\n  [inst_2 : Algebra R A] [inst_3 : IsLocalization S A] (x : R) (s : ↥S),\n  IsLocalization.mk' A x s = IsLocalizedModule.mk' (Algebra.linearMap R A) x s","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) (A : Type u_2) [inst_1 : CommSemiring A]\n  [inst_2 : Algebra R A] [inst_3 : IsLocalization S A] (x : R) (s : ↥S),\n  IsLocalization.mk' A x s = IsLocalizedModule.mk' (Algebra.linearMap R A) x s","typeReferences":[["instIsLocalizedModuleLinearMapOfIsLocalization"],["Subtype"],["SetLike","instMembership"],["Submonoid","instSetLike"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring"],["CommSemiring","toSemiring"],["Membership","mem"],["IsLocalization","mk'"],["Algebra","linearMap"],["MulZeroOneClass","toMulOneClass"],["Algebra"],["IsLocalization"],["Submonoid"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["Semiring","toModule"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["IsLocalizedModule","mk'"]],"valueReferences":[["instIsLocalizedModuleLinearMapOfIsLocalization"],["Subtype"],["LinearMap","instFunLike"],["Membership","mem"],["Submonoid","smul"],["MulZeroOneClass","toMulOneClass"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["congrArg"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Algebra","toModule"],["IsLocalization","smul_mk'_self"],["MonoidWithZero","toMonoid"],["Eq","symm"],["instHSMul"],["Semiring","toModule"],["Eq"],["IsLocalizedModule","mk'_cancel'"],["propext"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["IsLocalizedModule","mk'"],["SetLike","instMembership"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["IsLocalization","mk'"],["Semiring","toMonoidWithZero"],["Algebra","linearMap"],["LinearMap"],["AddZeroClass","toAddZero"],["Submonoid"],["Module","toDistribMulAction"],["HSMul","hSMul"],["id"],["Eq","mpr"],["AddZero","toZero"],["IsLocalizedModule","smul_inj"],["AddMonoid","toAddZeroClass"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.Rat.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["map_nnrat_smul"],"typeFallback":"forall {M : Type.{u_1}} {M₂ : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Rat.3031445689._hygCtx._hyg.4 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Rat.3031445689._hygCtx._hyg.7 : AddCommMonoid.{u_2} M₂] [_instM : Module.{0, u_1} NNRat M (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield)) inst._@.Mathlib.Algebra.Module.Rat.3031445689._hygCtx._hyg.4] [_instM₂ : Module.{0, u_2} NNRat M₂ (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield)) inst._@.Mathlib.Algebra.Module.Rat.3031445689._hygCtx._hyg.7] {F : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Rat.3031445689._hygCtx._hyg.21 : FunLike.{succ u_3, succ u_1, succ u_2} F M M₂] [inst._@.Mathlib.Algebra.Module.Rat.3031445689._hygCtx._hyg.26 : AddMonoidHomClass.{u_3, u_1, u_2} F M M₂ (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Rat.3031445689._hygCtx._hyg.4))) (AddZeroClass.toAddZero.{u_2} M₂ (AddMonoid.toAddZeroClass.{u_2} M₂ (AddCommMonoid.toAddMonoid.{u_2} M₂ inst._@.Mathlib.Algebra.Module.Rat.3031445689._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Rat.3031445689._hygCtx._hyg.21] (f : F) (c : NNRat) (x : M), Eq.{succ u_2} M₂ (DFunLike.coe.{succ u_3, succ u_1, succ u_2} F M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) inst._@.Mathlib.Algebra.Module.Rat.3031445689._hygCtx._hyg.21 f (HSMul.hSMul.{0, u_1, u_1} NNRat M M (instHSMul.{0, u_1} NNRat M (SMulZeroClass.toSMul.{0, u_1} NNRat 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.Rat.3031445689._hygCtx._hyg.4)))) (DistribSMul.toSMulZeroClass.{0, u_1} NNRat M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Rat.3031445689._hygCtx._hyg.4)) (DistribMulAction.toDistribSMul.{0, u_1} NNRat M (MonoidWithZero.toMonoid.{0} NNRat (Semiring.toMonoidWithZero.{0} NNRat (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield)))) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Rat.3031445689._hygCtx._hyg.4) (Module.toDistribMulAction.{0, u_1} NNRat M (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield)) inst._@.Mathlib.Algebra.Module.Rat.3031445689._hygCtx._hyg.4 _instM))))) c x)) (HSMul.hSMul.{0, u_2, u_2} NNRat M₂ M₂ (instHSMul.{0, u_2} NNRat M₂ (SMulZeroClass.toSMul.{0, u_2} NNRat M₂ (AddZero.toZero.{u_2} M₂ (AddZeroClass.toAddZero.{u_2} M₂ (AddMonoid.toAddZeroClass.{u_2} M₂ (AddCommMonoid.toAddMonoid.{u_2} M₂ inst._@.Mathlib.Algebra.Module.Rat.3031445689._hygCtx._hyg.7)))) (DistribSMul.toSMulZeroClass.{0, u_2} NNRat M₂ (AddMonoid.toAddZeroClass.{u_2} M₂ (AddCommMonoid.toAddMonoid.{u_2} M₂ inst._@.Mathlib.Algebra.Module.Rat.3031445689._hygCtx._hyg.7)) (DistribMulAction.toDistribSMul.{0, u_2} NNRat M₂ (MonoidWithZero.toMonoid.{0} NNRat (Semiring.toMonoidWithZero.{0} NNRat (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield)))) (AddCommMonoid.toAddMonoid.{u_2} M₂ inst._@.Mathlib.Algebra.Module.Rat.3031445689._hygCtx._hyg.7) (Module.toDistribMulAction.{0, u_2} NNRat M₂ (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield)) inst._@.Mathlib.Algebra.Module.Rat.3031445689._hygCtx._hyg.7 _instM₂))))) c (DFunLike.coe.{succ u_3, succ u_1, succ u_2} F M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) inst._@.Mathlib.Algebra.Module.Rat.3031445689._hygCtx._hyg.21 f x))","typeFull":"∀ {M : Type u_1} {M₂ : Type u_2} [inst : AddCommMonoid M] [inst_1 : AddCommMonoid M₂] [_instM : Module ℚ≥0 M]\n  [_instM₂ : Module ℚ≥0 M₂] {F : Type u_3} [inst_2 : FunLike F M M₂] [AddMonoidHomClass F M M₂] (f : F) (c : ℚ≥0)\n  (x : M), f (c • x) = c • f x","typeReadable":"∀ {M : Type u_1} {M₂ : Type u_2} [inst : AddCommMonoid M] [inst_1 : AddCommMonoid M₂] [_instM : Module ℚ≥0 M]\n  [_instM₂ : Module ℚ≥0 M₂] {F : Type u_3} [inst_2 : FunLike F M M₂] [AddMonoidHomClass F M M₂] (f : F) (c : ℚ≥0)\n  (x : M), f (c • x) = c • f x","typeReferences":[["FunLike"],["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["DivisionSemiring","toSemiring"],["AddCommMonoid","toAddMonoid"],["AddMonoidHomClass"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["Module","toDistribMulAction"],["AddCommMonoid"],["NNRat"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["instHSMul"],["Semifield","toDivisionSemiring"],["Eq"],["AddZero","toZero"],["NNRat","instSemifield"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["map_nnratCast_smul"],["NNRat"],["Semifield","toDivisionSemiring"],["NNRat","instSemifield"]]},{"isProp":true,"kind":"theorem","name":["SMulCommClass","rat'"],"typeFallback":"forall {α : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Rat.2674285734._hygCtx._hyg.6 : AddCommGroup.{v} M] [inst._@.Mathlib.Algebra.Module.Rat.2674285734._hygCtx._hyg.9 : DistribSMul.{u, v} α M (AddMonoid.toAddZeroClass.{v} M (SubNegMonoid.toAddMonoid.{v} M (AddGroup.toSubNegMonoid.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Rat.2674285734._hygCtx._hyg.6))))] [inst._@.Mathlib.Algebra.Module.Rat.2674285734._hygCtx._hyg.13 : Module.{0, v} Rat M Rat.semiring (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Rat.2674285734._hygCtx._hyg.6)], SMulCommClass.{u, 0, v} α Rat M (SMulZeroClass.toSMul.{u, v} α M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (SubNegMonoid.toAddMonoid.{v} M (AddGroup.toSubNegMonoid.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Rat.2674285734._hygCtx._hyg.6)))))) (DistribSMul.toSMulZeroClass.{u, v} α M (AddMonoid.toAddZeroClass.{v} M (SubNegMonoid.toAddMonoid.{v} M (AddGroup.toSubNegMonoid.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Rat.2674285734._hygCtx._hyg.6)))) inst._@.Mathlib.Algebra.Module.Rat.2674285734._hygCtx._hyg.9)) (SMulZeroClass.toSMul.{0, v} Rat M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (SubNegMonoid.toAddMonoid.{v} M (AddGroup.toSubNegMonoid.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Rat.2674285734._hygCtx._hyg.6)))))) (DistribSMul.toSMulZeroClass.{0, v} Rat M (AddMonoid.toAddZeroClass.{v} M (SubNegMonoid.toAddMonoid.{v} M (AddGroup.toSubNegMonoid.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Rat.2674285734._hygCtx._hyg.6)))) (DistribMulAction.toDistribSMul.{0, v} Rat M Rat.monoid (SubNegMonoid.toAddMonoid.{v} M (AddGroup.toSubNegMonoid.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Rat.2674285734._hygCtx._hyg.6))) (Module.toDistribMulAction.{0, v} Rat M Rat.semiring (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Rat.2674285734._hygCtx._hyg.6) inst._@.Mathlib.Algebra.Module.Rat.2674285734._hygCtx._hyg.13))))","typeFull":"∀ {α : Type u} {M : Type v} [inst : AddCommGroup M] [inst_1 : DistribSMul α M] [inst_2 : Module ℚ M],\n  SMulCommClass α ℚ M","typeReadable":"∀ {α : Type u} {M : Type v} [inst : AddCommGroup M] [inst_1 : DistribSMul α M] [inst_2 : Module ℚ M],\n  SMulCommClass α ℚ M","typeReferences":[["Rat","semiring"],["Module"],["AddCommGroup","toAddGroup"],["Rat","monoid"],["DistribMulAction","toDistribSMul"],["AddCommGroup"],["SMulZeroClass","toSMul"],["AddZeroClass","toAddZero"],["DistribSMul"],["SMulCommClass"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["Rat"],["AddCommGroup","toAddCommMonoid"],["AddGroup","toSubNegMonoid"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Rat","semiring"],["SMulCommClass","symm"],["AddCommGroup","toAddGroup"],["Rat","monoid"],["DistribMulAction","toDistribSMul"],["SMulCommClass","rat"],["SMulZeroClass","toSMul"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["AddCommGroup","toAddCommMonoid"],["Rat"],["AddGroup","toSubNegMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Rat","cast_smul_eq_qsmul"],"typeFallback":"forall {M : Type.{u_1}} (R : Type.{u_3}) [inst._@.Mathlib.Algebra.Module.Rat.3467823910._hygCtx._hyg.5 : DivisionRing.{u_3} R] [inst._@.Mathlib.Algebra.Module.Rat.3467823910._hygCtx._hyg.8 : MulAction.{u_3, u_1} R M (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R (DivisionSemiring.toSemiring.{u_3} R (DivisionRing.toDivisionSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.Rat.3467823910._hygCtx._hyg.5))))] [inst._@.Mathlib.Algebra.Module.Rat.3467823910._hygCtx._hyg.12 : MulAction.{0, u_1} Rat M Rat.monoid] [inst._@.Mathlib.Algebra.Module.Rat.3467823910._hygCtx._hyg.18 : IsScalarTower.{0, u_3, u_1} Rat R M (Rat.smulDivisionRing.{u_3} R inst._@.Mathlib.Algebra.Module.Rat.3467823910._hygCtx._hyg.5) (SemigroupAction.toSMul.{u_3, u_1} R M (Monoid.toSemigroup.{u_3} R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R (DivisionSemiring.toSemiring.{u_3} R (DivisionRing.toDivisionSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.Rat.3467823910._hygCtx._hyg.5))))) (MulAction.toSemigroupAction.{u_3, u_1} R M (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R (DivisionSemiring.toSemiring.{u_3} R (DivisionRing.toDivisionSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.Rat.3467823910._hygCtx._hyg.5)))) inst._@.Mathlib.Algebra.Module.Rat.3467823910._hygCtx._hyg.8)) (SemigroupAction.toSMul.{0, u_1} Rat M (Monoid.toSemigroup.{0} Rat Rat.monoid) (MulAction.toSemigroupAction.{0, u_1} Rat M Rat.monoid inst._@.Mathlib.Algebra.Module.Rat.3467823910._hygCtx._hyg.12))] (q : Rat) (x : M), Eq.{succ u_1} M (HSMul.hSMul.{u_3, u_1, u_1} R M M (instHSMul.{u_3, u_1} R M (SemigroupAction.toSMul.{u_3, u_1} R M (Monoid.toSemigroup.{u_3} R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R (DivisionSemiring.toSemiring.{u_3} R (DivisionRing.toDivisionSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.Rat.3467823910._hygCtx._hyg.5))))) (MulAction.toSemigroupAction.{u_3, u_1} R M (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R (DivisionSemiring.toSemiring.{u_3} R (DivisionRing.toDivisionSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.Rat.3467823910._hygCtx._hyg.5)))) inst._@.Mathlib.Algebra.Module.Rat.3467823910._hygCtx._hyg.8))) (Rat.cast.{u_3} R (DivisionRing.toRatCast.{u_3} R inst._@.Mathlib.Algebra.Module.Rat.3467823910._hygCtx._hyg.5) q) x) (HSMul.hSMul.{0, u_1, u_1} Rat M M (instHSMul.{0, u_1} Rat M (SemigroupAction.toSMul.{0, u_1} Rat M (Monoid.toSemigroup.{0} Rat Rat.monoid) (MulAction.toSemigroupAction.{0, u_1} Rat M Rat.monoid inst._@.Mathlib.Algebra.Module.Rat.3467823910._hygCtx._hyg.12))) q x)","typeFull":"∀ {M : Type u_1} (R : Type u_3) [inst : DivisionRing R] [inst_1 : MulAction R M] [inst_2 : MulAction ℚ M]\n  [IsScalarTower ℚ R M] (q : ℚ) (x : M), ↑q • x = q • x","typeReadable":"∀ {M : Type u_1} (R : Type u_3) [inst : DivisionRing R] [inst_1 : MulAction R M] [inst_2 : MulAction ℚ M]\n  [IsScalarTower ℚ R M] (q : ℚ) (x : M), ↑q • x = q • x","typeReferences":[["IsScalarTower"],["Rat","monoid"],["Semiring","toMonoidWithZero"],["SemigroupAction","toSMul"],["DivisionSemiring","toSemiring"],["DivisionRing","toRatCast"],["MulAction"],["DivisionRing","toDivisionSemiring"],["Rat","cast"],["HSMul","hSMul"],["MonoidWithZero","toMonoid"],["Rat"],["instHSMul"],["Monoid","toSemigroup"],["Eq"],["DivisionRing"],["MulAction","toSemigroupAction"],["Rat","smulDivisionRing"]],"valueReferences":[["Ring","toNonAssocRing"],["Eq","trans"],["SemigroupAction","toSMul"],["AddGroupWithOne","toAddMonoidWithOne"],["SMulZeroClass","toSMul"],["AddMonoidWithOne","toAddMonoid"],["instDistribSMul"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["Ring","toAddGroupWithOne"],["Eq","symm"],["Monoid","toSemigroup"],["NonAssocRing","toNonUnitalNonAssocRing"],["Rat","smulDivisionRing"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["MulOne","toOne"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["DivisionRing","toRatCast"],["one_smul"],["HSMul","hSMul"],["id"],["Rat"],["Eq","mpr"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["MulOneClass","toMulOne"],["MulZeroOneClass","toMulOneClass"],["congrArg"],["DivisionRing","toDivisionSemiring"],["congr"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["Rat","cast"],["instHSMul"],["congrFun'"],["Eq"],["IsScalarTower","left"],["True"],["Rat","monoid"],["DivisionRing","toRing"],["Semiring","toMonoidWithZero"],["mul_one"],["OfNat","ofNat"],["eq_self"],["Rat","smul_one_eq_cast"],["One","toOfNat1"],["of_eq_true"],["smul_assoc"],["MulAction","toSemigroupAction"]]},{"isProp":true,"kind":"theorem","name":["map_nnratCast_smul"],"typeFallback":"forall {M : Type.{u_1}} {M₂ : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.4 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.7 : AddCommMonoid.{u_2} M₂] {F : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.11 : FunLike.{succ u_3, succ u_1, succ u_2} F M M₂] [inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.16 : AddMonoidHomClass.{u_3, u_1, u_2} F M M₂ (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.4))) (AddZeroClass.toAddZero.{u_2} M₂ (AddMonoid.toAddZeroClass.{u_2} M₂ (AddCommMonoid.toAddMonoid.{u_2} M₂ inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.11] (f : F) (R : Type.{u_4}) (S : Type.{u_5}) [inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.24 : DivisionSemiring.{u_4} R] [inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.27 : DivisionSemiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.30 : Module.{u_4, u_1} R M (DivisionSemiring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.24) inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.4] [inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.34 : Module.{u_5, u_2} S M₂ (DivisionSemiring.toSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.7] (c : NNRat) (x : M), Eq.{succ u_2} M₂ (DFunLike.coe.{succ u_3, succ u_1, succ u_2} F M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.11 f (HSMul.hSMul.{u_4, u_1, u_1} R M M (instHSMul.{u_4, u_1} R M (SMulZeroClass.toSMul.{u_4, 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.Rat.564440112._hygCtx._hyg.4)))) (DistribSMul.toSMulZeroClass.{u_4, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.4)) (DistribMulAction.toDistribSMul.{u_4, u_1} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R (DivisionSemiring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.24))) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.4) (Module.toDistribMulAction.{u_4, u_1} R M (DivisionSemiring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.24) inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.30))))) (NNRat.cast.{u_4} R (DivisionSemiring.toNNRatCast.{u_4} R inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.24) c) x)) (HSMul.hSMul.{u_5, u_2, u_2} S M₂ M₂ (instHSMul.{u_5, u_2} S M₂ (SMulZeroClass.toSMul.{u_5, u_2} S M₂ (AddZero.toZero.{u_2} M₂ (AddZeroClass.toAddZero.{u_2} M₂ (AddMonoid.toAddZeroClass.{u_2} M₂ (AddCommMonoid.toAddMonoid.{u_2} M₂ inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.7)))) (DistribSMul.toSMulZeroClass.{u_5, u_2} S M₂ (AddMonoid.toAddZeroClass.{u_2} M₂ (AddCommMonoid.toAddMonoid.{u_2} M₂ inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.7)) (DistribMulAction.toDistribSMul.{u_5, u_2} S M₂ (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S (DivisionSemiring.toSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.27))) (AddCommMonoid.toAddMonoid.{u_2} M₂ inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.7) (Module.toDistribMulAction.{u_5, u_2} S M₂ (DivisionSemiring.toSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.34))))) (NNRat.cast.{u_5} S (DivisionSemiring.toNNRatCast.{u_5} S inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.27) c) (DFunLike.coe.{succ u_3, succ u_1, succ u_2} F M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) inst._@.Mathlib.Algebra.Module.Rat.564440112._hygCtx._hyg.11 f x))","typeFull":"∀ {M : Type u_1} {M₂ : Type u_2} [inst : AddCommMonoid M] [inst_1 : AddCommMonoid M₂] {F : Type u_3}\n  [inst_2 : FunLike F M M₂] [AddMonoidHomClass F M M₂] (f : F) (R : Type u_4) (S : Type u_5)\n  [inst_4 : DivisionSemiring R] [inst_5 : DivisionSemiring S] [inst_6 : Module R M] [inst_7 : Module S M₂] (c : ℚ≥0)\n  (x : M), f (↑c • x) = ↑c • f x","typeReadable":"∀ {M : Type u_1} {M₂ : Type u_2} [inst : AddCommMonoid M] [inst_1 : AddCommMonoid M₂] {F : Type u_3}\n  [inst_2 : FunLike F M M₂] [AddMonoidHomClass F M M₂] (f : F) (R : Type u_4) (S : Type u_5)\n  [inst_4 : DivisionSemiring R] [inst_5 : DivisionSemiring S] [inst_6 : Module R M] [inst_7 : Module S M₂] (c : ℚ≥0)\n  (x : M), f (↑c • x) = ↑c • f x","typeReferences":[["FunLike"],["DivisionSemiring","toNNRatCast"],["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DivisionSemiring","toSemiring"],["AddMonoidHomClass"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["DivisionSemiring"],["Module","toDistribMulAction"],["AddCommMonoid"],["NNRat"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["NNRat","cast"],["instHSMul"],["Eq"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["DivInvMonoid","toInv"],["div_eq_mul_inv"],["DivisionSemiring","toNNRatCast"],["NNRat","den"],["GroupWithZero","toDivInvMonoid"],["HMul","hMul"],["SemigroupAction","toSMul"],["NonUnitalSemiring","toSemigroupWithZero"],["SMulZeroClass","toSMul"],["HDiv","hDiv"],["Semiring","toNonAssocSemiring"],["NNRat","num"],["DistribMulAction","toMulAction"],["NNRat","cast"],["NonAssocSemiring","toAddCommMonoidWithOne"],["Semiring","toNonUnitalSemiring"],["DistribSMul","toSMulZeroClass"],["InvOneClass","toInv"],["DistribMulAction","toDistribSMul"],["NNRat","cast_def"],["DivisionSemiring","toGroupWithZero"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["AddMonoidWithOne","toNatCast"],["DivInvMonoid","toMonoid"],["Eq","refl"],["HSMul","hSMul"],["id"],["instHMul"],["Eq","mpr"],["AddZero","toZero"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"],["map_natCast_smul"],["AddMonoid","toAddZeroClass"],["MulOneClass","toMulOne"],["Nat","cast"],["AddCommMonoid","toAddMonoid"],["instHDiv"],["DFunLike","coe"],["map_inv_natCast_smul"],["congrArg"],["Semigroup","toMul"],["MulOne","toMul"],["MonoidWithZero","toMonoid"],["Monoid","toMulOneClass"],["instHSMul"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["SemigroupAction","mul_smul"],["Inv","inv"],["Semiring","toMonoidWithZero"],["DivInvMonoid","toDiv"],["Module","toDistribMulAction"],["DivInvOneMonoid","toInvOneClass"],["SemigroupWithZero","toSemigroup"],["MulAction","toSemigroupAction"]]},{"isProp":true,"kind":"theorem","name":["subsingleton_nnrat_module"],"typeFallback":"forall (E : Type.{u_3}) [inst._@.Mathlib.Algebra.Module.Rat.4246216625._hygCtx._hyg.5 : AddCommMonoid.{u_3} E], Subsingleton.{max (succ u_3) 1} (Module.{0, u_3} NNRat E (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield)) inst._@.Mathlib.Algebra.Module.Rat.4246216625._hygCtx._hyg.5)","typeFull":"∀ (E : Type u_3) [inst : AddCommMonoid E], Subsingleton (Module ℚ≥0 E)","typeReadable":"∀ (E : Type u_3) [inst : AddCommMonoid E], Subsingleton (Module ℚ≥0 E)","typeReferences":[["Subsingleton"],["AddCommMonoid"],["NNRat"],["Module"],["Semifield","toDivisionSemiring"],["DivisionSemiring","toSemiring"],["NNRat","instSemifield"]],"valueReferences":[["Subsingleton","intro"],["Module"],["AddMonoidHom","instAddMonoidHomClass"],["DivisionSemiring","toSemiring"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddMonoidHom","id"],["map_nnrat_smul"],["NNRat"],["Module","ext'"],["AddMonoidHom"],["AddMonoidHom","instFunLike"],["Semifield","toDivisionSemiring"],["NNRat","instSemifield"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["SMulCommClass","nnrat'"],"typeFallback":"forall {α : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Rat.3700740835._hygCtx._hyg.6 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Rat.3700740835._hygCtx._hyg.9 : DistribSMul.{u, v} α M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Rat.3700740835._hygCtx._hyg.6))] [inst._@.Mathlib.Algebra.Module.Rat.3700740835._hygCtx._hyg.13 : Module.{0, v} NNRat M (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield)) inst._@.Mathlib.Algebra.Module.Rat.3700740835._hygCtx._hyg.6], SMulCommClass.{u, 0, v} α NNRat M (SMulZeroClass.toSMul.{u, v} α M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Rat.3700740835._hygCtx._hyg.6)))) (DistribSMul.toSMulZeroClass.{u, v} α M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Rat.3700740835._hygCtx._hyg.6)) inst._@.Mathlib.Algebra.Module.Rat.3700740835._hygCtx._hyg.9)) (SMulZeroClass.toSMul.{0, v} NNRat M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Rat.3700740835._hygCtx._hyg.6)))) (DistribSMul.toSMulZeroClass.{0, v} NNRat M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Rat.3700740835._hygCtx._hyg.6)) (DistribMulAction.toDistribSMul.{0, v} NNRat M (MonoidWithZero.toMonoid.{0} NNRat (Semiring.toMonoidWithZero.{0} NNRat (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield)))) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Rat.3700740835._hygCtx._hyg.6) (Module.toDistribMulAction.{0, v} NNRat M (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield)) inst._@.Mathlib.Algebra.Module.Rat.3700740835._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Module.Rat.3700740835._hygCtx._hyg.13))))","typeFull":"∀ {α : Type u} {M : Type v} [inst : AddCommMonoid M] [inst_1 : DistribSMul α M] [inst_2 : Module ℚ≥0 M],\n  SMulCommClass α ℚ≥0 M","typeReadable":"∀ {α : Type u} {M : Type v} [inst : AddCommMonoid M] [inst_1 : DistribSMul α M] [inst_2 : Module ℚ≥0 M],\n  SMulCommClass α ℚ≥0 M","typeReferences":[["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["DistribSMul"],["SMulCommClass"],["Module","toDistribMulAction"],["AddCommMonoid"],["NNRat"],["MonoidWithZero","toMonoid"],["Semifield","toDivisionSemiring"],["AddZero","toZero"],["NNRat","instSemifield"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["SMulCommClass","symm"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["SMulCommClass","nnrat"],["Module","toDistribMulAction"],["NNRat"],["MonoidWithZero","toMonoid"],["Semifield","toDivisionSemiring"],["AddZero","toZero"],["NNRat","instSemifield"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["IsScalarTower","rat"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Rat.4188825137._hygCtx._hyg.6 : Ring.{u} R] [inst._@.Mathlib.Algebra.Module.Rat.4188825137._hygCtx._hyg.9 : AddCommGroup.{v} M] [inst._@.Mathlib.Algebra.Module.Rat.4188825137._hygCtx._hyg.12 : Module.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Rat.4188825137._hygCtx._hyg.6) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Rat.4188825137._hygCtx._hyg.9)] [inst._@.Mathlib.Algebra.Module.Rat.4188825137._hygCtx._hyg.16 : Module.{0, u} Rat R Rat.semiring (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R (NonAssocRing.toNonUnitalNonAssocRing.{u} R (Ring.toNonAssocRing.{u} R inst._@.Mathlib.Algebra.Module.Rat.4188825137._hygCtx._hyg.6))))] [inst._@.Mathlib.Algebra.Module.Rat.4188825137._hygCtx._hyg.22 : Module.{0, v} Rat M Rat.semiring (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Rat.4188825137._hygCtx._hyg.9)], IsScalarTower.{0, u, v} Rat R M (SMulZeroClass.toSMul.{0, u} Rat R (AddZero.toZero.{u} R (AddZeroClass.toAddZero.{u} R (AddMonoid.toAddZeroClass.{u} R (AddMonoidWithOne.toAddMonoid.{u} R (AddGroupWithOne.toAddMonoidWithOne.{u} R (Ring.toAddGroupWithOne.{u} R inst._@.Mathlib.Algebra.Module.Rat.4188825137._hygCtx._hyg.6)))))) (DistribSMul.toSMulZeroClass.{0, u} Rat R (AddMonoid.toAddZeroClass.{u} R (AddMonoidWithOne.toAddMonoid.{u} R (AddGroupWithOne.toAddMonoidWithOne.{u} R (Ring.toAddGroupWithOne.{u} R inst._@.Mathlib.Algebra.Module.Rat.4188825137._hygCtx._hyg.6)))) (DistribMulAction.toDistribSMul.{0, u} Rat R Rat.monoid (AddMonoidWithOne.toAddMonoid.{u} R (AddGroupWithOne.toAddMonoidWithOne.{u} R (Ring.toAddGroupWithOne.{u} R inst._@.Mathlib.Algebra.Module.Rat.4188825137._hygCtx._hyg.6))) (Module.toDistribMulAction.{0, u} Rat R Rat.semiring (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R (NonAssocRing.toNonUnitalNonAssocRing.{u} R (Ring.toNonAssocRing.{u} R inst._@.Mathlib.Algebra.Module.Rat.4188825137._hygCtx._hyg.6)))) inst._@.Mathlib.Algebra.Module.Rat.4188825137._hygCtx._hyg.16)))) (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (SubNegMonoid.toAddMonoid.{v} M (AddGroup.toSubNegMonoid.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Rat.4188825137._hygCtx._hyg.9)))))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (SubNegMonoid.toAddMonoid.{v} M (AddGroup.toSubNegMonoid.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Rat.4188825137._hygCtx._hyg.9)))) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Rat.4188825137._hygCtx._hyg.6))) (SubNegMonoid.toAddMonoid.{v} M (AddGroup.toSubNegMonoid.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Rat.4188825137._hygCtx._hyg.9))) (Module.toDistribMulAction.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Rat.4188825137._hygCtx._hyg.6) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Rat.4188825137._hygCtx._hyg.9) inst._@.Mathlib.Algebra.Module.Rat.4188825137._hygCtx._hyg.12)))) (SMulZeroClass.toSMul.{0, v} Rat M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (SubNegMonoid.toAddMonoid.{v} M (AddGroup.toSubNegMonoid.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Rat.4188825137._hygCtx._hyg.9)))))) (DistribSMul.toSMulZeroClass.{0, v} Rat M (AddMonoid.toAddZeroClass.{v} M (SubNegMonoid.toAddMonoid.{v} M (AddGroup.toSubNegMonoid.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Rat.4188825137._hygCtx._hyg.9)))) (DistribMulAction.toDistribSMul.{0, v} Rat M Rat.monoid (SubNegMonoid.toAddMonoid.{v} M (AddGroup.toSubNegMonoid.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Rat.4188825137._hygCtx._hyg.9))) (Module.toDistribMulAction.{0, v} Rat M Rat.semiring (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Rat.4188825137._hygCtx._hyg.9) inst._@.Mathlib.Algebra.Module.Rat.4188825137._hygCtx._hyg.22))))","typeFull":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] [inst_3 : Module ℚ R]\n  [inst_4 : Module ℚ M], IsScalarTower ℚ R M","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] [inst_3 : Module ℚ R]\n  [inst_4 : Module ℚ M], IsScalarTower ℚ R M","typeReferences":[["Rat","semiring"],["Ring","toNonAssocRing"],["IsScalarTower"],["Module"],["AddCommGroup","toAddGroup"],["AddGroupWithOne","toAddMonoidWithOne"],["SMulZeroClass","toSMul"],["AddMonoidWithOne","toAddMonoid"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Ring","toAddGroupWithOne"],["MonoidWithZero","toMonoid"],["AddGroup","toSubNegMonoid"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Rat","monoid"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["Rat"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Rat","semiring"],["smulAddHom"],["Ring","toNonAssocRing"],["AddCommGroup","toAddGroup"],["map_rat_smul"],["AddGroupWithOne","toAddMonoidWithOne"],["SMulZeroClass","toSMul"],["AddMonoidHom","instAddMonoidHomClass"],["AddCommMonoid","toAddMonoid"],["AddMonoidWithOne","toAddMonoid"],["DFunLike","coe"],["IsScalarTower","mk"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Ring","toAddGroupWithOne"],["AddMonoidHom"],["MonoidWithZero","toMonoid"],["AddMonoidHom","instFunLike"],["AddGroup","toSubNegMonoid"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["Rat","monoid"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["Ring","toAddCommGroup"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["AddMonoidHom","flip"],["Rat"],["AddCommGroup","toAddCommMonoid"],["AddMonoidHom","instAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["ratCast_smul_eq"],"typeFallback":"forall {E : Type.{u_3}} (R : Type.{u_4}) (S : Type.{u_5}) [inst._@.Mathlib.Algebra.Module.Rat.3003578451._hygCtx._hyg.7 : AddCommGroup.{u_3} E] [inst._@.Mathlib.Algebra.Module.Rat.3003578451._hygCtx._hyg.10 : DivisionRing.{u_4} R] [inst._@.Mathlib.Algebra.Module.Rat.3003578451._hygCtx._hyg.13 : DivisionRing.{u_5} S] [inst._@.Mathlib.Algebra.Module.Rat.3003578451._hygCtx._hyg.16 : Module.{u_4, u_3} R E (DivisionSemiring.toSemiring.{u_4} R (DivisionRing.toDivisionSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Rat.3003578451._hygCtx._hyg.10)) (AddCommGroup.toAddCommMonoid.{u_3} E inst._@.Mathlib.Algebra.Module.Rat.3003578451._hygCtx._hyg.7)] [inst._@.Mathlib.Algebra.Module.Rat.3003578451._hygCtx._hyg.20 : Module.{u_5, u_3} S E (DivisionSemiring.toSemiring.{u_5} S (DivisionRing.toDivisionSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.Rat.3003578451._hygCtx._hyg.13)) (AddCommGroup.toAddCommMonoid.{u_3} E inst._@.Mathlib.Algebra.Module.Rat.3003578451._hygCtx._hyg.7)] (r : Rat) (x : E), Eq.{succ u_3} E (HSMul.hSMul.{u_4, u_3, u_3} R E E (instHSMul.{u_4, u_3} R E (SMulZeroClass.toSMul.{u_4, u_3} R E (AddZero.toZero.{u_3} E (AddZeroClass.toAddZero.{u_3} E (AddMonoid.toAddZeroClass.{u_3} E (SubNegMonoid.toAddMonoid.{u_3} E (AddGroup.toSubNegMonoid.{u_3} E (AddCommGroup.toAddGroup.{u_3} E inst._@.Mathlib.Algebra.Module.Rat.3003578451._hygCtx._hyg.7)))))) (DistribSMul.toSMulZeroClass.{u_4, u_3} R E (AddMonoid.toAddZeroClass.{u_3} E (SubNegMonoid.toAddMonoid.{u_3} E (AddGroup.toSubNegMonoid.{u_3} E (AddCommGroup.toAddGroup.{u_3} E inst._@.Mathlib.Algebra.Module.Rat.3003578451._hygCtx._hyg.7)))) (DistribMulAction.toDistribSMul.{u_4, u_3} R E (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R (DivisionSemiring.toSemiring.{u_4} R (DivisionRing.toDivisionSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Rat.3003578451._hygCtx._hyg.10)))) (SubNegMonoid.toAddMonoid.{u_3} E (AddGroup.toSubNegMonoid.{u_3} E (AddCommGroup.toAddGroup.{u_3} E inst._@.Mathlib.Algebra.Module.Rat.3003578451._hygCtx._hyg.7))) (Module.toDistribMulAction.{u_4, u_3} R E (DivisionSemiring.toSemiring.{u_4} R (DivisionRing.toDivisionSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Rat.3003578451._hygCtx._hyg.10)) (AddCommGroup.toAddCommMonoid.{u_3} E inst._@.Mathlib.Algebra.Module.Rat.3003578451._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Rat.3003578451._hygCtx._hyg.16))))) (Rat.cast.{u_4} R (DivisionRing.toRatCast.{u_4} R inst._@.Mathlib.Algebra.Module.Rat.3003578451._hygCtx._hyg.10) r) x) (HSMul.hSMul.{u_5, u_3, u_3} S E E (instHSMul.{u_5, u_3} S E (SMulZeroClass.toSMul.{u_5, u_3} S E (AddZero.toZero.{u_3} E (AddZeroClass.toAddZero.{u_3} E (AddMonoid.toAddZeroClass.{u_3} E (SubNegMonoid.toAddMonoid.{u_3} E (AddGroup.toSubNegMonoid.{u_3} E (AddCommGroup.toAddGroup.{u_3} E inst._@.Mathlib.Algebra.Module.Rat.3003578451._hygCtx._hyg.7)))))) (DistribSMul.toSMulZeroClass.{u_5, u_3} S E (AddMonoid.toAddZeroClass.{u_3} E (SubNegMonoid.toAddMonoid.{u_3} E (AddGroup.toSubNegMonoid.{u_3} E (AddCommGroup.toAddGroup.{u_3} E inst._@.Mathlib.Algebra.Module.Rat.3003578451._hygCtx._hyg.7)))) (DistribMulAction.toDistribSMul.{u_5, u_3} S E (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S (DivisionSemiring.toSemiring.{u_5} S (DivisionRing.toDivisionSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.Rat.3003578451._hygCtx._hyg.13)))) (SubNegMonoid.toAddMonoid.{u_3} E (AddGroup.toSubNegMonoid.{u_3} E (AddCommGroup.toAddGroup.{u_3} E inst._@.Mathlib.Algebra.Module.Rat.3003578451._hygCtx._hyg.7))) (Module.toDistribMulAction.{u_5, u_3} S E (DivisionSemiring.toSemiring.{u_5} S (DivisionRing.toDivisionSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.Rat.3003578451._hygCtx._hyg.13)) (AddCommGroup.toAddCommMonoid.{u_3} E inst._@.Mathlib.Algebra.Module.Rat.3003578451._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Rat.3003578451._hygCtx._hyg.20))))) (Rat.cast.{u_5} S (DivisionRing.toRatCast.{u_5} S inst._@.Mathlib.Algebra.Module.Rat.3003578451._hygCtx._hyg.13) r) x)","typeFull":"�� {E : Type u_3} (R : Type u_4) (S : Type u_5) [inst : AddCommGroup E] [inst_1 : DivisionRing R]\n  [inst_2 : DivisionRing S] [inst_3 : Module R E] [inst_4 : Module S E] (r : ℚ) (x : E), ↑r • x = ↑r • x","typeReadable":"∀ {E : Type u_3} (R : Type u_4) (S : Type u_5) [inst : AddCommGroup E] [inst_1 : DivisionRing R]\n  [inst_2 : DivisionRing S] [inst_3 : Module R E] [inst_4 : Module S E] (r : ℚ) (x : E), ↑r • x = ↑r • x","typeReferences":[["Module"],["AddCommGroup","toAddGroup"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["SMulZeroClass","toSMul"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["DivisionRing","toRatCast"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["DivisionRing","toDivisionSemiring"],["Rat","cast"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["AddCommGroup","toAddCommMonoid"],["Rat"],["instHSMul"],["AddGroup","toSubNegMonoid"],["DivisionRing"],["Eq"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["SubNegMonoid","toAddMonoid"],["AddCommGroup","toAddGroup"],["AddMonoidHom"],["map_ratCast_smul"],["AddMonoidHom","instFunLike"],["AddMonoidHom","instAddMonoidHomClass"],["AddGroup","toSubNegMonoid"],["AddZeroClass","toAddZero"],["AddMonoidHom","id"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["subsingleton_rat_module"],"typeFallback":"forall (E : Type.{u_3}) [inst._@.Mathlib.Algebra.Module.Rat.62501518._hygCtx._hyg.5 : AddCommGroup.{u_3} E], Subsingleton.{max (succ u_3) 1} (Module.{0, u_3} Rat E Rat.semiring (AddCommGroup.toAddCommMonoid.{u_3} E inst._@.Mathlib.Algebra.Module.Rat.62501518._hygCtx._hyg.5))","typeFull":"∀ (E : Type u_3) [inst : AddCommGroup E], Subsingleton (Module ℚ E)","typeReadable":"∀ (E : Type u_3) [inst : AddCommGroup E], Subsingleton (Module ℚ E)","typeReferences":[["Rat","semiring"],["Subsingleton"],["Module"],["AddCommGroup","toAddCommMonoid"],["Rat"],["AddCommGroup"]],"valueReferences":[["Rat","semiring"],["Subsingleton","intro"],["Module"],["AddCommGroup","toAddGroup"],["map_rat_smul"],["AddMonoidHom","instAddMonoidHomClass"],["AddZeroClass","toAddZero"],["AddMonoidHom","id"],["Module","ext'"],["SubNegMonoid","toAddMonoid"],["AddMonoidHom"],["Rat"],["AddCommGroup","toAddCommMonoid"],["AddMonoidHom","instFunLike"],["AddGroup","toSubNegMonoid"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["SMulCommClass","rat"],"typeFallback":"forall {α : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Rat.4188825138._hygCtx._hyg.6 : AddCommGroup.{v} M] [inst._@.Mathlib.Algebra.Module.Rat.4188825138._hygCtx._hyg.9 : DistribSMul.{u, v} α M (AddMonoid.toAddZeroClass.{v} M (SubNegMonoid.toAddMonoid.{v} M (AddGroup.toSubNegMonoid.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Rat.4188825138._hygCtx._hyg.6))))] [inst._@.Mathlib.Algebra.Module.Rat.4188825138._hygCtx._hyg.13 : Module.{0, v} Rat M Rat.semiring (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Rat.4188825138._hygCtx._hyg.6)], SMulCommClass.{0, u, v} Rat α M (SMulZeroClass.toSMul.{0, v} Rat M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (SubNegMonoid.toAddMonoid.{v} M (AddGroup.toSubNegMonoid.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Rat.4188825138._hygCtx._hyg.6)))))) (DistribSMul.toSMulZeroClass.{0, v} Rat M (AddMonoid.toAddZeroClass.{v} M (SubNegMonoid.toAddMonoid.{v} M (AddGroup.toSubNegMonoid.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Rat.4188825138._hygCtx._hyg.6)))) (DistribMulAction.toDistribSMul.{0, v} Rat M Rat.monoid (SubNegMonoid.toAddMonoid.{v} M (AddGroup.toSubNegMonoid.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Rat.4188825138._hygCtx._hyg.6))) (Module.toDistribMulAction.{0, v} Rat M Rat.semiring (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Rat.4188825138._hygCtx._hyg.6) inst._@.Mathlib.Algebra.Module.Rat.4188825138._hygCtx._hyg.13)))) (SMulZeroClass.toSMul.{u, v} α M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (SubNegMonoid.toAddMonoid.{v} M (AddGroup.toSubNegMonoid.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Rat.4188825138._hygCtx._hyg.6)))))) (DistribSMul.toSMulZeroClass.{u, v} α M (AddMonoid.toAddZeroClass.{v} M (SubNegMonoid.toAddMonoid.{v} M (AddGroup.toSubNegMonoid.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Rat.4188825138._hygCtx._hyg.6)))) inst._@.Mathlib.Algebra.Module.Rat.4188825138._hygCtx._hyg.9))","typeFull":"∀ {α : Type u} {M : Type v} [inst : AddCommGroup M] [inst_1 : DistribSMul α M] [inst_2 : Module ℚ M],\n  SMulCommClass ℚ α M","typeReadable":"∀ {α : Type u} {M : Type v} [inst : AddCommGroup M] [inst_1 : DistribSMul α M] [inst_2 : Module ℚ M],\n  SMulCommClass ℚ α M","typeReferences":[["Rat","semiring"],["Module"],["AddCommGroup","toAddGroup"],["Rat","monoid"],["DistribMulAction","toDistribSMul"],["AddCommGroup"],["SMulZeroClass","toSMul"],["AddZeroClass","toAddZero"],["DistribSMul"],["SMulCommClass"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["Rat"],["AddCommGroup","toAddCommMonoid"],["AddGroup","toSubNegMonoid"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Rat","semiring"],["SMulCommClass","mk"],["AddCommGroup","toAddGroup"],["Rat","monoid"],["map_rat_smul"],["DistribMulAction","toDistribSMul"],["SMulZeroClass","toSMul"],["AddMonoidHom","instAddMonoidHomClass"],["DistribSMul","toAddMonoidHom"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["AddMonoidHom"],["HSMul","hSMul"],["Eq","symm"],["AddCommGroup","toAddCommMonoid"],["Rat"],["instHSMul"],["AddMonoidHom","instFunLike"],["AddGroup","toSubNegMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["IsScalarTower","nnrat"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Rat.2735027304._hygCtx._hyg.6 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Rat.2735027304._hygCtx._hyg.9 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Rat.2735027304._hygCtx._hyg.12 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Rat.2735027304._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Module.Rat.2735027304._hygCtx._hyg.9] [inst._@.Mathlib.Algebra.Module.Rat.2735027304._hygCtx._hyg.16 : Module.{0, u} NNRat R (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Module.Rat.2735027304._hygCtx._hyg.6)))] [inst._@.Mathlib.Algebra.Module.Rat.2735027304._hygCtx._hyg.22 : Module.{0, v} NNRat M (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield)) 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(AddMonoidWithOne.toAddMonoid.{u} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u} R (NonAssocSemiring.toAddCommMonoidWithOne.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Module.Rat.2735027304._hygCtx._hyg.6)))) (Module.toDistribMulAction.{0, u} NNRat R (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Module.Rat.2735027304._hygCtx._hyg.6))) inst._@.Mathlib.Algebra.Module.Rat.2735027304._hygCtx._hyg.16)))) (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.Rat.2735027304._hygCtx._hyg.9)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Rat.2735027304._hygCtx._hyg.9)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Rat.2735027304._hygCtx._hyg.6)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Rat.2735027304._hygCtx._hyg.9) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Rat.2735027304._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Module.Rat.2735027304._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.Rat.2735027304._hygCtx._hyg.12)))) (SMulZeroClass.toSMul.{0, v} NNRat M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Rat.2735027304._hygCtx._hyg.9)))) (DistribSMul.toSMulZeroClass.{0, v} NNRat M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Rat.2735027304._hygCtx._hyg.9)) (DistribMulAction.toDistribSMul.{0, v} NNRat M (MonoidWithZero.toMonoid.{0} NNRat (Semiring.toMonoidWithZero.{0} NNRat (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield)))) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Rat.2735027304._hygCtx._hyg.9) (Module.toDistribMulAction.{0, v} NNRat M (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield)) inst._@.Mathlib.Algebra.Module.Rat.2735027304._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.Rat.2735027304._hygCtx._hyg.22))))","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [inst_3 : Module ℚ≥0 R]\n  [inst_4 : Module ℚ≥0 M], IsScalarTower ℚ≥0 R M","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [inst_3 : Module ℚ≥0 R]\n  [inst_4 : Module ℚ≥0 M], IsScalarTower ℚ≥0 R M","typeReferences":[["Module"],["IsScalarTower"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DivisionSemiring","toSemiring"],["AddMonoidWithOne","toAddMonoid"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["AddCommMonoid"],["NNRat"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Semifield","toDivisionSemiring"],["NonAssocSemiring","toAddCommMonoidWithOne"],["AddZero","toZero"],["NNRat","instSemifield"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["smulAddHom"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddMonoidHom","instAddMonoidHomClass"],["AddMonoidWithOne","toAddMonoid"],["DFunLike","coe"],["IsScalarTower","mk"],["Semiring","toNonAssocSemiring"],["NNRat"],["AddMonoidHom"],["MonoidWithZero","toMonoid"],["AddMonoidHom","instFunLike"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Semifield","toDivisionSemiring"],["NonAssocSemiring","toAddCommMonoidWithOne"],["NNRat","instSemifield"],["DistribSMul","toSMulZeroClass"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["map_nnrat_smul"],["Module","toDistribMulAction"],["AddMonoidHom","flip"],["AddMonoidHom","instAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["nnratCast_smul_eq"],"typeFallback":"forall {E : Type.{u_3}} (R : Type.{u_4}) (S : Type.{u_5}) [inst._@.Mathlib.Algebra.Module.Rat.2670750015._hygCtx._hyg.7 : AddCommMonoid.{u_3} E] [inst._@.Mathlib.Algebra.Module.Rat.2670750015._hygCtx._hyg.10 : DivisionSemiring.{u_4} R] [inst._@.Mathlib.Algebra.Module.Rat.2670750015._hygCtx._hyg.13 : DivisionSemiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.Rat.2670750015._hygCtx._hyg.16 : Module.{u_4, u_3} R E (DivisionSemiring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Rat.2670750015._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.Rat.2670750015._hygCtx._hyg.7] [inst._@.Mathlib.Algebra.Module.Rat.2670750015._hygCtx._hyg.20 : Module.{u_5, u_3} S E (DivisionSemiring.toSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.Rat.2670750015._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Module.Rat.2670750015._hygCtx._hyg.7] (r : NNRat) (x : E), Eq.{succ u_3} E (HSMul.hSMul.{u_4, u_3, u_3} R E E (instHSMul.{u_4, u_3} R E (SMulZeroClass.toSMul.{u_4, u_3} R E (AddZero.toZero.{u_3} E 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inst._@.Mathlib.Algebra.Module.Rat.2670750015._hygCtx._hyg.10) r) x) (HSMul.hSMul.{u_5, u_3, u_3} S E E (instHSMul.{u_5, u_3} S E (SMulZeroClass.toSMul.{u_5, u_3} S E (AddZero.toZero.{u_3} E (AddZeroClass.toAddZero.{u_3} E (AddMonoid.toAddZeroClass.{u_3} E (AddCommMonoid.toAddMonoid.{u_3} E inst._@.Mathlib.Algebra.Module.Rat.2670750015._hygCtx._hyg.7)))) (DistribSMul.toSMulZeroClass.{u_5, u_3} S E (AddMonoid.toAddZeroClass.{u_3} E (AddCommMonoid.toAddMonoid.{u_3} E inst._@.Mathlib.Algebra.Module.Rat.2670750015._hygCtx._hyg.7)) (DistribMulAction.toDistribSMul.{u_5, u_3} S E (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S (DivisionSemiring.toSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.Rat.2670750015._hygCtx._hyg.13))) (AddCommMonoid.toAddMonoid.{u_3} E inst._@.Mathlib.Algebra.Module.Rat.2670750015._hygCtx._hyg.7) (Module.toDistribMulAction.{u_5, u_3} S E (DivisionSemiring.toSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.Rat.2670750015._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Module.Rat.2670750015._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Rat.2670750015._hygCtx._hyg.20))))) (NNRat.cast.{u_5} S (DivisionSemiring.toNNRatCast.{u_5} S inst._@.Mathlib.Algebra.Module.Rat.2670750015._hygCtx._hyg.13) r) x)","typeFull":"∀ {E : Type u_3} (R : Type u_4) (S : Type u_5) [inst : AddCommMonoid E] [inst_1 : DivisionSemiring R]\n  [inst_2 : DivisionSemiring S] [inst_3 : Module R E] [inst_4 : Module S E] (r : ℚ≥0) (x : E), ↑r • x = ↑r • x","typeReadable":"∀ {E : Type u_3} (R : Type u_4) (S : Type u_5) [inst : AddCommMonoid E] [inst_1 : DivisionSemiring R]\n  [inst_2 : DivisionSemiring S] [inst_3 : Module R E] [inst_4 : Module S E] (r : ℚ≥0) (x : E), ↑r • x = ↑r • x","typeReferences":[["DivisionSemiring","toNNRatCast"],["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["DivisionSemiring","toSemiring"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["DivisionSemiring"],["Module","toDistribMulAction"],["AddCommMonoid"],["NNRat"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["NNRat","cast"],["instHSMul"],["Eq"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["map_nnratCast_smul"],["AddMonoidHom"],["AddMonoidHom","instFunLike"],["AddMonoidHom","instAddMonoidHomClass"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddMonoidHom","id"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["map_ratCast_smul"],"typeFallback":"forall {M : Type.{u_1}} {M₂ : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.4 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.7 : AddCommGroup.{u_2} M₂] {F : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.11 : FunLike.{succ u_3, succ u_1, succ u_2} F M M₂] [inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.16 : AddMonoidHomClass.{u_3, u_1, u_2} F M M₂ (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.4))))) (AddZeroClass.toAddZero.{u_2} M₂ (AddMonoid.toAddZeroClass.{u_2} M₂ (SubNegMonoid.toAddMonoid.{u_2} M₂ (AddGroup.toSubNegMonoid.{u_2} M₂ (AddCommGroup.toAddGroup.{u_2} M₂ inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.7))))) inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.11] (f : F) (R : Type.{u_4}) (S : Type.{u_5}) [inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.24 : DivisionRing.{u_4} R] [inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.27 : DivisionRing.{u_5} S] [inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.30 : Module.{u_4, u_1} R M (DivisionSemiring.toSemiring.{u_4} R (DivisionRing.toDivisionSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.24)) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.4)] [inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.34 : Module.{u_5, u_2} S M₂ (DivisionSemiring.toSemiring.{u_5} S (DivisionRing.toDivisionSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.27)) (AddCommGroup.toAddCommMonoid.{u_2} M₂ inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.7)] (c : Rat) (x : M), Eq.{succ u_2} M₂ (DFunLike.coe.{succ u_3, succ u_1, succ u_2} F M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.11 f (HSMul.hSMul.{u_4, u_1, u_1} R M M (instHSMul.{u_4, u_1} R M (SMulZeroClass.toSMul.{u_4, u_1} R M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.4)))))) (DistribSMul.toSMulZeroClass.{u_4, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.4)))) (DistribMulAction.toDistribSMul.{u_4, u_1} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R (DivisionSemiring.toSemiring.{u_4} R (DivisionRing.toDivisionSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.24)))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.4))) (Module.toDistribMulAction.{u_4, u_1} R M (DivisionSemiring.toSemiring.{u_4} R (DivisionRing.toDivisionSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.24)) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.30))))) (Rat.cast.{u_4} R (DivisionRing.toRatCast.{u_4} R inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.24) c) x)) (HSMul.hSMul.{u_5, u_2, u_2} S M₂ M₂ (instHSMul.{u_5, u_2} S M₂ (SMulZeroClass.toSMul.{u_5, u_2} S M₂ (AddZero.toZero.{u_2} M₂ (AddZeroClass.toAddZero.{u_2} M₂ (AddMonoid.toAddZeroClass.{u_2} M₂ (SubNegMonoid.toAddMonoid.{u_2} M₂ (AddGroup.toSubNegMonoid.{u_2} M₂ (AddCommGroup.toAddGroup.{u_2} M₂ inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.7)))))) (DistribSMul.toSMulZeroClass.{u_5, u_2} S M₂ (AddMonoid.toAddZeroClass.{u_2} M₂ (SubNegMonoid.toAddMonoid.{u_2} M₂ (AddGroup.toSubNegMonoid.{u_2} M₂ (AddCommGroup.toAddGroup.{u_2} M₂ inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.7)))) (DistribMulAction.toDistribSMul.{u_5, u_2} S M₂ (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S (DivisionSemiring.toSemiring.{u_5} S (DivisionRing.toDivisionSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.27)))) (SubNegMonoid.toAddMonoid.{u_2} M₂ (AddGroup.toSubNegMonoid.{u_2} M₂ (AddCommGroup.toAddGroup.{u_2} M₂ inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.7))) (Module.toDistribMulAction.{u_5, u_2} S M₂ (DivisionSemiring.toSemiring.{u_5} S (DivisionRing.toDivisionSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.27)) (AddCommGroup.toAddCommMonoid.{u_2} M₂ inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.34))))) (Rat.cast.{u_5} S (DivisionRing.toRatCast.{u_5} S inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.27) c) (DFunLike.coe.{succ u_3, succ u_1, succ u_2} F M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) inst._@.Mathlib.Algebra.Module.Rat.3254073199._hygCtx._hyg.11 f x))","typeFull":"∀ {M : Type u_1} {M₂ : Type u_2} [inst : AddCommGroup M] [inst_1 : AddCommGroup M₂] {F : Type u_3}\n  [inst_2 : FunLike F M M₂] [AddMonoidHomClass F M M₂] (f : F) (R : Type u_4) (S : Type u_5) [inst_4 : DivisionRing R]\n  [inst_5 : DivisionRing S] [inst_6 : Module R M] [inst_7 : Module S M₂] (c : ℚ) (x : M), f (↑c • x) = ↑c • f x","typeReadable":"∀ {M : Type u_1} {M₂ : Type u_2} [inst : AddCommGroup M] [inst_1 : AddCommGroup M₂] {F : Type u_3}\n  [inst_2 : FunLike F M M₂] [AddMonoidHomClass F M M₂] (f : F) (R : Type u_4) (S : Type u_5) [inst_4 : DivisionRing R]\n  [inst_5 : DivisionRing S] [inst_6 : Module R M] [inst_7 : Module S M₂] (c : ℚ) (x : M), f (↑c • x) = ↑c • f x","typeReferences":[["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["AddMonoidHomClass"],["DFunLike","coe"],["DivisionRing","toDivisionSemiring"],["Rat","cast"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddGroup","toSubNegMonoid"],["DivisionRing"],["Eq"],["DistribSMul","toSMulZeroClass"],["FunLike"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["DivisionRing","toRatCast"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["HSMul","hSMul"],["AddCommGroup","toAddCommMonoid"],["Rat"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["DivInvMonoid","toInv"],["div_eq_mul_inv"],["AddCommGroup","toAddGroup"],["NonUnitalSemiring","toSemigroupWithZero"],["AddGroupWithOne","toAddMonoidWithOne"],["HMul","hMul"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["HDiv","hDiv"],["Semiring","toNonAssocSemiring"],["Ring","toAddGroupWithOne"],["DistribMulAction","toMulAction"],["Rat","cast_def"],["AddGroup","toSubNegMonoid"],["NonAssocSemiring","toAddCommMonoidWithOne"],["Semiring","toNonUnitalSemiring"],["DistribSMul","toSMulZeroClass"],["InvOneClass","toInv"],["DistribMulAction","toDistribSMul"],["DivisionSemiring","toGroupWithZero"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["DivisionRing","toDivInvMonoid"],["Ring","toSemiring"],["DivisionRing","toRatCast"],["AddMonoidWithOne","toNatCast"],["DivInvMonoid","toMonoid"],["Eq","refl"],["HSMul","hSMul"],["AddCommGroup","toAddCommMonoid"],["id"],["instHMul"],["Eq","mpr"],["AddZero","toZero"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"],["AddMonoid","toAddZeroClass"],["MulOneClass","toMulOne"],["Nat","cast"],["AddCommMonoid","toAddMonoid"],["instHDiv"],["DFunLike","coe"],["Int","cast"],["map_inv_natCast_smul"],["Rat","num"],["Semigroup","toMul"],["congrArg"],["MulOne","toMul"],["DivisionRing","toDivisionSemiring"],["Rat","cast"],["MonoidWithZero","toMonoid"],["Monoid","toMulOneClass"],["instHSMul"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["SemigroupAction","mul_smul"],["Rat","den"],["Inv","inv"],["DivisionRing","toRing"],["map_intCast_smul"],["Semiring","toMonoidWithZero"],["DivInvMonoid","toDiv"],["Module","toDistribMulAction"],["DivInvOneMonoid","toInvOneClass"],["SubNegMonoid","toAddMonoid"],["AddGroupWithOne","toIntCast"],["SemigroupWithZero","toSemigroup"],["MulAction","toSemigroupAction"]]},{"isProp":true,"kind":"theorem","name":["SMulCommClass","nnrat"],"typeFallback":"forall {α : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Rat.2735027305._hygCtx._hyg.6 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Rat.2735027305._hygCtx._hyg.9 : DistribSMul.{u, v} α M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Rat.2735027305._hygCtx._hyg.6))] [inst._@.Mathlib.Algebra.Module.Rat.2735027305._hygCtx._hyg.13 : Module.{0, v} NNRat M (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield)) inst._@.Mathlib.Algebra.Module.Rat.2735027305._hygCtx._hyg.6], SMulCommClass.{0, u, v} NNRat α M (SMulZeroClass.toSMul.{0, v} NNRat M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Rat.2735027305._hygCtx._hyg.6)))) (DistribSMul.toSMulZeroClass.{0, v} NNRat M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Rat.2735027305._hygCtx._hyg.6)) (DistribMulAction.toDistribSMul.{0, v} NNRat M (MonoidWithZero.toMonoid.{0} NNRat (Semiring.toMonoidWithZero.{0} NNRat (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield)))) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Rat.2735027305._hygCtx._hyg.6) (Module.toDistribMulAction.{0, v} NNRat M (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield)) inst._@.Mathlib.Algebra.Module.Rat.2735027305._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Module.Rat.2735027305._hygCtx._hyg.13)))) (SMulZeroClass.toSMul.{u, v} α M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Rat.2735027305._hygCtx._hyg.6)))) (DistribSMul.toSMulZeroClass.{u, v} α M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Rat.2735027305._hygCtx._hyg.6)) inst._@.Mathlib.Algebra.Module.Rat.2735027305._hygCtx._hyg.9))","typeFull":"∀ {α : Type u} {M : Type v} [inst : AddCommMonoid M] [inst_1 : DistribSMul α M] [inst_2 : Module ℚ≥0 M],\n  SMulCommClass ℚ≥0 α M","typeReadable":"∀ {α : Type u} {M : Type v} [inst : AddCommMonoid M] [inst_1 : DistribSMul α M] [inst_2 : Module ℚ≥0 M],\n  SMulCommClass ℚ≥0 α 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(M : Type.{u_1}) [inst._@.Mathlib.Algebra.Module.Rat.4274353584._hygCtx._hyg.4 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Module.Rat.4274353584._hygCtx._hyg.7 : Module.{0, u_1} Rat M Rat.semiring (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Rat.4274353584._hygCtx._hyg.4)], IsAddTorsionFree.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Module.Rat.4274353584._hygCtx._hyg.4)))","typeFull":"∀ (M : Type u_1) [inst : AddCommGroup M] [Module ℚ M], IsAddTorsionFree M","typeReadable":"∀ (M : Type u_1) [inst : AddCommGroup M] [Module ℚ M], IsAddTorsionFree 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{M : Type.{u_1}} {M₂ : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Rat.3930294174._hygCtx._hyg.4 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Module.Rat.3930294174._hygCtx._hyg.7 : AddCommGroup.{u_2} M₂] [_instM : Module.{0, u_1} Rat M Rat.semiring (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Rat.3930294174._hygCtx._hyg.4)] [_instM₂ : Module.{0, u_2} Rat M₂ Rat.semiring (AddCommGroup.toAddCommMonoid.{u_2} M₂ inst._@.Mathlib.Algebra.Module.Rat.3930294174._hygCtx._hyg.7)] {F : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Rat.3930294174._hygCtx._hyg.21 : FunLike.{succ u_3, succ u_1, succ u_2} F M M₂] [inst._@.Mathlib.Algebra.Module.Rat.3930294174._hygCtx._hyg.26 : AddMonoidHomClass.{u_3, u_1, u_2} F M M₂ (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Module.Rat.3930294174._hygCtx._hyg.4))))) (AddZeroClass.toAddZero.{u_2} 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 f (c • x) = c • f x","typeReadable":"∀ {M : Type u_1} {M₂ : Type u_2} [inst : AddCommGroup M] [inst_1 : AddCommGroup M₂] [_instM : Module ℚ M]\n  [_instM₂ : Module ℚ M₂] {F : Type u_3} [inst_2 : FunLike F M M₂] [AddMonoidHomClass F M M₂] (f : F) (c : ℚ) (x : M),\n  f (c • x) = c • f x","typeReferences":[["Rat","semiring"],["FunLike"],["Module"],["AddCommGroup","toAddGroup"],["Rat","monoid"],["DistribMulAction","toDistribSMul"],["SMulZeroClass","toSMul"],["AddCommGroup"],["AddMonoidHomClass"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["HSMul","hSMul"],["Rat"],["AddCommGroup","toAddCommMonoid"],["instHSMul"],["AddGroup","toSubNegMonoid"],["Eq"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["map_ratCast_smul"],["Rat"],["Rat","instDivisionRing"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Order.Hom.Units.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["OrderMonoidIso","val_unitsCongr_symm_apply"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.Order.Hom.Units.609721182._hygCtx._hyg.4 : Preorder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Hom.Units.609721182._hygCtx._hyg.7 : Monoid.{u_1} α] [inst._@.Mathlib.Algebra.Order.Hom.Units.609721182._hygCtx._hyg.10 : Preorder.{u_2} β] [inst._@.Mathlib.Algebra.Order.Hom.Units.609721182._hygCtx._hyg.13 : Monoid.{u_2} β] (e : OrderMonoidIso.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Hom.Units.609721182._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Order.Hom.Units.609721182._hygCtx._hyg.10 (MulOne.toMul.{u_1} α (MulOneClass.toMulOne.{u_1} α (Monoid.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Order.Hom.Units.609721182._hygCtx._hyg.7))) (MulOne.toMul.{u_2} β (MulOneClass.toMulOne.{u_2} β (Monoid.toMulOneClass.{u_2} β inst._@.Mathlib.Algebra.Order.Hom.Units.609721182._hygCtx._hyg.13)))) (a : Units.{u_2} β 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inst._@.Mathlib.Algebra.Order.Hom.Units.609721182._hygCtx._hyg.10 (MulOne.toMul.{u_1} α (MulOneClass.toMulOne.{u_1} α (Monoid.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Order.Hom.Units.609721182._hygCtx._hyg.7))) (MulOne.toMul.{u_2} β (MulOneClass.toMulOne.{u_2} β (Monoid.toMulOneClass.{u_2} β inst._@.Mathlib.Algebra.Order.Hom.Units.609721182._hygCtx._hyg.13)))) e)) (Units.val.{u_2} β inst._@.Mathlib.Algebra.Order.Hom.Units.609721182._hygCtx._hyg.13 a))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst_1 : Monoid α] [inst_2 : Preorder β] [inst_3 : Monoid β]\n  (e : α ≃*o β) (a : βˣ), ↑(e.unitsCongr.symm a) = (↑e).symm ↑a","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst_1 : Monoid α] [inst_2 : Preorder β] [inst_3 : Monoid β]\n  (e : α ≃*o β) (a : βˣ), ↑(e.unitsCongr.symm a) = (↑e).symm ↑a","typeReferences":[["MulOneClass","toMulOne"],["Units","instMul"],["MulEquiv"],["Units","instPreorder"],["OrderMonoidIso","instMulEquivClass"],["MulEquivClass","toMulEquiv"],["MulEquiv","symm"],["MulEquiv","instEquivLike"],["DFunLike","coe"],["OrderMonoidIso"],["Preorder"],["MulOne","toMul"],["OrderMonoidIso","instEquivLike"],["Units","val"],["Monoid","toMulOneClass"],["EquivLike","toFunLike"],["OrderMonoidIso","unitsCongr"],["Monoid"],["OrderMonoidIso","symm"],["Eq"],["Units"]],"valueReferences":[["OrderMonoidIso"],["Units","instMul"],["OrderMonoidIso","instEquivLike"],["Units","instPreorder"],["Units","val"],["Eq","refl"],["EquivLike","toFunLike"],["OrderMonoidIso","unitsCongr"],["OrderMonoidIso","symm"],["DFunLike","coe"],["Units"]]},{"isProp":true,"kind":"theorem","name":["OrderMonoidIso","val_inv_unitsCongr_symm_apply"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.Order.Hom.Units.609721182._hygCtx._hyg.4 : Preorder.{u_1} α] 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Order.Ring.Int.sym.json

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