diff --git "a/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.Basic.sym.json" "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.Basic.sym.json"
new file mode 100644--- /dev/null
+++ "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.Basic.sym.json"
@@ -0,0 +1 @@
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(fun x => r • f x) a","typeReadable":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Zero M] [inst_1 : SMulZeroClass R M] (s : Set α) (r : R)\n  (f : α → M) (a : α), r • s.indicator f a = s.indicator (fun x => r • f x) a","typeReferences":[["Set"],["HSMul","hSMul"],["SMulZeroClass"],["instHSMul"],["SMulZeroClass","toSMul"],["Zero"],["Eq"],["Set","indicator"]],"valueReferences":[["Set","indicator_const_smul_apply"],["HSMul","hSMul"],["Eq","symm"],["instHSMul"],["SMulZeroClass","toSMul"],["Set","indicator"]]},{"isProp":true,"kind":"theorem","name":["Function","support_smul_subset_left"],"typeFallback":"forall {α : Type.{u_1}} {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Basic.1345628246._hygCtx._hyg.6 : Zero.{u_2} R] [inst._@.Mathlib.Algebra.Module.Basic.1345628246._hygCtx._hyg.9 : Zero.{u_3} M] [inst._@.Mathlib.Algebra.Module.Basic.1345628246._hygCtx._hyg.12 : SMulWithZero.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Basic.1345628246._hygCtx._hyg.6 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[inst._@.Mathlib.Algebra.Module.Basic.4162339609._hygCtx._hyg.6 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Basic.4162339609._hygCtx._hyg.9 : IsDomain.{u_2} R inst._@.Mathlib.Algebra.Module.Basic.4162339609._hygCtx._hyg.6] [inst._@.Mathlib.Algebra.Module.Basic.4162339609._hygCtx._hyg.12 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Basic.4162339609._hygCtx._hyg.15 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Basic.4162339609._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Module.Basic.4162339609._hygCtx._hyg.12] [inst._@.Mathlib.Algebra.Module.Basic.4162339609._hygCtx._hyg.19 : Module.IsTorsionFree.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Basic.4162339609._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Module.Basic.4162339609._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.Basic.4162339609._hygCtx._hyg.15] (f : α -> R) (g : α -> M), Eq.{succ u_1} (Set.{u_1} α) (Function.support.{u_1, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M 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{E : Type.{u_5}} (R : Type.{u_6}) (S : Type.{u_7}) [inst._@.Mathlib.Algebra.Module.Basic.3283922001._hygCtx._hyg.9 : AddCommGroup.{u_5} E] [inst._@.Mathlib.Algebra.Module.Basic.3283922001._hygCtx._hyg.12 : DivisionRing.{u_6} R] [inst._@.Mathlib.Algebra.Module.Basic.3283922001._hygCtx._hyg.15 : DivisionRing.{u_7} S] [inst._@.Mathlib.Algebra.Module.Basic.3283922001._hygCtx._hyg.18 : Module.{u_6, u_5} R E (DivisionSemiring.toSemiring.{u_6} R (DivisionRing.toDivisionSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.Basic.3283922001._hygCtx._hyg.12)) (AddCommGroup.toAddCommMonoid.{u_5} E inst._@.Mathlib.Algebra.Module.Basic.3283922001._hygCtx._hyg.9)] [inst._@.Mathlib.Algebra.Module.Basic.3283922001._hygCtx._hyg.22 : Module.{u_7, u_5} S E (DivisionSemiring.toSemiring.{u_7} S (DivisionRing.toDivisionSemiring.{u_7} S inst._@.Mathlib.Algebra.Module.Basic.3283922001._hygCtx._hyg.15)) (AddCommGroup.toAddCommMonoid.{u_5} E inst._@.Mathlib.Algebra.Module.Basic.3283922001._hygCtx._hyg.9)] (n : Int) 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(SMulZeroClass.toSMul.{u_7, u_5} S E (AddZero.toZero.{u_5} E (AddZeroClass.toAddZero.{u_5} E (AddMonoid.toAddZeroClass.{u_5} E (SubNegMonoid.toAddMonoid.{u_5} E (AddGroup.toSubNegMonoid.{u_5} E (AddCommGroup.toAddGroup.{u_5} E inst._@.Mathlib.Algebra.Module.Basic.3283922001._hygCtx._hyg.9)))))) (DistribSMul.toSMulZeroClass.{u_7, u_5} S E (AddMonoid.toAddZeroClass.{u_5} E (SubNegMonoid.toAddMonoid.{u_5} E (AddGroup.toSubNegMonoid.{u_5} E (AddCommGroup.toAddGroup.{u_5} E inst._@.Mathlib.Algebra.Module.Basic.3283922001._hygCtx._hyg.9)))) (DistribMulAction.toDistribSMul.{u_7, u_5} S E (MonoidWithZero.toMonoid.{u_7} S (Semiring.toMonoidWithZero.{u_7} S (DivisionSemiring.toSemiring.{u_7} S (DivisionRing.toDivisionSemiring.{u_7} S inst._@.Mathlib.Algebra.Module.Basic.3283922001._hygCtx._hyg.15)))) (SubNegMonoid.toAddMonoid.{u_5} E (AddGroup.toSubNegMonoid.{u_5} E (AddCommGroup.toAddGroup.{u_5} E inst._@.Mathlib.Algebra.Module.Basic.3283922001._hygCtx._hyg.9))) (Module.toDistribMulAction.{u_7, u_5} S E (DivisionSemiring.toSemiring.{u_7} S (DivisionRing.toDivisionSemiring.{u_7} S inst._@.Mathlib.Algebra.Module.Basic.3283922001._hygCtx._hyg.15)) (AddCommGroup.toAddCommMonoid.{u_5} E inst._@.Mathlib.Algebra.Module.Basic.3283922001._hygCtx._hyg.9) inst._@.Mathlib.Algebra.Module.Basic.3283922001._hygCtx._hyg.22))))) (Inv.inv.{u_7} S (InvOneClass.toInv.{u_7} S (DivInvOneMonoid.toInvOneClass.{u_7} S (DivisionMonoid.toDivInvOneMonoid.{u_7} S (GroupWithZero.toDivisionMonoid.{u_7} S (DivisionSemiring.toGroupWithZero.{u_7} S (DivisionRing.toDivisionSemiring.{u_7} S inst._@.Mathlib.Algebra.Module.Basic.3283922001._hygCtx._hyg.15)))))) (Int.cast.{u_7} S (AddGroupWithOne.toIntCast.{u_7} S (Ring.toAddGroupWithOne.{u_7} S (DivisionRing.toRing.{u_7} S inst._@.Mathlib.Algebra.Module.Basic.3283922001._hygCtx._hyg.15))) n)) x)","typeFull":"∀ {E : Type u_5} (R : Type u_6) (S : Type u_7) [inst : AddCommGroup E] [inst_1 : DivisionRing R]\n  [inst_2 : DivisionRing S] [inst_3 : Module R E] [inst_4 : Module S E] (n : ℤ) (x : E), (↑n)⁻¹ • x = (↑n)⁻¹ • x","typeReadable":"∀ {E : Type u_5} (R : Type u_6) (S : Type u_7) [inst : AddCommGroup E] [inst_1 : DivisionRing R]\n  [inst_2 : DivisionRing S] [inst_3 : Module R E] [inst_4 : Module S E] (n : ℤ) (x : E), (↑n)⁻¹ • x = (↑n)⁻¹ • x","typeReferences":[["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["Int","cast"],["Ring","toAddGroupWithOne"],["DivisionRing","toDivisionSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddGroup","toSubNegMonoid"],["Eq"],["DivisionRing"],["DistribSMul","toSMulZeroClass"],["Inv","inv"],["InvOneClass","toInv"],["DivisionRing","toRing"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["DivisionSemiring","toGroupWithZero"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["Int"],["Module","toDistribMulAction"],["DivInvOneMonoid","toInvOneClass"],["SubNegMonoid","toAddMonoid"],["HSMul","hSMul"],["AddGroupWithOne","toIntCast"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["SubNegMonoid","toAddMonoid"],["AddCommGroup","toAddGroup"],["AddMonoidHom"],["AddMonoidHom","instFunLike"],["AddMonoidHom","instAddMonoidHomClass"],["AddGroup","toSubNegMonoid"],["map_inv_intCast_smul"],["AddZeroClass","toAddZero"],["AddMonoidHom","id"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["inv_natCast_smul_comm"],"typeFallback":"forall {α : Type.{u_5}} {E : Type.{u_6}} (R : Type.{u_7}) [inst._@.Mathlib.Algebra.Module.Basic.1735039143._hygCtx._hyg.9 : AddCommMonoid.{u_6} E] [inst._@.Mathlib.Algebra.Module.Basic.1735039143._hygCtx._hyg.12 : DivisionSemiring.{u_7} R] [inst._@.Mathlib.Algebra.Module.Basic.1735039143._hygCtx._hyg.15 : Module.{u_7, u_6} R E (DivisionSemiring.toSemiring.{u_7} R inst._@.Mathlib.Algebra.Module.Basic.1735039143._hygCtx._hyg.12) inst._@.Mathlib.Algebra.Module.Basic.1735039143._hygCtx._hyg.9] [inst._@.Mathlib.Algebra.Module.Basic.1735039143._hygCtx._hyg.19 : DistribSMul.{u_5, u_6} α E (AddMonoid.toAddZeroClass.{u_6} E (AddCommMonoid.toAddMonoid.{u_6} E inst._@.Mathlib.Algebra.Module.Basic.1735039143._hygCtx._hyg.9))] (n : Nat) (s : α) (x : E), Eq.{succ u_6} E (HSMul.hSMul.{u_7, u_6, u_6} R E E (instHSMul.{u_7, u_6} R E (SMulZeroClass.toSMul.{u_7, u_6} R E (AddZero.toZero.{u_6} E (AddZeroClass.toAddZero.{u_6} E (AddMonoid.toAddZeroClass.{u_6} E (AddCommMonoid.toAddMonoid.{u_6} E 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(HSMul.hSMul.{u_5, u_6, u_6} α E E (instHSMul.{u_5, u_6} α E (SMulZeroClass.toSMul.{u_5, u_6} α E (AddZero.toZero.{u_6} E (AddZeroClass.toAddZero.{u_6} E (AddMonoid.toAddZeroClass.{u_6} E (AddCommMonoid.toAddMonoid.{u_6} E inst._@.Mathlib.Algebra.Module.Basic.1735039143._hygCtx._hyg.9)))) (DistribSMul.toSMulZeroClass.{u_5, u_6} α E (AddMonoid.toAddZeroClass.{u_6} E (AddCommMonoid.toAddMonoid.{u_6} E inst._@.Mathlib.Algebra.Module.Basic.1735039143._hygCtx._hyg.9)) inst._@.Mathlib.Algebra.Module.Basic.1735039143._hygCtx._hyg.19))) s (HSMul.hSMul.{u_7, u_6, u_6} R E E (instHSMul.{u_7, u_6} R E (SMulZeroClass.toSMul.{u_7, u_6} R E (AddZero.toZero.{u_6} E (AddZeroClass.toAddZero.{u_6} E (AddMonoid.toAddZeroClass.{u_6} E (AddCommMonoid.toAddMonoid.{u_6} E inst._@.Mathlib.Algebra.Module.Basic.1735039143._hygCtx._hyg.9)))) (DistribSMul.toSMulZeroClass.{u_7, u_6} R E (AddMonoid.toAddZeroClass.{u_6} E (AddCommMonoid.toAddMonoid.{u_6} E 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(AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} R (Semiring.toNonAssocSemiring.{u_7} R (DivisionSemiring.toSemiring.{u_7} R inst._@.Mathlib.Algebra.Module.Basic.1735039143._hygCtx._hyg.12))))) n)) x))","typeFull":"∀ {α : Type u_5} {E : Type u_6} (R : Type u_7) [inst : AddCommMonoid E] [inst_1 : DivisionSemiring R]\n  [inst_2 : Module R E] [inst_3 : DistribSMul α E] (n : ℕ) (s : α) (x : E), (↑n)⁻¹ • s • x = s • (↑n)⁻¹ • x","typeReadable":"∀ {α : Type u_5} {E : Type u_6} (R : Type u_7) [inst : AddCommMonoid E] [inst_1 : DivisionSemiring R]\n  [inst_2 : Module R E] [inst_3 : DistribSMul α E] (n : ℕ) (s : α) (x : E), (↑n)⁻¹ • s • x = s • (↑n)⁻¹ • x","typeReferences":[["Nat","cast"],["Module"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DistribSMul"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["Inv","inv"],["InvOneClass","toInv"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["DivisionSemiring","toSemiring"],["DivisionSemiring","toGroupWithZero"],["AddZeroClass","toAddZero"],["DivisionSemiring"],["Nat"],["Module","toDistribMulAction"],["AddMonoidWithOne","toNatCast"],["DivInvOneMonoid","toInvOneClass"],["HSMul","hSMul"],["AddZero","toZero"],["DivisionMonoid","toDivInvOneMonoid"],["AddMonoid","toAddZeroClass"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["Nat","cast"],["SMulZeroClass","toSMul"],["AddMonoidHom","instAddMonoidHomClass"],["DistribSMul","toAddMonoidHom"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["map_inv_natCast_smul"],["Semiring","toNonAssocSemiring"],["AddMonoidHom"],["MonoidWithZero","toMonoid"],["Eq","symm"],["instHSMul"],["AddMonoidHom","instFunLike"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["Inv","inv"],["InvOneClass","toInv"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["DivisionSemiring","toSemiring"],["DivisionSemiring","toGroupWithZero"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["AddMonoidWithOne","toNatCast"],["DivInvOneMonoid","toInvOneClass"],["HSMul","hSMul"],["AddZero","toZero"],["DivisionMonoid","toDivInvOneMonoid"],["AddMonoid","toAddZeroClass"],["GroupWithZero","toDivisionMonoid"]]},{"isProp":true,"kind":"theorem","name":["map_inv_intCast_smul"],"typeFallback":"forall {M : Type.{u_3}} {M₂ : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.6 : AddCommGroup.{u_3} M] [inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.9 : AddCommGroup.{u_4} M₂] {F : Type.{u_5}} [inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.13 : FunLike.{succ u_5, succ u_3, succ u_4} F M M₂] [inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.18 : AddMonoidHomClass.{u_5, u_3, u_4} F M M₂ (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (SubNegMonoid.toAddMonoid.{u_3} M (AddGroup.toSubNegMonoid.{u_3} M (AddCommGroup.toAddGroup.{u_3} M inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.6))))) (AddZeroClass.toAddZero.{u_4} M₂ (AddMonoid.toAddZeroClass.{u_4} M₂ (SubNegMonoid.toAddMonoid.{u_4} M₂ (AddGroup.toSubNegMonoid.{u_4} M₂ (AddCommGroup.toAddGroup.{u_4} M₂ inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.9))))) inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.13] (f : F) (R : Type.{u_6}) (S : Type.{u_7}) [inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.26 : DivisionRing.{u_6} R] [inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.29 : DivisionRing.{u_7} S] [inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.32 : Module.{u_6, u_3} R M (DivisionSemiring.toSemiring.{u_6} R (DivisionRing.toDivisionSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.26)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.6)] [inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.36 : Module.{u_7, u_4} S M₂ (DivisionSemiring.toSemiring.{u_7} S (DivisionRing.toDivisionSemiring.{u_7} S inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.29)) (AddCommGroup.toAddCommMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.9)] (z : Int) (x : M), Eq.{succ u_4} M₂ (DFunLike.coe.{succ u_5, succ u_3, succ u_4} F M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.13 f (HSMul.hSMul.{u_6, u_3, u_3} R M M (instHSMul.{u_6, u_3} R M (SMulZeroClass.toSMul.{u_6, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (SubNegMonoid.toAddMonoid.{u_3} M (AddGroup.toSubNegMonoid.{u_3} M (AddCommGroup.toAddGroup.{u_3} M inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.6)))))) (DistribSMul.toSMulZeroClass.{u_6, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (SubNegMonoid.toAddMonoid.{u_3} M (AddGroup.toSubNegMonoid.{u_3} M (AddCommGroup.toAddGroup.{u_3} M inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.6)))) (DistribMulAction.toDistribSMul.{u_6, u_3} R M (MonoidWithZero.toMonoid.{u_6} R (Semiring.toMonoidWithZero.{u_6} R (DivisionSemiring.toSemiring.{u_6} R (DivisionRing.toDivisionSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.26)))) (SubNegMonoid.toAddMonoid.{u_3} M (AddGroup.toSubNegMonoid.{u_3} M (AddCommGroup.toAddGroup.{u_3} M inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.6))) (Module.toDistribMulAction.{u_6, u_3} R M (DivisionSemiring.toSemiring.{u_6} R (DivisionRing.toDivisionSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.26)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.6) inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.32))))) (Inv.inv.{u_6} R (InvOneClass.toInv.{u_6} R (DivInvOneMonoid.toInvOneClass.{u_6} R (DivisionMonoid.toDivInvOneMonoid.{u_6} R (GroupWithZero.toDivisionMonoid.{u_6} R (DivisionSemiring.toGroupWithZero.{u_6} R (DivisionRing.toDivisionSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.26)))))) (Int.cast.{u_6} R (AddGroupWithOne.toIntCast.{u_6} R (Ring.toAddGroupWithOne.{u_6} R (DivisionRing.toRing.{u_6} R inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.26))) z)) x)) (HSMul.hSMul.{u_7, u_4, u_4} S M₂ M₂ (instHSMul.{u_7, u_4} S M₂ (SMulZeroClass.toSMul.{u_7, u_4} S M₂ (AddZero.toZero.{u_4} M₂ (AddZeroClass.toAddZero.{u_4} M₂ (AddMonoid.toAddZeroClass.{u_4} M₂ (SubNegMonoid.toAddMonoid.{u_4} M₂ (AddGroup.toSubNegMonoid.{u_4} M₂ (AddCommGroup.toAddGroup.{u_4} M₂ inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.9)))))) (DistribSMul.toSMulZeroClass.{u_7, u_4} S M₂ (AddMonoid.toAddZeroClass.{u_4} M₂ (SubNegMonoid.toAddMonoid.{u_4} M₂ (AddGroup.toSubNegMonoid.{u_4} M₂ (AddCommGroup.toAddGroup.{u_4} M₂ inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.9)))) (DistribMulAction.toDistribSMul.{u_7, u_4} S M₂ (MonoidWithZero.toMonoid.{u_7} S (Semiring.toMonoidWithZero.{u_7} S (DivisionSemiring.toSemiring.{u_7} S (DivisionRing.toDivisionSemiring.{u_7} S inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.29)))) (SubNegMonoid.toAddMonoid.{u_4} M₂ (AddGroup.toSubNegMonoid.{u_4} M₂ (AddCommGroup.toAddGroup.{u_4} M₂ inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.9))) (Module.toDistribMulAction.{u_7, u_4} S M₂ (DivisionSemiring.toSemiring.{u_7} S (DivisionRing.toDivisionSemiring.{u_7} S inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.29)) (AddCommGroup.toAddCommMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.9) inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.36))))) (Inv.inv.{u_7} S (InvOneClass.toInv.{u_7} S (DivInvOneMonoid.toInvOneClass.{u_7} S (DivisionMonoid.toDivInvOneMonoid.{u_7} S (GroupWithZero.toDivisionMonoid.{u_7} S (DivisionSemiring.toGroupWithZero.{u_7} S (DivisionRing.toDivisionSemiring.{u_7} S inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.29)))))) (Int.cast.{u_7} S (AddGroupWithOne.toIntCast.{u_7} S (Ring.toAddGroupWithOne.{u_7} S (DivisionRing.toRing.{u_7} S inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.29))) z)) (DFunLike.coe.{succ u_5, succ u_3, succ u_4} F M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) inst._@.Mathlib.Algebra.Module.Basic.848232645._hygCtx._hyg.13 f x))","typeFull":"∀ {M : Type u_3} {M₂ : Type u_4} [inst : AddCommGroup M] [inst_1 : AddCommGroup M₂] {F : Type u_5}\n  [inst_2 : FunLike F M M₂] [AddMonoidHomClass F M M₂] (f : F) (R : Type u_6) (S : Type u_7) [inst_4 : DivisionRing R]\n  [inst_5 : DivisionRing S] [inst_6 : Module R M] [inst_7 : Module S M₂] (z : ℤ) (x : M), f ((↑z)⁻¹ • x) = (↑z)⁻¹ • f x","typeReadable":"∀ {M : Type u_3} {M₂ : Type u_4} [inst : AddCommGroup M] [inst_1 : AddCommGroup M₂] {F : Type u_5}\n  [inst_2 : FunLike F M M₂] [AddMonoidHomClass F M M₂] (f : F) (R : Type u_6) (S : Type u_7) [inst_4 : DivisionRing R]\n  [inst_5 : DivisionRing S] [inst_6 : Module R M] [inst_7 : Module S M₂] (z : ℤ) (x : M), f ((↑z)⁻¹ • x) = (↑z)⁻¹ • f x","typeReferences":[["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["AddMonoidHomClass"],["Int","cast"],["DFunLike","coe"],["Ring","toAddGroupWithOne"],["DivisionRing","toDivisionSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddGroup","toSubNegMonoid"],["DivisionRing"],["Eq"],["DistribSMul","toSMulZeroClass"],["FunLike"],["Inv","inv"],["InvOneClass","toInv"],["DivisionRing","toRing"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["DivisionSemiring","toGroupWithZero"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["Int"],["Module","toDistribMulAction"],["DivInvOneMonoid","toInvOneClass"],["SubNegMonoid","toAddMonoid"],["HSMul","hSMul"],["AddGroupWithOne","toIntCast"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Ring","toNonAssocRing"],["Eq","trans"],["map_neg"],["AddCommGroup","toAddGroup"],["AddGroupWithOne","toAddMonoidWithOne"],["SMulZeroClass","toSMul"],["AddGroup","toSubtractionMonoid"],["DivisionMonoid","toDivInvMonoid"],["Semiring","toNonAssocSemiring"],["Or"],["Ring","toAddGroupWithOne"],["Eq","symm"],["NonUnitalNonAssocRing","toHasDistribNeg"],["InvolutiveNeg","toNeg"],["AddGroup","toSubNegMonoid"],["Eq","ndrec"],["NonAssocSemiring","toAddCommMonoidWithOne"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["InvOneClass","toInv"],["Neg","neg"],["DistribMulAction","toDistribSMul"],["DivisionSemiring","toSemiring"],["DivisionSemiring","toGroupWithZero"],["AddZeroClass","toAddZero"],["Int","instNegInt"],["Ring","toSemiring"],["Exists","casesOn"],["Nat"],["AddMonoidWithOne","toNatCast"],["DivInvMonoid","toMonoid"],["SubtractionMonoid","toSubNegMonoid"],["Eq","refl"],["HSMul","hSMul"],["AddCommGroup","toAddCommMonoid"],["id"],["Eq","mpr"],["Int","cast_natCast"],["AddZero","toZero"],["DivisionMonoid","toDivInvOneMonoid"],["AddMonoid","toAddZeroClass"],["GroupWithZero","toDivisionMonoid"],["MulOneClass","toMulOne"],["Nat","cast"],["inv_neg"],["SubtractionCommMonoid","toSubtractionMonoid"],["AddCommMonoid","toAddMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["DFunLike","coe"],["Int","cast"],["map_inv_natCast_smul"],["congrArg"],["MulOne","toMul"],["DivisionRing","toDivisionSemiring"],["congr"],["Int","eq_nat_or_neg"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["instHSMul"],["congrFun'"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["instNatCastInt"],["neg_smul"],["Inv","inv"],["True"],["SubNegMonoid","toNeg"],["DivisionRing","toRing"],["Semiring","toMonoidWithZero"],["Int","cast_neg"],["Int"],["Or","casesOn"],["eq_self"],["Module","toDistribMulAction"],["AddGroupWithOne","toAddGroup"],["NegZeroClass","toNeg"],["DivInvOneMonoid","toInvOneClass"],["of_eq_true"],["AddCommGroup","toDivisionAddCommMonoid"],["HasDistribNeg","toInvolutiveNeg"],["SubNegMonoid","toAddMonoid"],["AddGroupWithOne","toIntCast"]]},{"isProp":true,"kind":"theorem","name":["map_inv_natCast_smul"],"typeFallback":"forall {M : Type.{u_3}} {M₂ : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.6 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.9 : AddCommMonoid.{u_4} M₂] {F : Type.{u_5}} [inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.13 : FunLike.{succ u_5, succ u_3, succ u_4} F M M₂] [inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.18 : AddMonoidHomClass.{u_5, u_3, u_4} F M M₂ (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.6))) (AddZeroClass.toAddZero.{u_4} M₂ (AddMonoid.toAddZeroClass.{u_4} M₂ (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.9))) inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.13] (f : F) (R : Type.{u_6}) (S : Type.{u_7}) [inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.26 : DivisionSemiring.{u_6} R] [inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.29 : DivisionSemiring.{u_7} S] [inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.32 : Module.{u_6, u_3} R M (DivisionSemiring.toSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.26) inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.6] [inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.36 : Module.{u_7, u_4} S M₂ (DivisionSemiring.toSemiring.{u_7} S inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.9] (n : Nat) (x : M), Eq.{succ u_4} M₂ (DFunLike.coe.{succ u_5, succ u_3, succ u_4} F M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.13 f (HSMul.hSMul.{u_6, u_3, u_3} R M M (instHSMul.{u_6, u_3} R M (SMulZeroClass.toSMul.{u_6, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.6)))) (DistribSMul.toSMulZeroClass.{u_6, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.6)) (DistribMulAction.toDistribSMul.{u_6, u_3} R M (MonoidWithZero.toMonoid.{u_6} R (Semiring.toMonoidWithZero.{u_6} R (DivisionSemiring.toSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.26))) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.6) (Module.toDistribMulAction.{u_6, u_3} R M (DivisionSemiring.toSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.26) inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.32))))) (Inv.inv.{u_6} R (InvOneClass.toInv.{u_6} R (DivInvOneMonoid.toInvOneClass.{u_6} R (DivisionMonoid.toDivInvOneMonoid.{u_6} R (GroupWithZero.toDivisionMonoid.{u_6} R (DivisionSemiring.toGroupWithZero.{u_6} R inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.26))))) (Nat.cast.{u_6} R (AddMonoidWithOne.toNatCast.{u_6} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_6} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R (DivisionSemiring.toSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.26))))) n)) x)) (HSMul.hSMul.{u_7, u_4, u_4} S M₂ M₂ (instHSMul.{u_7, u_4} S M₂ (SMulZeroClass.toSMul.{u_7, u_4} S M₂ (AddZero.toZero.{u_4} M₂ (AddZeroClass.toAddZero.{u_4} M₂ (AddMonoid.toAddZeroClass.{u_4} M₂ (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.9)))) (DistribSMul.toSMulZeroClass.{u_7, u_4} S M₂ (AddMonoid.toAddZeroClass.{u_4} M₂ (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.9)) (DistribMulAction.toDistribSMul.{u_7, u_4} S M₂ (MonoidWithZero.toMonoid.{u_7} S (Semiring.toMonoidWithZero.{u_7} S (DivisionSemiring.toSemiring.{u_7} S inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.29))) (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.9) (Module.toDistribMulAction.{u_7, u_4} S M₂ (DivisionSemiring.toSemiring.{u_7} S inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.29) inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.36))))) (Inv.inv.{u_7} S (InvOneClass.toInv.{u_7} S (DivInvOneMonoid.toInvOneClass.{u_7} S (DivisionMonoid.toDivInvOneMonoid.{u_7} S (GroupWithZero.toDivisionMonoid.{u_7} S (DivisionSemiring.toGroupWithZero.{u_7} S inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.29))))) (Nat.cast.{u_7} S (AddMonoidWithOne.toNatCast.{u_7} S (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} S (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} S (Semiring.toNonAssocSemiring.{u_7} S (DivisionSemiring.toSemiring.{u_7} S inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.29))))) n)) (DFunLike.coe.{succ u_5, succ u_3, succ u_4} F M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) inst._@.Mathlib.Algebra.Module.Basic.4176078827._hygCtx._hyg.13 f x))","typeFull":"∀ {M : Type u_3} {M₂ : Type u_4} [inst : AddCommMonoid M] [inst_1 : AddCommMonoid M₂] {F : Type u_5}\n  [inst_2 : FunLike F M M₂] [AddMonoidHomClass F M M₂] (f : F) (R : Type u_6) (S : Type u_7)\n  [inst_4 : DivisionSemiring R] [inst_5 : DivisionSemiring S] [inst_6 : Module R M] [inst_7 : Module S M₂] (n : ℕ)\n  (x : M), f ((↑n)⁻¹ • x) = (↑n)⁻¹ • f x","typeReadable":"∀ {M : Type u_3} {M₂ : Type u_4} [inst : AddCommMonoid M] [inst_1 : AddCommMonoid M₂] {F : Type u_5}\n  [inst_2 : FunLike F M M₂] [AddMonoidHomClass F M M₂] (f : F) (R : Type u_6) (S : Type u_7)\n  [inst_4 : DivisionSemiring R] [inst_5 : DivisionSemiring S] [inst_6 : Module R M] [inst_7 : Module S M₂] (n : ℕ)\n  (x : M), f ((↑n)⁻¹ • x) = (↑n)⁻¹ • f x","typeReferences":[["Nat","cast"],["Module"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddMonoidHomClass"],["DFunLike","coe"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["FunLike"],["Inv","inv"],["InvOneClass","toInv"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["DivisionSemiring","toGroupWithZero"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["DivisionSemiring"],["Nat"],["Module","toDistribMulAction"],["AddMonoidWithOne","toNatCast"],["DivInvOneMonoid","toInvOneClass"],["HSMul","hSMul"],["AddZero","toZero"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["inv_zero"],["Eq","trans"],["Classical","propDecidable"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["MonoidWithZero","toMulZeroOneClass"],["Semiring","toNonAssocSemiring"],["Eq","symm"],["DistribMulAction","toMulAction"],["Monoid","toSemigroup"],["NonAssocSemiring","toAddCommMonoidWithOne"],["smul_inv_smul₀"],["DistribSMul","toSMulZeroClass"],["InvOneClass","toInv"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["DivisionSemiring","toGroupWithZero"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["SMulWithZero","toSMulZeroClass"],["AddMonoidWithOne","toNatCast"],["MulZeroOneClass","toMulZeroClass"],["Eq","refl"],["HSMul","hSMul"],["id"],["zero_smul"],["Eq","mpr"],["AddZero","toZero"],["DivisionMonoid","toDivInvOneMonoid"],["map_natCast_smul"],["GroupWithZero","toDivisionMonoid"],["AddMonoid","toAddZeroClass"],["smul_zero"],["Nat","cast"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["congrArg"],["inv_smul_smul₀"],["congr"],["MonoidWithZero","toMonoid"],["GroupWithZero","toMonoidWithZero"],["instHSMul"],["Zero","toOfNat0"],["congrFun'"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["map_zero"],["MulActionWithZero","toSMulWithZero"],["Inv","inv"],["True"],["AddMonoidHomClass","toZeroHomClass"],["Semiring","toMonoidWithZero"],["Module","toMulActionWithZero"],["OfNat","ofNat"],["eq_self"],["Module","toDistribMulAction"],["DivInvOneMonoid","toInvOneClass"],["of_eq_true"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["dite"],["MulAction","toSemigroupAction"]]},{"isProp":true,"kind":"theorem","name":["inv_intCast_smul_comm"],"typeFallback":"forall {α : Type.{u_5}} {E : Type.{u_6}} (R : Type.{u_7}) [inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.9 : AddCommGroup.{u_6} E] [inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.12 : DivisionRing.{u_7} R] [inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.15 : Module.{u_7, u_6} R E (DivisionSemiring.toSemiring.{u_7} R (DivisionRing.toDivisionSemiring.{u_7} R inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.12)) (AddCommGroup.toAddCommMonoid.{u_6} E inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.9)] [inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.19 : DistribSMul.{u_5, u_6} α E (AddMonoid.toAddZeroClass.{u_6} E (SubNegMonoid.toAddMonoid.{u_6} E (AddGroup.toSubNegMonoid.{u_6} E (AddCommGroup.toAddGroup.{u_6} E inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.9))))] (n : Int) (s : α) (x : E), Eq.{succ u_6} E (HSMul.hSMul.{u_7, u_6, u_6} R E E (instHSMul.{u_7, u_6} R E (SMulZeroClass.toSMul.{u_7, u_6} R E (AddZero.toZero.{u_6} E (AddZeroClass.toAddZero.{u_6} E (AddMonoid.toAddZeroClass.{u_6} E (SubNegMonoid.toAddMonoid.{u_6} E (AddGroup.toSubNegMonoid.{u_6} E (AddCommGroup.toAddGroup.{u_6} E inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.9)))))) (DistribSMul.toSMulZeroClass.{u_7, u_6} R E (AddMonoid.toAddZeroClass.{u_6} E (SubNegMonoid.toAddMonoid.{u_6} E (AddGroup.toSubNegMonoid.{u_6} E (AddCommGroup.toAddGroup.{u_6} E inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.9)))) (DistribMulAction.toDistribSMul.{u_7, u_6} R E (MonoidWithZero.toMonoid.{u_7} R (Semiring.toMonoidWithZero.{u_7} R (DivisionSemiring.toSemiring.{u_7} R (DivisionRing.toDivisionSemiring.{u_7} R inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.12)))) (SubNegMonoid.toAddMonoid.{u_6} E (AddGroup.toSubNegMonoid.{u_6} E (AddCommGroup.toAddGroup.{u_6} E inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.9))) (Module.toDistribMulAction.{u_7, u_6} R E (DivisionSemiring.toSemiring.{u_7} R (DivisionRing.toDivisionSemiring.{u_7} R inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.12)) (AddCommGroup.toAddCommMonoid.{u_6} E inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.9) inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.15))))) (Inv.inv.{u_7} R (InvOneClass.toInv.{u_7} R (DivInvOneMonoid.toInvOneClass.{u_7} R (DivisionMonoid.toDivInvOneMonoid.{u_7} R (GroupWithZero.toDivisionMonoid.{u_7} R (DivisionSemiring.toGroupWithZero.{u_7} R (DivisionRing.toDivisionSemiring.{u_7} R inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.12)))))) (Int.cast.{u_7} R (AddGroupWithOne.toIntCast.{u_7} R (Ring.toAddGroupWithOne.{u_7} R (DivisionRing.toRing.{u_7} R inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.12))) n)) (HSMul.hSMul.{u_5, u_6, u_6} α E E (instHSMul.{u_5, u_6} α E (SMulZeroClass.toSMul.{u_5, u_6} α E (AddZero.toZero.{u_6} E (AddZeroClass.toAddZero.{u_6} E (AddMonoid.toAddZeroClass.{u_6} E (SubNegMonoid.toAddMonoid.{u_6} E (AddGroup.toSubNegMonoid.{u_6} E (AddCommGroup.toAddGroup.{u_6} E inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.9)))))) (DistribSMul.toSMulZeroClass.{u_5, u_6} α E (AddMonoid.toAddZeroClass.{u_6} E (SubNegMonoid.toAddMonoid.{u_6} E (AddGroup.toSubNegMonoid.{u_6} E (AddCommGroup.toAddGroup.{u_6} E inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.9)))) inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.19))) s x)) (HSMul.hSMul.{u_5, u_6, u_6} α E E (instHSMul.{u_5, u_6} α E (SMulZeroClass.toSMul.{u_5, u_6} α E (AddZero.toZero.{u_6} E (AddZeroClass.toAddZero.{u_6} E (AddMonoid.toAddZeroClass.{u_6} E (SubNegMonoid.toAddMonoid.{u_6} E (AddGroup.toSubNegMonoid.{u_6} E (AddCommGroup.toAddGroup.{u_6} E inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.9)))))) (DistribSMul.toSMulZeroClass.{u_5, u_6} α E (AddMonoid.toAddZeroClass.{u_6} E (SubNegMonoid.toAddMonoid.{u_6} E (AddGroup.toSubNegMonoid.{u_6} E (AddCommGroup.toAddGroup.{u_6} E inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.9)))) inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.19))) s (HSMul.hSMul.{u_7, u_6, u_6} R E E (instHSMul.{u_7, u_6} R E (SMulZeroClass.toSMul.{u_7, u_6} R E (AddZero.toZero.{u_6} E (AddZeroClass.toAddZero.{u_6} E (AddMonoid.toAddZeroClass.{u_6} E (SubNegMonoid.toAddMonoid.{u_6} E (AddGroup.toSubNegMonoid.{u_6} E (AddCommGroup.toAddGroup.{u_6} E inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.9)))))) (DistribSMul.toSMulZeroClass.{u_7, u_6} R E (AddMonoid.toAddZeroClass.{u_6} E (SubNegMonoid.toAddMonoid.{u_6} E (AddGroup.toSubNegMonoid.{u_6} E (AddCommGroup.toAddGroup.{u_6} E inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.9)))) (DistribMulAction.toDistribSMul.{u_7, u_6} R E (MonoidWithZero.toMonoid.{u_7} R (Semiring.toMonoidWithZero.{u_7} R (DivisionSemiring.toSemiring.{u_7} R (DivisionRing.toDivisionSemiring.{u_7} R inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.12)))) (SubNegMonoid.toAddMonoid.{u_6} E (AddGroup.toSubNegMonoid.{u_6} E (AddCommGroup.toAddGroup.{u_6} E inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.9))) (Module.toDistribMulAction.{u_7, u_6} R E (DivisionSemiring.toSemiring.{u_7} R (DivisionRing.toDivisionSemiring.{u_7} R inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.12)) (AddCommGroup.toAddCommMonoid.{u_6} E inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.9) inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.15))))) (Inv.inv.{u_7} R (InvOneClass.toInv.{u_7} R (DivInvOneMonoid.toInvOneClass.{u_7} R (DivisionMonoid.toDivInvOneMonoid.{u_7} R (GroupWithZero.toDivisionMonoid.{u_7} R (DivisionSemiring.toGroupWithZero.{u_7} R (DivisionRing.toDivisionSemiring.{u_7} R inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.12)))))) (Int.cast.{u_7} R (AddGroupWithOne.toIntCast.{u_7} R (Ring.toAddGroupWithOne.{u_7} R (DivisionRing.toRing.{u_7} R inst._@.Mathlib.Algebra.Module.Basic.2046414032._hygCtx._hyg.12))) n)) x))","typeFull":"∀ {α : Type u_5} {E : Type u_6} (R : Type u_7) [inst : AddCommGroup E] [inst_1 : DivisionRing R] [inst_2 : Module R E]\n  [inst_3 : DistribSMul α E] (n : ℤ) (s : α) (x : E), (↑n)⁻¹ • s • x = s • (↑n)⁻¹ • x","typeReadable":"∀ {α : Type u_5} {E : Type u_6} (R : Type u_7) [inst : AddCommGroup E] [inst_1 : DivisionRing R] [inst_2 : Module R E]\n  [inst_3 : DistribSMul α E] (n : ℤ) (s : α) (x : E), (↑n)⁻¹ • s • x = s • (↑n)⁻¹ • x","typeReferences":[["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["Int","cast"],["DistribSMul"],["Ring","toAddGroupWithOne"],["DivisionRing","toDivisionSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddGroup","toSubNegMonoid"],["Eq"],["DivisionRing"],["DistribSMul","toSMulZeroClass"],["Inv","inv"],["InvOneClass","toInv"],["DivisionRing","toRing"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["DivisionSemiring","toGroupWithZero"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["Int"],["Module","toDistribMulAction"],["DivInvOneMonoid","toInvOneClass"],["SubNegMonoid","toAddMonoid"],["HSMul","hSMul"],["AddGroupWithOne","toIntCast"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["AddMonoidHom","instAddMonoidHomClass"],["DistribSMul","toAddMonoidHom"],["Int","cast"],["DFunLike","coe"],["Ring","toAddGroupWithOne"],["DivisionRing","toDivisionSemiring"],["AddMonoidHom"],["MonoidWithZero","toMonoid"],["Eq","symm"],["instHSMul"],["AddMonoidHom","instFunLike"],["AddGroup","toSubNegMonoid"],["DistribSMul","toSMulZeroClass"],["Inv","inv"],["InvOneClass","toInv"],["DivisionRing","toRing"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["map_inv_intCast_smul"],["DivisionSemiring","toGroupWithZero"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["DivInvOneMonoid","toInvOneClass"],["SubNegMonoid","toAddMonoid"],["HSMul","hSMul"],["AddGroupWithOne","toIntCast"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Units","neg_smul"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Basic.2907391035._hygCtx._hyg.6 : Ring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Basic.2907391035._hygCtx._hyg.9 : AddCommGroup.{u_3} M] [inst._@.Mathlib.Algebra.Module.Basic.2907391035._hygCtx._hyg.12 : Module.{u_2, u_3} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Basic.2907391035._hygCtx._hyg.6) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Basic.2907391035._hygCtx._hyg.9)] (u : Units.{u_2} R (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Basic.2907391035._hygCtx._hyg.6)))) (x : M), Eq.{succ u_3} M (HSMul.hSMul.{u_2, u_3, u_3} (Units.{u_2} R (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Basic.2907391035._hygCtx._hyg.6)))) M M (instHSMul.{u_2, u_3} (Units.{u_2} R (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Basic.2907391035._hygCtx._hyg.6)))) M (Units.instSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Basic.2907391035._hygCtx._hyg.6))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (SubNegMonoid.toAddMonoid.{u_3} M (AddGroup.toSubNegMonoid.{u_3} M (AddCommGroup.toAddGroup.{u_3} M inst._@.Mathlib.Algebra.Module.Basic.2907391035._hygCtx._hyg.9)))))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (SubNegMonoid.toAddMonoid.{u_3} M (AddGroup.toSubNegMonoid.{u_3} M (AddCommGroup.toAddGroup.{u_3} M inst._@.Mathlib.Algebra.Module.Basic.2907391035._hygCtx._hyg.9)))) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R (Ring.toSemiring.{u_2} R 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(Ring.toNonAssocRing.{u_2} R inst._@.Mathlib.Algebra.Module.Basic.2907391035._hygCtx._hyg.6)))) u) x) (Neg.neg.{u_3} M (NegZeroClass.toNeg.{u_3} M (SubNegZeroMonoid.toNegZeroClass.{u_3} M (SubtractionMonoid.toSubNegZeroMonoid.{u_3} M (SubtractionCommMonoid.toSubtractionMonoid.{u_3} M (AddCommGroup.toDivisionAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Basic.2907391035._hygCtx._hyg.9))))) (HSMul.hSMul.{u_2, u_3, u_3} (Units.{u_2} R (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Basic.2907391035._hygCtx._hyg.6)))) M M (instHSMul.{u_2, u_3} (Units.{u_2} R (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Basic.2907391035._hygCtx._hyg.6)))) M (Units.instSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Basic.2907391035._hygCtx._hyg.6))) 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inst._@.Mathlib.Algebra.Module.Basic.2907391035._hygCtx._hyg.6) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Basic.2907391035._hygCtx._hyg.9) inst._@.Mathlib.Algebra.Module.Basic.2907391035._hygCtx._hyg.12)))))) u x))","typeFull":"∀ {R : Type u_2} {M : Type u_3} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] (u : Rˣ) (x : M),\n  -u • x = -(u • x)","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] (u : Rˣ) (x : M),\n  -u • x = -(u • x)","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Ring","toNonAssocRing"],["Units","instNeg"],["Module"],["AddCommGroup","toAddGroup"],["SubtractionCommMonoid","toSubtractionMonoid"],["SMulZeroClass","toSMul"],["SubNegZeroMonoid","toNegZeroClass"],["MonoidWithZero","toMonoid"],["instHSMul"],["NonUnitalNonAssocRing","toHasDistribNeg"],["AddGroup","toSubNegMonoid"],["Units","instSMul"],["Eq"],["NonAssocRing","toNonUnitalNonAssocRing"],["Units"],["DistribSMul","toSMulZeroClass"],["Neg","neg"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["Module","toDistribMulAction"],["NegZeroClass","toNeg"],["AddCommGroup","toDivisionAddCommMonoid"],["SubNegMonoid","toAddMonoid"],["HSMul","hSMul"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["MulOneClass","toMulOne"],["SubtractionMonoid","toSubNegZeroMonoid"],["Units","instNeg"],["Ring","toNonAssocRing"],["AddCommGroup","toAddGroup"],["SubtractionCommMonoid","toSubtractionMonoid"],["SMulZeroClass","toSMul"],["SubNegZeroMonoid","toNegZeroClass"],["Units","smul_def"],["congrArg"],["MulOne","toMul"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["instHSMul"],["NonUnitalNonAssocRing","toHasDistribNeg"],["InvolutiveNeg","toNeg"],["AddGroup","toSubNegMonoid"],["Units","instSMul"],["Eq"],["Units"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["neg_smul"],["Neg","neg"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["Ring","toAddCommGroup"],["Units","val_neg"],["Module","toDistribMulAction"],["NegZeroClass","toNeg"],["HasDistribNeg","toInvolutiveNeg"],["AddCommGroup","toDivisionAddCommMonoid"],["SubNegMonoid","toAddMonoid"],["Units","val"],["Eq","refl"],["HSMul","hSMul"],["id"],["AddCommGroup","toAddCommMonoid"],["Eq","mpr"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Set","indicator_smul_apply_left"],"typeFallback":"forall {α : Type.{u_1}} {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Basic.2960584617._hygCtx._hyg.6 : Zero.{u_2} R] [inst._@.Mathlib.Algebra.Module.Basic.2960584617._hygCtx._hyg.9 : Zero.{u_3} M] [inst._@.Mathlib.Algebra.Module.Basic.2960584617._hygCtx._hyg.12 : SMulWithZero.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Basic.2960584617._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Module.Basic.2960584617._hygCtx._hyg.9] (s : Set.{u_1} α) (r : α -> R) (f : α -> M) (a : α), Eq.{succ u_3} M (Set.indicator.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Basic.2960584617._hygCtx._hyg.9 s (fun (a : α) => HSMul.hSMul.{u_2, u_3, u_3} R M M (instHSMul.{u_2, u_3} R M (SMulZeroClass.toSMul.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Basic.2960584617._hygCtx._hyg.9 (SMulWithZero.toSMulZeroClass.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Basic.2960584617._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Module.Basic.2960584617._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.Basic.2960584617._hygCtx._hyg.12))) (r a) (f a)) a) (HSMul.hSMul.{u_2, u_3, u_3} R M M (instHSMul.{u_2, u_3} R M (SMulZeroClass.toSMul.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Basic.2960584617._hygCtx._hyg.9 (SMulWithZero.toSMulZeroClass.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Basic.2960584617._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Module.Basic.2960584617._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.Basic.2960584617._hygCtx._hyg.12))) (Set.indicator.{u_1, u_2} α R inst._@.Mathlib.Algebra.Module.Basic.2960584617._hygCtx._hyg.6 s r a) (f a))","typeFull":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Zero R] [inst_1 : Zero M] [inst_2 : SMulWithZero R M] (s : Set α)\n  (r : α → R) (f : α → M) (a : α), s.indicator (fun a => r a • f a) a = s.indicator r a • f a","typeReadable":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Zero R] [inst_1 : Zero M] [inst_2 : SMulWithZero R M] (s : Set α)\n  (r : α → R) (f : α → M) (a : α), s.indicator (fun a => r a • f a) a = s.indicator r a • f a","typeReferences":[["Set"],["HSMul","hSMul"],["instHSMul"],["SMulZeroClass","toSMul"],["Zero"],["Eq"],["SMulWithZero","toSMulZeroClass"],["Set","indicator"],["SMulWithZero"]],"valueReferences":[["Membership","mem"],["Classical","propDecidable"],["SMulZeroClass","toSMul"],["congrArg"],["Classical","decPred"],["congr"],["Eq","symm"],["instHSMul"],["Zero","toOfNat0"],["congrFun'"],["Eq"],["rfl"],["Set"],["ite"],["SMulWithZero","toSMulZeroClass"],["Set","indicator"],["OfNat","ofNat"],["Set","instMembership"],["if_neg"],["HSMul","hSMul"],["id"],["zero_smul"],["Eq","mpr"],["dite"],["if_pos"]]},{"isProp":true,"kind":"theorem","name":["inv_natCast_smul_eq"],"typeFallback":"forall {E : Type.{u_5}} (R : Type.{u_6}) (S : Type.{u_7}) [inst._@.Mathlib.Algebra.Module.Basic.1617908598._hygCtx._hyg.9 : AddCommMonoid.{u_5} E] [inst._@.Mathlib.Algebra.Module.Basic.1617908598._hygCtx._hyg.12 : DivisionSemiring.{u_6} R] [inst._@.Mathlib.Algebra.Module.Basic.1617908598._hygCtx._hyg.15 : DivisionSemiring.{u_7} S] [inst._@.Mathlib.Algebra.Module.Basic.1617908598._hygCtx._hyg.18 : Module.{u_6, u_5} R E (DivisionSemiring.toSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.Basic.1617908598._hygCtx._hyg.12) inst._@.Mathlib.Algebra.Module.Basic.1617908598._hygCtx._hyg.9] [inst._@.Mathlib.Algebra.Module.Basic.1617908598._hygCtx._hyg.22 : Module.{u_7, u_5} S E (DivisionSemiring.toSemiring.{u_7} S inst._@.Mathlib.Algebra.Module.Basic.1617908598._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Basic.1617908598._hygCtx._hyg.9] (n : Nat) (x : E), Eq.{succ u_5} E (HSMul.hSMul.{u_6, u_5, u_5} R E E (instHSMul.{u_6, u_5} R E (SMulZeroClass.toSMul.{u_6, u_5} R E (AddZero.toZero.{u_5} E (AddZeroClass.toAddZero.{u_5} E (AddMonoid.toAddZeroClass.{u_5} E (AddCommMonoid.toAddMonoid.{u_5} E inst._@.Mathlib.Algebra.Module.Basic.1617908598._hygCtx._hyg.9)))) (DistribSMul.toSMulZeroClass.{u_6, u_5} R E 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inst._@.Mathlib.Algebra.Module.Basic.1617908598._hygCtx._hyg.12))))) (Nat.cast.{u_6} R (AddMonoidWithOne.toNatCast.{u_6} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_6} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R (DivisionSemiring.toSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.Basic.1617908598._hygCtx._hyg.12))))) n)) x) (HSMul.hSMul.{u_7, u_5, u_5} S E E (instHSMul.{u_7, u_5} S E (SMulZeroClass.toSMul.{u_7, u_5} S E (AddZero.toZero.{u_5} E (AddZeroClass.toAddZero.{u_5} E (AddMonoid.toAddZeroClass.{u_5} E (AddCommMonoid.toAddMonoid.{u_5} E inst._@.Mathlib.Algebra.Module.Basic.1617908598._hygCtx._hyg.9)))) (DistribSMul.toSMulZeroClass.{u_7, u_5} S E (AddMonoid.toAddZeroClass.{u_5} E (AddCommMonoid.toAddMonoid.{u_5} E inst._@.Mathlib.Algebra.Module.Basic.1617908598._hygCtx._hyg.9)) (DistribMulAction.toDistribSMul.{u_7, u_5} S E (MonoidWithZero.toMonoid.{u_7} S (Semiring.toMonoidWithZero.{u_7} S (DivisionSemiring.toSemiring.{u_7} S 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inst._@.Mathlib.Algebra.Module.Basic.1617908598._hygCtx._hyg.15))))) n)) x)","typeFull":"∀ {E : Type u_5} (R : Type u_6) (S : Type u_7) [inst : AddCommMonoid E] [inst_1 : DivisionSemiring R]\n  [inst_2 : DivisionSemiring S] [inst_3 : Module R E] [inst_4 : Module S E] (n : ℕ) (x : E), (↑n)⁻¹ • x = (↑n)⁻¹ • x","typeReadable":"∀ {E : Type u_5} (R : Type u_6) (S : Type u_7) [inst : AddCommMonoid E] [inst_1 : DivisionSemiring R]\n  [inst_2 : DivisionSemiring S] [inst_3 : Module R E] [inst_4 : Module S E] (n : ℕ) (x : E), (↑n)⁻¹ • x = (↑n)⁻¹ • x","typeReferences":[["Nat","cast"],["Module"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["Inv","inv"],["InvOneClass","toInv"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["DivisionSemiring","toSemiring"],["DivisionSemiring","toGroupWithZero"],["AddZeroClass","toAddZero"],["DivisionSemiring"],["Nat"],["Module","toDistribMulAction"],["AddMonoidWithOne","toNatCast"],["DivInvOneMonoid","toInvOneClass"],["HSMul","hSMul"],["AddZero","toZero"],["DivisionMonoid","toDivInvOneMonoid"],["AddMonoid","toAddZeroClass"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["AddMonoidHom"],["AddMonoidHom","instFunLike"],["AddMonoidHom","instAddMonoidHomClass"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddMonoidHom","id"],["map_inv_natCast_smul"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["invOf_two_smul_add_invOf_two_smul"],"typeFallback":"forall 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