diff --git "a/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.LocalizedModule.Int.sym.json" "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.LocalizedModule.Int.sym.json" new file mode 100644--- /dev/null +++ "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.LocalizedModule.Int.sym.json" @@ -0,0 +1 @@ +[{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","isInteger_add"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3483575899._hygCtx._hyg.3 : CommSemiring.{u_1} R] {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3483575899._hygCtx._hyg.9 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3483575899._hygCtx._hyg.12 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3483575899._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3483575899._hygCtx._hyg.9] {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3483575899._hygCtx._hyg.17 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3483575899._hygCtx._hyg.20 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3483575899._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3483575899._hygCtx._hyg.17] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3483575899._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3483575899._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3483575899._hygCtx._hyg.3))) M M' 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: Type u_2} [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {M' : Type u_3}\n [inst_3 : AddCommMonoid M'] [inst_4 : Module R M'] (f : M →ₗ[R] M') {x y : M'},\n IsLocalizedModule.IsInteger f x → IsLocalizedModule.IsInteger f y → IsLocalizedModule.IsInteger f (x + y)","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {M : Type u_2} [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {M' : Type u_3}\n [inst_3 : AddCommMonoid M'] [inst_4 : Module R M'] (f : M →ₗ[R] M') {x y : M'},\n IsLocalizedModule.IsInteger f x → IsLocalizedModule.IsInteger f y → IsLocalizedModule.IsInteger f (x + y)","typeReferences":[["Module"],["instHAdd"],["CommSemiring"],["CommSemiring","toSemiring"],["LinearMap"],["IsLocalizedModule","IsInteger"],["HAdd","hAdd"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["AddCommMonoid","toAddCommSemigroup"],["RingHom","id"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"]],"valueReferences":[["Semiring","toNonAssocSemiring"],["RingHom","id"],["LinearMap","range"],["CommSemiring","toSemiring"],["IsLocalizedModule","IsInteger","_proof_1"],["Submodule","add_mem"]]},{"isProp":false,"kind":"definition","name":["IsLocalizedModule","commonDenom"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4184893337._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.Int.4184893337._hygCtx._hyg.3))))) {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4184893337._hygCtx._hyg.9 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4184893337._hygCtx._hyg.12 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4184893337._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4184893337._hygCtx._hyg.9] {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4184893337._hygCtx._hyg.17 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4184893337._hygCtx._hyg.20 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4184893337._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4184893337._hygCtx._hyg.17] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4184893337._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4184893337._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4184893337._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4184893337._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4184893337._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4184893337._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4184893337._hygCtx._hyg.20) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4184893337._hygCtx._hyg.33 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4184893337._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4184893337._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4184893337._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4184893337._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4184893337._hygCtx._hyg.20 S f] {ι : Type.{u_4}}, (Finset.{u_4} ι) -> (ι -> M') -> (Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4184893337._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.Int.4184893337._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.Int.4184893337._hygCtx._hyg.3)))))) S x))","typeFull":"{R : Type u_1} →\n [inst : CommSemiring R] →\n (S : Submonoid R) →\n {M : Type u_2} →\n [inst_1 : AddCommMonoid M] →\n [inst_2 : Module R M] →\n {M' : Type u_3} →\n [inst_3 : AddCommMonoid M'] →\n [inst_4 : Module R M'] →\n (f : M →ₗ[R] M') → [IsLocalizedModule S f] → {ι : Type u_4} → Finset ι → (ι → M') → ↥S","typeReadable":"{R : Type u_1} →\n [inst : CommSemiring R] →\n (S : Submonoid R) →\n {M : Type u_2} →\n [inst_1 : AddCommMonoid M] →\n [inst_2 : Module R M] →\n {M' : Type u_3} →\n [inst_3 : AddCommMonoid M'] →\n [inst_4 : Module R M'] →\n (f : M →ₗ[R] M') → [IsLocalizedModule S f] → {ι : Type u_4} → Finset ι → (ι → M') → ↥S","typeReferences":[["IsLocalizedModule"],["SetLike","instMembership"],["Finset"],["Subtype"],["Submonoid","instSetLike"],["Module"],["Membership","mem"],["CommSemiring"],["CommSemiring","toSemiring"],["MulZeroOneClass","toMulOneClass"],["LinearMap"],["Submonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["Finset","instSetLike"],["Finset"],["Subtype"],["Membership","mem"],["MulZeroOneClass","toMulOneClass"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Subtype","val"],["IsLocalizedModule","IsInteger"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["IsLocalizedModule","exist_integer_multiples"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["Exists","choose"],["AddZeroClass","toAddZero"],["Submonoid"],["Module","toDistribMulAction"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"definition","name":["IsLocalizedModule","finsetIntegerMultiple"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3149876861._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.Int.3149876861._hygCtx._hyg.3))))) {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3149876861._hygCtx._hyg.9 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3149876861._hygCtx._hyg.12 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3149876861._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3149876861._hygCtx._hyg.9] {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3149876861._hygCtx._hyg.17 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3149876861._hygCtx._hyg.20 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3149876861._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3149876861._hygCtx._hyg.17] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3149876861._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3149876861._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3149876861._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3149876861._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3149876861._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3149876861._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3149876861._hygCtx._hyg.20) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3149876861._hygCtx._hyg.33 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3149876861._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3149876861._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3149876861._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3149876861._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3149876861._hygCtx._hyg.20 S f] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3149876861._hygCtx._hyg.37 : DecidableEq.{succ u_2} M], (Finset.{u_3} M') -> (Finset.{u_2} M)","typeFull":"{R : Type u_1} →\n [inst : CommSemiring R] →\n (S : Submonoid R) →\n {M : Type u_2} →\n [inst_1 : AddCommMonoid M] →\n [inst_2 : Module R M] →\n {M' : Type u_3} →\n [inst_3 : AddCommMonoid M'] →\n [inst_4 : Module R M'] →\n (f : M →ₗ[R] M') → [IsLocalizedModule S f] → [DecidableEq M] → Finset M' → Finset M","typeReadable":"{R : Type u_1} →\n [inst : CommSemiring R] →\n (S : Submonoid R) →\n {M : Type u_2} →\n [inst_1 : AddCommMonoid M] →\n [inst_2 : Module R M] →\n {M' : Type u_3} →\n [inst_3 : AddCommMonoid M'] →\n [inst_4 : Module R M'] →\n (f : M →ₗ[R] M') → [IsLocalizedModule S f] → [DecidableEq M] → Finset M' → Finset M","typeReferences":[["IsLocalizedModule"],["Finset"],["Module"],["DecidableEq"],["CommSemiring"],["CommSemiring","toSemiring"],["MulZeroOneClass","toMulOneClass"],["LinearMap"],["Submonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["IsLocalizedModule","integerMultiple"],["Finset","instSetLike"],["SetLike","instMembership"],["Finset"],["Subtype"],["Finset","image"],["Membership","mem"],["id"],["Finset","attach"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","smul_mem_finsetIntegerMultiple_span"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2237106176._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.Int.2237106176._hygCtx._hyg.3))))) {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2237106176._hygCtx._hyg.9 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2237106176._hygCtx._hyg.12 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2237106176._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2237106176._hygCtx._hyg.9] {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2237106176._hygCtx._hyg.17 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2237106176._hygCtx._hyg.20 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2237106176._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2237106176._hygCtx._hyg.17] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2237106176._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2237106176._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2237106176._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2237106176._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2237106176._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2237106176._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2237106176._hygCtx._hyg.20) 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.484884849._hygCtx._hyg.3 : CommSemiring.{u_1} R] {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.484884849._hygCtx._hyg.9 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.484884849._hygCtx._hyg.12 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.484884849._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.484884849._hygCtx._hyg.9] {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.484884849._hygCtx._hyg.17 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.484884849._hygCtx._hyg.20 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.484884849._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.484884849._hygCtx._hyg.17], (LinearMap.{u_1, u_1, u_2, u_3} R R 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(AddZeroClass.toAddZero.{u_3} M' (AddMonoid.toAddZeroClass.{u_3} M' (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.295907554._hygCtx._hyg.17))))))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] {M : Type u_2} [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {M' : Type u_3}\n [inst_3 : AddCommMonoid M'] [inst_4 : Module R M'] (f : M →ₗ[R] M'), IsLocalizedModule.IsInteger f 0","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {M : Type u_2} [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {M' : Type u_3}\n [inst_3 : AddCommMonoid M'] [inst_4 : Module R M'] (f : M →ₗ[R] M'), IsLocalizedModule.IsInteger f 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2145823666._hygCtx._hyg.3))))) {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2145823666._hygCtx._hyg.9 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2145823666._hygCtx._hyg.12 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2145823666._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2145823666._hygCtx._hyg.9] {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2145823666._hygCtx._hyg.17 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2145823666._hygCtx._hyg.20 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2145823666._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2145823666._hygCtx._hyg.17] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2145823666._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2145823666._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2145823666._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2145823666._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2145823666._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2145823666._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2145823666._hygCtx._hyg.20) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2145823666._hygCtx._hyg.33 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2145823666._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2145823666._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2145823666._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2145823666._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2145823666._hygCtx._hyg.20 S f], (Finset.{u_3} M') -> (Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2145823666._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.Int.2145823666._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.Int.2145823666._hygCtx._hyg.3)))))) S x))","typeFull":"{R : Type u_1} →\n [inst : CommSemiring R] →\n (S : Submonoid R) →\n {M : Type u_2} →\n [inst_1 : AddCommMonoid M] →\n [inst_2 : Module R M] →\n {M' : Type u_3} →\n [inst_3 : AddCommMonoid M'] →\n [inst_4 : Module R M'] → (f : M →ₗ[R] M') → [IsLocalizedModule S f] → Finset M' → ↥S","typeReadable":"{R : Type u_1} →\n [inst : CommSemiring R] →\n (S : Submonoid R) →\n {M : Type u_2} →\n [inst_1 : AddCommMonoid M] →\n [inst_2 : Module R M] →\n {M' : Type u_3} →\n [inst_3 : AddCommMonoid M'] →\n [inst_4 : Module R M'] → (f : M →ₗ[R] M') → [IsLocalizedModule S f] → Finset M' → ↥S","typeReferences":[["IsLocalizedModule"],["SetLike","instMembership"],["Finset"],["Subtype"],["Submonoid","instSetLike"],["Module"],["Membership","mem"],["CommSemiring"],["CommSemiring","toSemiring"],["MulZeroOneClass","toMulOneClass"],["LinearMap"],["Submonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["NonAssocSemiring","toMulZeroOneClass"]],"valueReferences":[["id"],["IsLocalizedModule","commonDenom"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","exist_integer_multiples"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._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.Int.2842273061._hygCtx._hyg.3))))) {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.9 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.12 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.9] {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.17 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.20 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.17] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.20) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.33 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.20 S f] {ι : Type.{u_4}} (s : Finset.{u_4} ι) (g : ι -> M'), Exists.{succ u_1} (Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._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.Int.2842273061._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.Int.2842273061._hygCtx._hyg.3)))))) S x)) (fun (b : 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.Int.2842273061._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.Int.2842273061._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.Int.2842273061._hygCtx._hyg.3)))))) S x)) => forall (i : ι), (Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι) (SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (Finset.instSetLike.{u_4} ι)) s i) -> (IsLocalizedModule.IsInteger.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.3 M inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.12 M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.20 f (HSMul.hSMul.{u_1, u_3, u_3} R M' M' (instHSMul.{u_1, u_3} R M' (SMulZeroClass.toSMul.{u_1, u_3} R M' (AddZero.toZero.{u_3} M' (AddZeroClass.toAddZero.{u_3} M' (AddMonoid.toAddZeroClass.{u_3} M' (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.17)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R M' (AddMonoid.toAddZeroClass.{u_3} M' (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.17)) (DistribMulAction.toDistribSMul.{u_1, u_3} R M' (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.17) (Module.toDistribMulAction.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2842273061._hygCtx._hyg.20))))) (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 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IsLocalizedModule.IsInteger f (↑b • g i)","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} [inst_1 : AddCommMonoid M]\n [inst_2 : Module R M] {M' : Type u_3} [inst_3 : AddCommMonoid M'] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n [IsLocalizedModule S f] {ι : Type u_4} (s : Finset ι) (g : ι → M'),\n ∃ b, ∀ i ∈ s, IsLocalizedModule.IsInteger f (↑b • g 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4168942277._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.Int.4168942277._hygCtx._hyg.3))))) {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4168942277._hygCtx._hyg.9 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4168942277._hygCtx._hyg.12 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4168942277._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4168942277._hygCtx._hyg.9] {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4168942277._hygCtx._hyg.17 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4168942277._hygCtx._hyg.20 : 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(Module.toDistribMulAction.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4168942277._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4168942277._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4168942277._hygCtx._hyg.20)))) S)) (IsLocalizedModule.commonDenom.{u_1, u_2, u_3, u_4} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4168942277._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4168942277._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4168942277._hygCtx._hyg.12 M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4168942277._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4168942277._hygCtx._hyg.20 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4168942277._hygCtx._hyg.33 ι s g) (g (Subtype.val.{succ u_4} ι (fun (x : ι) => Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι) (SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (Finset.instSetLike.{u_4} ι)) s x) i)))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} [inst_1 : AddCommMonoid M]\n [inst_2 : Module R M] {M' : Type u_3} [inst_3 : AddCommMonoid M'] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n [inst_5 : IsLocalizedModule S f] {ι : Type u_4} (s : Finset ι) (g : ι → M') (i : ↥s),\n f (IsLocalizedModule.integerMultiple S f s g i) = IsLocalizedModule.commonDenom S f s g • g ↑i","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} [inst_1 : AddCommMonoid M]\n [inst_2 : Module R M] {M' : Type u_3} [inst_3 : AddCommMonoid M'] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n [inst_5 : IsLocalizedModule S f] {ι : Type u_4} (s : Finset ι) (g : ι → M') (i : ↥s),\n f (IsLocalizedModule.integerMultiple S f s g i) = IsLocalizedModule.commonDenom S f s g • g ↑i","typeReferences":[["IsLocalizedModule"],["Finset","instSetLike"],["Subtype"],["Finset"],["LinearMap","instFunLike"],["Module"],["CommSemiring"],["Membership","mem"],["Submonoid","smul"],["MulZeroOneClass","toMulOneClass"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Subtype","val"],["DFunLike","coe"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["IsLocalizedModule","integerMultiple"],["SetLike","instMembership"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["IsLocalizedModule","commonDenom"],["AddZeroClass","toAddZero"],["LinearMap"],["Submonoid"],["Module","toDistribMulAction"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Finset","instSetLike"],["Finset"],["Subtype"],["LinearMap","instFunLike"],["Membership","mem"],["SMulZeroClass","toSMul"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["Subtype","val"],["IsLocalizedModule","IsInteger"],["Semiring","toNonAssocSemiring"],["Subtype","prop"],["RingHom","id"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["Exists","choose_spec"],["IsLocalizedModule","exist_integer_multiples"],["Submonoid","instSetLike"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["Exists","choose"],["AddZeroClass","toAddZero"],["LinearMap"],["Submonoid"],["Module","toDistribMulAction"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["IsLocalizedModule","exist_integer_multiples_of_finite"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4160604971._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.Int.4160604971._hygCtx._hyg.3))))) {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4160604971._hygCtx._hyg.9 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4160604971._hygCtx._hyg.12 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4160604971._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4160604971._hygCtx._hyg.9] {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4160604971._hygCtx._hyg.17 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4160604971._hygCtx._hyg.20 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4160604971._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4160604971._hygCtx._hyg.17] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4160604971._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4160604971._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4160604971._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4160604971._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4160604971._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4160604971._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4160604971._hygCtx._hyg.20) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4160604971._hygCtx._hyg.33 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4160604971._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4160604971._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4160604971._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4160604971._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4160604971._hygCtx._hyg.20 S f] {ι : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4160604971._hygCtx._hyg.38 : Finite.{succ u_4} ι] (g : ι -> M'), Exists.{succ u_1} (Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4160604971._hygCtx._hyg.3))))) 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.4160604971._hygCtx._hyg.3)))))) S x) b) (g i)))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} [inst_1 : AddCommMonoid M]\n [inst_2 : Module R M] {M' : Type u_3} [inst_3 : AddCommMonoid M'] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n [IsLocalizedModule S f] {ι : Type u_4} [Finite ι] (g : ι → M'),\n ∃ b, ∀ (i : ι), IsLocalizedModule.IsInteger f (↑b • g i)","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} [inst_1 : AddCommMonoid M]\n [inst_2 : Module R M] {M' : Type u_3} [inst_3 : AddCommMonoid M'] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n [IsLocalizedModule S f] {ι : Type u_4} [Finite ι] (g : ι → M'),\n ∃ b, ∀ (i : ι), IsLocalizedModule.IsInteger f (↑b • g 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.1449739664._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.1449739664._hygCtx._hyg.12 M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.1449739664._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.1449739664._hygCtx._hyg.20 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.1449739664._hygCtx._hyg.33 s) (SetLike.coe.{u_3, u_3} (Finset.{u_3} M') M' (Finset.instSetLike.{u_3} M') s))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} [inst_1 : AddCommMonoid M]\n [inst_2 : Module R M] {M' : Type u_3} [inst_3 : AddCommMonoid M'] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n [inst_5 : IsLocalizedModule S f] [inst_6 : DecidableEq M] (s : Finset M'),\n ⇑f '' ↑(IsLocalizedModule.finsetIntegerMultiple S f s) = IsLocalizedModule.commonDenomOfFinset S f s • ↑s","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} [inst_1 : AddCommMonoid 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.778302324._hygCtx._hyg.3 : CommSemiring.{u_1} R] {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.778302324._hygCtx._hyg.9 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.778302324._hygCtx._hyg.12 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.778302324._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.778302324._hygCtx._hyg.9] {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.778302324._hygCtx._hyg.17 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.778302324._hygCtx._hyg.20 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.778302324._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.778302324._hygCtx._hyg.17] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.778302324._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.778302324._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.778302324._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.778302324._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.778302324._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.778302324._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.778302324._hygCtx._hyg.20) {a : R} {x : M'}, (IsLocalizedModule.IsInteger.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.778302324._hygCtx._hyg.3 M inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.778302324._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.778302324._hygCtx._hyg.12 M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.778302324._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.778302324._hygCtx._hyg.20 f x) -> (IsLocalizedModule.IsInteger.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.778302324._hygCtx._hyg.3 M inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.778302324._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.778302324._hygCtx._hyg.12 M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.778302324._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.778302324._hygCtx._hyg.20 f (HSMul.hSMul.{u_1, u_3, u_3} R M' M' (instHSMul.{u_1, u_3} R M' (SMulZeroClass.toSMul.{u_1, u_3} R M' (AddZero.toZero.{u_3} M' (AddZeroClass.toAddZero.{u_3} M' (AddMonoid.toAddZeroClass.{u_3} M' (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.778302324._hygCtx._hyg.17)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R M' (AddMonoid.toAddZeroClass.{u_3} M' 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IsLocalizedModule.IsInteger f x → IsLocalizedModule.IsInteger f (a • x)","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] {M : Type u_2} [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {M' : Type u_3}\n [inst_3 : AddCommMonoid M'] [inst_4 : Module R M'] (f : M →ₗ[R] M') {a : R} {x : M'},\n IsLocalizedModule.IsInteger f x → IsLocalizedModule.IsInteger f (a • 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._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.Int.3035498878._hygCtx._hyg.3))))) {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.9 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.12 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.9] {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.17 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.20 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.17] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.20) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.33 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.20 S f] (s : Finset.{u_3} M'), Exists.{succ u_1} (Subtype.{succ u_1} R (fun (x : R) => Membership.mem.{u_1, u_1} R (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.3))))) (SetLike.instMembership.{u_1, u_1} (Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R 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(CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._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.Int.3035498878._hygCtx._hyg.3)))))) S x)) => forall (a : M'), (Membership.mem.{u_3, u_3} M' (Finset.{u_3} M') (SetLike.instMembership.{u_3, u_3} (Finset.{u_3} M') M' (Finset.instSetLike.{u_3} M')) s a) -> (IsLocalizedModule.IsInteger.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.3 M inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.12 M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.20 f (HSMul.hSMul.{u_1, u_3, u_3} R M' M' (instHSMul.{u_1, u_3} R M' (SMulZeroClass.toSMul.{u_1, u_3} R M' (AddZero.toZero.{u_3} M' (AddZeroClass.toAddZero.{u_3} M' (AddMonoid.toAddZeroClass.{u_3} M' (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.17)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R M' (AddMonoid.toAddZeroClass.{u_3} M' (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.17)) (DistribMulAction.toDistribSMul.{u_1, u_3} R M' (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.3035498878._hygCtx._hyg.17) (Module.toDistribMulAction.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R 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[inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.33 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.20 S f] {ι : Type.{u_4}} (s : Finset.{u_4} ι), (ι -> M') -> (Subtype.{succ u_4} ι (fun (x : ι) => Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι) (SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (Finset.instSetLike.{u_4} ι)) s x)) -> M","typeFull":"{R : Type u_1} →\n [inst : CommSemiring R] →\n (S : Submonoid R) →\n {M : Type u_2} →\n [inst_1 : AddCommMonoid M] →\n [inst_2 : Module R M] →\n {M' : Type u_3} →\n [inst_3 : AddCommMonoid M'] →\n [inst_4 : Module R M'] →\n (f : M →ₗ[R] M') → 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._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.Int.2573599350._hygCtx._hyg.3))))) {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.9 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.12 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.9] {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.17 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.20 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.17] (f : LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.3) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.3))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.20) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.33 : IsLocalizedModule.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.3 M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.20 S f] {ι : Type.{u_4}} (s : Finset.{u_4} ι) (g : ι -> M') (i : Subtype.{succ u_4} ι (fun (x : ι) => Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι) (SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (Finset.instSetLike.{u_4} ι)) s x)), IsLocalizedModule.IsInteger.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.3 M inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.12 M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.20 f (HSMul.hSMul.{u_1, u_3, u_3} R M' M' (instHSMul.{u_1, u_3} R M' (SMulZeroClass.toSMul.{u_1, u_3} R M' (AddZero.toZero.{u_3} M' (AddZeroClass.toAddZero.{u_3} M' (AddMonoid.toAddZeroClass.{u_3} M' (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.17)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R M' (AddMonoid.toAddZeroClass.{u_3} M' (AddCommMonoid.toAddMonoid.{u_3} M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.17)) (DistribMulAction.toDistribSMul.{u_1, u_3} R M' (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.3))) 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inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.3 S M inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.12 M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.20 f inst._@.Mathlib.Algebra.Module.LocalizedModule.Int.2573599350._hygCtx._hyg.33 ι s g))) (g (Subtype.val.{succ u_4} ι (fun (x : ι) => Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι) (SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (Finset.instSetLike.{u_4} ι)) s x) i)))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} [inst_1 : AddCommMonoid M]\n [inst_2 : Module R M] {M' : Type u_3} [inst_3 : AddCommMonoid M'] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n [inst_5 : IsLocalizedModule S f] {ι : Type u_4} (s : Finset ι) (g : ι → M') (i : ↥s),\n IsLocalizedModule.IsInteger f (↑⋯.choose • g ↑i)","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} [inst_1 : AddCommMonoid M]\n [inst_2 : Module R M] {M' : Type u_3} [inst_3 : AddCommMonoid M'] [inst_4 : Module R M'] (f : M →ₗ[R] M')\n [inst_5 : IsLocalizedModule S f] {ι : Type u_4} (s : Finset ι) (g : ι → M') (i : ↥s),\n IsLocalizedModule.IsInteger f (↑⋯.choose • g 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