Add files using upload-large-folder tool

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Algebra.IsSimpleRing.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.ArithmeticGeometric.sym.json

@@ -0,0 +1 @@
+[]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.BigOperators.Group.Multiset.Defs.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Category.FGModuleCat.Abelian.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Category.Grp.Injective.sym.json

@@ -0,0 +1 @@
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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Category.Grp.IsFinite.sym.json

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AddCommGrpCat.instCategory.{u_1} AddCommGrpCat.instAbelian.{u_1})) S)])) -> Sort.{u_2}) (x._@.Mathlib.Algebra.Category.Grp.IsFinite.3482765569._hygCtx.218.Mathlib.Algebra.Category.Grp.IsFinite.3482765569._hygCtx._hyg.230 : Prod.{u_1, u_1} ([mdata _patWithRef: AddCommGrpCat.carrier.{u_1} (CategoryTheory.ShortComplex.X₁.{u_1, succ u_1} AddCommGrpCat.{u_1} AddCommGrpCat.instCategory.{u_1} (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u_1, succ u_1} AddCommGrpCat.{u_1} AddCommGrpCat.instCategory.{u_1} (CategoryTheory.Abelian.toPreadditive.{u_1, succ u_1} AddCommGrpCat.{u_1} AddCommGrpCat.instCategory.{u_1} AddCommGrpCat.instAbelian.{u_1})) S)]) ([mdata _patWithRef: AddCommGrpCat.carrier.{u_1} (CategoryTheory.ShortComplex.X₃.{u_1, succ u_1} AddCommGrpCat.{u_1} AddCommGrpCat.instCategory.{u_1} (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u_1, succ u_1} AddCommGrpCat.{u_1} AddCommGrpCat.instCategory.{u_1} (CategoryTheory.Abelian.toPreadditive.{u_1, succ u_1} 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Category.Grp.LargeColimits.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.CharP.IntermediateField.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["IntermediateField","charP"],"typeFallback":"forall {F : Type.{u_1}} {E : Type.{u_2}} [inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.4 : Field.{u_1} F] [inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.7 : Field.{u_2} E] [inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.10 : Algebra.{u_1, u_2} F E (Semifield.toCommSemiring.{u_1} F (Field.toSemifield.{u_1} F inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.4)) (DivisionSemiring.toSemiring.{u_2} E (Semifield.toDivisionSemiring.{u_2} E (Field.toSemifield.{u_2} E inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.7)))] (L : IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.10) (p : Nat) 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inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.10)) L x)) (AddGroupWithOne.toAddMonoidWithOne.{u_2} (Subtype.{succ u_2} E (fun (x : E) => Membership.mem.{u_2, u_2} E (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.10) (SetLike.instMembership.{u_2, u_2} (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.10) E (IntermediateField.instSetLike.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.10)) L x)) (Ring.toAddGroupWithOne.{u_2} (Subtype.{succ u_2} E (fun (x : E) => Membership.mem.{u_2, u_2} E (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.10) (SetLike.instMembership.{u_2, u_2} (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.10) E (IntermediateField.instSetLike.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.10)) L x)) (DivisionRing.toRing.{u_2} (Subtype.{succ u_2} E (fun (x : E) => Membership.mem.{u_2, u_2} E (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.10) (SetLike.instMembership.{u_2, u_2} (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.10) E (IntermediateField.instSetLike.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.10)) L x)) (Field.toDivisionRing.{u_2} (Subtype.{succ u_2} E (fun (x : E) => Membership.mem.{u_2, u_2} E (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.10) (SetLike.instMembership.{u_2, u_2} (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.10) E (IntermediateField.instSetLike.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.10)) L x)) (IntermediateField.toField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.430573643._hygCtx._hyg.10 L))))) p","typeFull":"∀ {F : Type u_1} {E : Type u_2} [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] (L : IntermediateField F E)\n  (p : ℕ) [CharP F p], CharP (↥L) p","typeReadable":"∀ {F : Type u_1} {E : Type u_2} [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] (L : IntermediateField F E)\n  (p : ℕ) [CharP F p], CharP (↥L) p","typeReferences":[["Semifield","toCommSemiring"],["SetLike","instMembership"],["Subtype"],["Field"],["IntermediateField"],["CharP"],["IntermediateField","toField"],["Membership","mem"],["DivisionRing","toRing"],["AddGroupWithOne","toAddMonoidWithOne"],["Field","toDivisionRing"],["DivisionSemiring","toSemiring"],["Algebra"],["IntermediateField","instSetLike"],["Nat"],["Ring","toAddGroupWithOne"],["Field","toSemifield"],["Semifield","toDivisionSemiring"]],"valueReferences":[["Semifield","toCommSemiring"],["Subtype"],["IntermediateField"],["Membership","mem"],["RingHom","injective"],["AddCommMonoid","toAddMonoid"],["CommRing","toNonUnitalCommRing"],["charP_of_injective_algebraMap"],["Algebra","id"],["SubsemiringClass","nontrivial"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["Algebra","toSMul"],["MonoidWithZero","toMonoid"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["DistribMulAction","toMulAction"],["IntermediateField","algebra'"],["NonAssocCommRing","toNonAssocRing"],["Semifield","toDivisionSemiring"],["SubfieldClass","toSubringClass"],["Field","toCommRing"],["SetLike","instMembership"],["IsScalarTower","left"],["CommRing","toNonAssocCommRing"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["IntermediateField","toField"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["IsLocalRing","toNontrivial"],["Field","toDivisionRing"],["Semiring","toMonoidWithZero"],["SubringClass","toSubsemiringClass"],["DivisionSemiring","toSemiring"],["IntermediateField","instSetLike"],["Module","toDistribMulAction"],["IntermediateField","instSubfieldClass"],["Field","toSemifield"],["DivisionRing","isSimpleRing"],["Field","instIsLocalRing"],["algebraMap"]]},{"isProp":true,"kind":"theorem","name":["IntermediateField","charP'"],"typeFallback":"forall {F : Type.{u_1}} {E : Type.{u_2}} [inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.4 : Field.{u_1} F] [inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.7 : Field.{u_2} E] [inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.10 : Algebra.{u_1, u_2} F E (Semifield.toCommSemiring.{u_1} F (Field.toSemifield.{u_1} F inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.4)) (DivisionSemiring.toSemiring.{u_2} E (Semifield.toDivisionSemiring.{u_2} E (Field.toSemifield.{u_2} E inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.7)))] (L : IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.10) (p : Nat) [inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.20 : CharP.{u_2} E (AddGroupWithOne.toAddMonoidWithOne.{u_2} E (Ring.toAddGroupWithOne.{u_2} E (DivisionRing.toRing.{u_2} E (Field.toDivisionRing.{u_2} E inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.7)))) p], CharP.{u_2} (Subtype.{succ u_2} E (fun (x : E) => Membership.mem.{u_2, u_2} E (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.10) (SetLike.instMembership.{u_2, u_2} (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.10) E (IntermediateField.instSetLike.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.10)) L x)) (AddGroupWithOne.toAddMonoidWithOne.{u_2} (Subtype.{succ u_2} E (fun (x : E) => Membership.mem.{u_2, u_2} E (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.10) (SetLike.instMembership.{u_2, u_2} (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.10) E (IntermediateField.instSetLike.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.10)) L x)) (Ring.toAddGroupWithOne.{u_2} (Subtype.{succ u_2} E (fun (x : E) => Membership.mem.{u_2, u_2} E (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.10) (SetLike.instMembership.{u_2, u_2} (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.10) E (IntermediateField.instSetLike.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.10)) L x)) (DivisionRing.toRing.{u_2} (Subtype.{succ u_2} E (fun (x : E) => Membership.mem.{u_2, u_2} E (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.10) (SetLike.instMembership.{u_2, u_2} (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.10) E (IntermediateField.instSetLike.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.10)) L x)) (Field.toDivisionRing.{u_2} (Subtype.{succ u_2} E (fun (x : E) => Membership.mem.{u_2, u_2} E (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.10) (SetLike.instMembership.{u_2, u_2} (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.10) E (IntermediateField.instSetLike.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.10)) L x)) (IntermediateField.toField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.2356791895._hygCtx._hyg.10 L))))) p","typeFull":"∀ {F : Type u_1} {E : Type u_2} [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] (L : IntermediateField F E)\n  (p : ℕ) [CharP E p], CharP (↥L) p","typeReadable":"∀ {F : Type u_1} {E : Type u_2} [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] (L : IntermediateField F E)\n  (p : ℕ) [CharP E p], CharP (↥L) p","typeReferences":[["Semifield","toCommSemiring"],["SetLike","instMembership"],["Subtype"],["Field"],["IntermediateField"],["CharP"],["IntermediateField","toField"],["Membership","mem"],["DivisionRing","toRing"],["AddGroupWithOne","toAddMonoidWithOne"],["Field","toDivisionRing"],["DivisionSemiring","toSemiring"],["Algebra"],["IntermediateField","instSetLike"],["Nat"],["Ring","toAddGroupWithOne"],["Field","toSemifield"],["Semifield","toDivisionSemiring"]],"valueReferences":[["IntermediateField","toSubfield"],["Subfield","charP"],["Field","toDivisionRing"]]},{"isProp":true,"kind":"theorem","name":["IntermediateField","charZero"],"typeFallback":"forall {F : Type.{u_1}} {E : Type.{u_2}} [inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.4 : Field.{u_1} F] [inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.7 : Field.{u_2} E] [inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.10 : Algebra.{u_1, u_2} F E (Semifield.toCommSemiring.{u_1} F (Field.toSemifield.{u_1} F inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.4)) (DivisionSemiring.toSemiring.{u_2} E (Semifield.toDivisionSemiring.{u_2} E (Field.toSemifield.{u_2} E inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.7)))] (L : IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.10) [inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.20 : CharZero.{u_1} F (AddGroupWithOne.toAddMonoidWithOne.{u_1} F (Ring.toAddGroupWithOne.{u_1} F (DivisionRing.toRing.{u_1} F (Field.toDivisionRing.{u_1} F inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.4))))], CharZero.{u_2} (Subtype.{succ u_2} E (fun (x : E) => Membership.mem.{u_2, u_2} E (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.10) (SetLike.instMembership.{u_2, u_2} (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.10) E (IntermediateField.instSetLike.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.10)) L x)) (AddGroupWithOne.toAddMonoidWithOne.{u_2} (Subtype.{succ u_2} E (fun (x : E) => Membership.mem.{u_2, u_2} E (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.10) (SetLike.instMembership.{u_2, u_2} (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.10) E (IntermediateField.instSetLike.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.10)) L x)) (Ring.toAddGroupWithOne.{u_2} (Subtype.{succ u_2} E (fun (x : E) => Membership.mem.{u_2, u_2} E (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.10) (SetLike.instMembership.{u_2, u_2} (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.10) E (IntermediateField.instSetLike.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.10)) L x)) (DivisionRing.toRing.{u_2} (Subtype.{succ u_2} E (fun (x : E) => Membership.mem.{u_2, u_2} E (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.10) (SetLike.instMembership.{u_2, u_2} (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.10) E (IntermediateField.instSetLike.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.10)) L x)) (Field.toDivisionRing.{u_2} (Subtype.{succ u_2} E (fun (x : E) => Membership.mem.{u_2, u_2} E (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.10) (SetLike.instMembership.{u_2, u_2} (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.10) E (IntermediateField.instSetLike.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.10)) L x)) (IntermediateField.toField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.3024440656._hygCtx._hyg.10 L)))))","typeFull":"∀ {F : Type u_1} {E : Type u_2} [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] (L : IntermediateField F E)\n  [CharZero F], CharZero ↥L","typeReadable":"∀ {F : Type u_1} {E : Type u_2} [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] (L : IntermediateField F E)\n  [CharZero F], CharZero ↥L","typeReferences":[["Semifield","toCommSemiring"],["SetLike","instMembership"],["Subtype"],["Field"],["IntermediateField"],["IntermediateField","toField"],["Membership","mem"],["DivisionRing","toRing"],["AddGroupWithOne","toAddMonoidWithOne"],["Field","toDivisionRing"],["DivisionSemiring","toSemiring"],["Algebra"],["IntermediateField","instSetLike"],["Ring","toAddGroupWithOne"],["Field","toSemifield"],["Semifield","toDivisionSemiring"],["CharZero"]],"valueReferences":[["Semifield","toCommSemiring"],["Subtype"],["IntermediateField"],["Membership","mem"],["RingHom","injective"],["AddCommMonoid","toAddMonoid"],["CommRing","toNonUnitalCommRing"],["Algebra","id"],["SubsemiringClass","nontrivial"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["Algebra","toSMul"],["MonoidWithZero","toMonoid"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["DistribMulAction","toMulAction"],["IntermediateField","algebra'"],["NonAssocCommRing","toNonAssocRing"],["Semifield","toDivisionSemiring"],["SubfieldClass","toSubringClass"],["Field","toCommRing"],["SetLike","instMembership"],["charZero_of_injective_algebraMap"],["IsScalarTower","left"],["CommRing","toNonAssocCommRing"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["IntermediateField","toField"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["IsLocalRing","toNontrivial"],["Field","toDivisionRing"],["Semiring","toMonoidWithZero"],["SubringClass","toSubsemiringClass"],["DivisionSemiring","toSemiring"],["IntermediateField","instSetLike"],["Module","toDistribMulAction"],["IntermediateField","instSubfieldClass"],["Field","toSemifield"],["DivisionRing","isSimpleRing"],["Field","instIsLocalRing"],["algebraMap"]]},{"isProp":true,"kind":"theorem","name":["IntermediateField","expChar'"],"typeFallback":"forall {F : Type.{u_1}} {E : Type.{u_2}} [inst._@.Mathlib.Algebra.CharP.IntermediateField.1719933876._hygCtx._hyg.4 : Field.{u_1} F] [inst._@.Mathlib.Algebra.CharP.IntermediateField.1719933876._hygCtx._hyg.7 : Field.{u_2} E] [inst._@.Mathlib.Algebra.CharP.IntermediateField.1719933876._hygCtx._hyg.10 : Algebra.{u_1, u_2} F E (Semifield.toCommSemiring.{u_1} F (Field.toSemifield.{u_1} F inst._@.Mathlib.Algebra.CharP.IntermediateField.1719933876._hygCtx._hyg.4)) (DivisionSemiring.toSemiring.{u_2} E (Semifield.toDivisionSemiring.{u_2} E (Field.toSemifield.{u_2} E inst._@.Mathlib.Algebra.CharP.IntermediateField.1719933876._hygCtx._hyg.7)))] (L : IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.1719933876._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.1719933876._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.1719933876._hygCtx._hyg.10) (p : Nat) [inst._@.Mathlib.Algebra.CharP.IntermediateField.1719933876._hygCtx._hyg.20 : ExpChar.{u_2} E (AddGroupWithOne.toAddMonoidWithOne.{u_2} E (Ring.toAddGroupWithOne.{u_2} E (DivisionRing.toRing.{u_2} E (Field.toDivisionRing.{u_2} E inst._@.Mathlib.Algebra.CharP.IntermediateField.1719933876._hygCtx._hyg.7)))) p], ExpChar.{u_2} (Subtype.{succ u_2} E (fun (x : E) => Membership.mem.{u_2, u_2} E (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.1719933876._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.1719933876._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.1719933876._hygCtx._hyg.10) (SetLike.instMembership.{u_2, u_2} (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.1719933876._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.1719933876._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.1719933876._hygCtx._hyg.10) E (IntermediateField.instSetLike.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.1719933876._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.1719933876._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.1719933876._hygCtx._hyg.10)) L x)) (AddGroupWithOne.toAddMonoidWithOne.{u_2} (Subtype.{succ u_2} E (fun (x : E) => Membership.mem.{u_2, u_2} E (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.1719933876._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.1719933876._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.1719933876._hygCtx._hyg.10) (SetLike.instMembership.{u_2, u_2} (IntermediateField.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.1719933876._hygCtx._hyg.4 inst._@.Mathlib.Algebra.CharP.IntermediateField.1719933876._hygCtx._hyg.7 inst._@.Mathlib.Algebra.CharP.IntermediateField.1719933876._hygCtx._hyg.10) E (IntermediateField.instSetLike.{u_1, u_2} F E inst._@.Mathlib.Algebra.CharP.IntermediateField.1719933876._hygCtx._hyg.4 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Expr.sym.json

@@ -0,0 +1 @@
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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.GCDMonoid.PUnit.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Group.ConjFinite.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Group.EvenFunction.sym.json

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{β : Type u_2} [inst : Neg α] {γ : Type u_3} {g : α → γ} [inst_1 : Monoid β] [inst_2 : AddGroup γ]\n  [inst_3 : DistribMulAction β γ], Function.Odd g → ∀ (r : β), Function.Odd (r • g)","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Neg α] {γ : Type u_3} {g : α → γ} [inst_1 : Monoid β] [inst_2 : AddGroup γ]\n  [inst_3 : DistribMulAction β γ], Function.Odd g → ∀ (r : β), Function.Odd (r • 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inst._@.Mathlib.Algebra.Group.EvenFunction.2202578331._hygCtx._hyg.17)], (Function.Odd.{u_1, u_2} α β inst._@.Mathlib.Algebra.Group.EvenFunction.2202578331._hygCtx._hyg.4 (NegZeroClass.toNeg.{u_2} β (SubNegZeroMonoid.toNegZeroClass.{u_2} β (SubtractionMonoid.toSubNegZeroMonoid.{u_2} β (SubtractionCommMonoid.toSubtractionMonoid.{u_2} β (AddCommGroup.toDivisionAddCommMonoid.{u_2} β (Ring.toAddCommGroup.{u_2} β inst._@.Mathlib.Algebra.Group.EvenFunction.2202578331._hygCtx._hyg.14)))))) f) -> (Function.Even.{u_1, u_3} α γ inst._@.Mathlib.Algebra.Group.EvenFunction.2202578331._hygCtx._hyg.4 g) -> (Function.Odd.{u_1, u_3} α γ inst._@.Mathlib.Algebra.Group.EvenFunction.2202578331._hygCtx._hyg.4 (NegZeroClass.toNeg.{u_3} γ (SubNegZeroMonoid.toNegZeroClass.{u_3} γ (SubtractionMonoid.toSubNegZeroMonoid.{u_3} γ (SubtractionCommMonoid.toSubtractionMonoid.{u_3} γ (AddCommGroup.toDivisionAddCommMonoid.{u_3} γ inst._@.Mathlib.Algebra.Group.EvenFunction.2202578331._hygCtx._hyg.17))))) 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Function.Even g → Function.Odd (f • 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Group.Fin.Tuple.sym.json

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

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Submodule","isInternal_prime_power_torsion"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3 : CommRing.{u} R] [inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.6 : IsDomain.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))] {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.10 : AddCommGroup.{v} M] [inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.13 : Module.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.10)] [inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.17 : IsDedekindDomain.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3] [inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.20 : Module.Finite.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.13], (Module.IsTorsion.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.13) -> (DirectSum.IsInternal.{u, v, v} (Subtype.{succ u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R 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(Submodule.instTop.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.13))))) x)) (b : Subtype.{succ u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (fun (x : Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) => Membership.mem.{u, u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)))) (SetLike.instMembership.{u, u} (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)))) (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (Finset.instSetLike.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))))) (Multiset.toFinset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (Submodule.decidableEq.{u, u} R R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R (CommRing.toNonUnitalCommRing.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))))) (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)))) (UniqueFactorizationMonoid.factors.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (CommSemiring.toCommMonoidWithZero.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (IdemCommSemiring.toCommSemiring.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (Ideal.instIdemCommSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)))) (Ideal.uniqueFactorizationMonoid.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.17) (Submodule.annihilator.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.13 (Top.top.{v} (Submodule.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.13) (Submodule.instTop.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.13))))) x)) => Subtype.instDecidableEq.{succ u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (Membership.mem.{u, u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)))) (SetLike.instMembership.{u, u} (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)))) (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (Finset.instSetLike.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))))) (Multiset.toFinset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (Submodule.decidableEq.{u, u} R R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R (CommRing.toNonUnitalCommRing.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))))) (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)))) (UniqueFactorizationMonoid.factors.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (CommSemiring.toCommMonoidWithZero.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (IdemCommSemiring.toCommSemiring.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (Ideal.instIdemCommSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)))) (Ideal.uniqueFactorizationMonoid.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.17) (Submodule.annihilator.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.13 (Top.top.{v} (Submodule.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.13) (Submodule.instTop.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.13)))))) (Submodule.decidableEq.{u, u} R R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R (CommRing.toNonUnitalCommRing.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))))) (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)))) a b) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.10) (Submodule.setLike.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.13) (Submodule.addSubmonoidClass.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.13) (fun (p : Subtype.{succ u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (fun (x : Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) => Membership.mem.{u, u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)))) (SetLike.instMembership.{u, u} (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)))) (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (Finset.instSetLike.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))))) (Multiset.toFinset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (Submodule.decidableEq.{u, u} R R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R (CommRing.toNonUnitalCommRing.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))))) (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)))) (UniqueFactorizationMonoid.factors.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (CommSemiring.toCommMonoidWithZero.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (IdemCommSemiring.toCommSemiring.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (Ideal.instIdemCommSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)))) (Ideal.uniqueFactorizationMonoid.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.17) (Submodule.annihilator.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.13 (Top.top.{v} (Submodule.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.13) (Submodule.instTop.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.13))))) x)) => Submodule.torsionBySet.{u, v} R M (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.13 (SetLike.coe.{u, u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) R (Submodule.setLike.{u, u} R R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))))) (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)))) (HPow.hPow.{u, 0, u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) Nat (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (instHPow.{u, 0} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) Nat (Submodule.instPowNat.{u, u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (IsScalarTower.right.{u, u} R R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3) (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (Algebra.id.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))))) (Subtype.val.{succ u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (fun (x : Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) => Membership.mem.{u, u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)))) (SetLike.instMembership.{u, u} (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)))) (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (Finset.instSetLike.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))))) (Multiset.toFinset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (Submodule.decidableEq.{u, u} R R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R (CommRing.toNonUnitalCommRing.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))))) (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)))) (UniqueFactorizationMonoid.factors.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (CommSemiring.toCommMonoidWithZero.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (IdemCommSemiring.toCommSemiring.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (Ideal.instIdemCommSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)))) (Ideal.uniqueFactorizationMonoid.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.17) (Submodule.annihilator.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.13 (Top.top.{v} (Submodule.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.13) (Submodule.instTop.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.13))))) x) p) (Multiset.count.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (Submodule.decidableEq.{u, u} R R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R (CommRing.toNonUnitalCommRing.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))))) (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)))) (Subtype.val.{succ u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (fun (x : Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) => Membership.mem.{u, u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)))) (SetLike.instMembership.{u, u} (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)))) (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (Finset.instSetLike.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))))) (Multiset.toFinset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (Submodule.decidableEq.{u, u} R R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R (CommRing.toNonUnitalCommRing.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))))) (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)))) (UniqueFactorizationMonoid.factors.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (CommSemiring.toCommMonoidWithZero.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (IdemCommSemiring.toCommSemiring.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (Ideal.instIdemCommSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)))) (Ideal.uniqueFactorizationMonoid.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.17) (Submodule.annihilator.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.13 (Top.top.{v} (Submodule.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.13) (Submodule.instTop.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.13))))) x) p) (UniqueFactorizationMonoid.factors.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (CommSemiring.toCommMonoidWithZero.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (IdemCommSemiring.toCommSemiring.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3))) (Ideal.instIdemCommSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)))) (Ideal.uniqueFactorizationMonoid.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.17) (Submodule.annihilator.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.13 (Top.top.{v} (Submodule.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.13) (Submodule.instTop.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2401201452._hygCtx._hyg.13)))))))))","typeFull":"∀ {R : Type u} [inst : CommRing R] [IsDomain R] {M : Type v} [inst_2 : AddCommGroup M] [inst_3 : Module R M]\n  [inst_4 : IsDedekindDomain R] [Module.Finite R M],\n  Module.IsTorsion R M →\n    DirectSum.IsInternal fun p =>\n      Submodule.torsionBySet R M ↑(↑p ^ Multiset.count (↑p) (UniqueFactorizationMonoid.factors ⊤.annihilator))","typeReadable":"∀ {R : Type u} [inst : CommRing R] [IsDomain R] {M : Type v} [inst_2 : AddCommGroup M] [inst_3 : Module R M]\n  [inst_4 : IsDedekindDomain R] [Module.Finite R M],\n  Module.IsTorsion R M →\n    DirectSum.IsInternal fun p =>\n      Submodule.torsionBySet R M ↑(↑p ^ Multiset.count (↑p) (UniqueFactorizationMonoid.factors ⊤.annihilator))","typeReferences":[["Module","IsTorsion"],["UniqueFactorizationMonoid","factors"],["Finset"],["Multiset","count"],["Membership","mem"],["IsDomain"],["Algebra","id"],["Subtype","val"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["Ideal","instIdemCommSemiring"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["IsScalarTower","right"],["Semiring","toModule"],["Submodule","annihilator"],["Submodule","addSubmonoidClass"],["instHPow"],["SetLike","instMembership"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["CommSemiring","toCommMonoidWithZero"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Submodule","decidableEq"],["Nat"],["SetLike","coe"],["AddCommGroup","toAddCommMonoid"],["Top","top"],["DirectSum","IsInternal"],["Finset","instSetLike"],["Subtype"],["Module"],["CommRing","toNonUnitalCommRing"],["Submodule","instPowNat"],["Submodule"],["Ideal"],["Submodule","torsionBySet"],["IdemCommSemiring","toCommSemiring"],["CommRing","toCommSemiring"],["Ideal","uniqueFactorizationMonoid"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["IsDedekindDomain"],["Submodule","instTop"],["AddCommGroup"],["Multiset","toFinset"],["CommRing"],["HPow","hPow"],["Submodule","setLike"],["Subtype","instDecidableEq"],["Module","Finite"]],"valueReferences":[["Membership","mem"],["Submodule","isInternal_prime_power_torsion_of_is_torsion_by_ideal"],["Inter","inter"],["Exists","intro"],["MonoidWithZero","toMulZeroOneClass"],["Submodule","ne_bot_iff"],["And","intro"],["Semiring","toNonAssocSemiring"],["Semiring","toModule"],["Submodule","annihilator"],["Set","instInter"],["SetLike","instMembership"],["Exists"],["Submonoid","instSetLike"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["And"],["AddZeroClass","toAddZero"],["Bot","bot"],["Set","instMembership"],["Exists","casesOn"],["Submonoid"],["SetLike","coe"],["Submodule","instBot"],["AddCommGroup","toAddCommMonoid"],["Top","top"],["id"],["Eq","mpr"],["AddZero","toZero"],["nonZeroDivisors","ne_zero"],["AddMonoid","toAddZeroClass"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["Submodule"],["congrArg"],["Ideal"],["Zero","toOfNat0"],["Eq"],["Module","isTorsionBySet_annihilator_top"],["propext"],["IsDomain","toNontrivial"],["CommRing","toCommSemiring"],["Set"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Submodule","instTop"],["Submodule","annihilator_top_inter_nonZeroDivisors"],["Semiring","toMonoidWithZero"],["OfNat","ofNat"],["Submodule","setLike"],["nonZeroDivisors"],["Ne"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["Submodule","isInternal_prime_power_torsion_of_is_torsion_by_ideal"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3 : CommRing.{u} R] [inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.6 : IsDomain.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))] {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.10 : AddCommGroup.{v} M] [inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.13 : Module.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.10)] [inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.17 : IsDedekindDomain.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3] {I : Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))}, (Ne.{succ u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) I (Bot.bot.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Submodule.instBot.{u, u} R R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))))) (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))))) -> (Module.IsTorsionBySet.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.13 (SetLike.coe.{u, u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) R (Submodule.setLike.{u, u} R R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))))) (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) I)) -> (DirectSum.IsInternal.{u, v, v} (Subtype.{succ u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (fun (x : Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) => Membership.mem.{u, u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (SetLike.instMembership.{u, u} (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Finset.instSetLike.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))))) (Multiset.toFinset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Submodule.decidableEq.{u, u} R R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R (CommRing.toNonUnitalCommRing.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))))) (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (UniqueFactorizationMonoid.factors.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (CommSemiring.toCommMonoidWithZero.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (IdemCommSemiring.toCommSemiring.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Ideal.instIdemCommSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (Ideal.uniqueFactorizationMonoid.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.17) I)) x)) M (Submodule.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.13) (fun (a : Subtype.{succ u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (fun (x : Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) => Membership.mem.{u, u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (SetLike.instMembership.{u, u} (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Finset.instSetLike.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))))) (Multiset.toFinset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Submodule.decidableEq.{u, u} R R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R (CommRing.toNonUnitalCommRing.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))))) (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (UniqueFactorizationMonoid.factors.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (CommSemiring.toCommMonoidWithZero.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (IdemCommSemiring.toCommSemiring.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Ideal.instIdemCommSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (Ideal.uniqueFactorizationMonoid.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.17) I)) x)) (b : Subtype.{succ u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (fun (x : Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) => Membership.mem.{u, u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (SetLike.instMembership.{u, u} (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Finset.instSetLike.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))))) (Multiset.toFinset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Submodule.decidableEq.{u, u} R R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R (CommRing.toNonUnitalCommRing.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))))) (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (UniqueFactorizationMonoid.factors.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (CommSemiring.toCommMonoidWithZero.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (IdemCommSemiring.toCommSemiring.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Ideal.instIdemCommSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (Ideal.uniqueFactorizationMonoid.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.17) I)) x)) => Subtype.instDecidableEq.{succ u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Membership.mem.{u, u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (SetLike.instMembership.{u, u} (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Finset.instSetLike.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))))) (Multiset.toFinset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Submodule.decidableEq.{u, u} R R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R (CommRing.toNonUnitalCommRing.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))))) (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (UniqueFactorizationMonoid.factors.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (CommSemiring.toCommMonoidWithZero.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (IdemCommSemiring.toCommSemiring.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Ideal.instIdemCommSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (Ideal.uniqueFactorizationMonoid.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.17) I))) (Submodule.decidableEq.{u, u} R R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R (CommRing.toNonUnitalCommRing.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))))) (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) a b) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.10) (Submodule.setLike.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.13) (Submodule.addSubmonoidClass.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.13) (fun (p : Subtype.{succ u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (fun (x : Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) => Membership.mem.{u, u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (SetLike.instMembership.{u, u} (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Finset.instSetLike.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))))) (Multiset.toFinset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Submodule.decidableEq.{u, u} R R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R (CommRing.toNonUnitalCommRing.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))))) (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (UniqueFactorizationMonoid.factors.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (CommSemiring.toCommMonoidWithZero.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (IdemCommSemiring.toCommSemiring.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Ideal.instIdemCommSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (Ideal.uniqueFactorizationMonoid.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.17) I)) x)) => Submodule.torsionBySet.{u, v} R M (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.13 (SetLike.coe.{u, u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) R (Submodule.setLike.{u, u} R R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))))) (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (HPow.hPow.{u, 0, u} (Ideal.{u} R 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inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (SetLike.instMembership.{u, u} (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Finset.instSetLike.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))))) (Multiset.toFinset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Submodule.decidableEq.{u, u} R R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R (CommRing.toNonUnitalCommRing.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))))) (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (UniqueFactorizationMonoid.factors.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (CommSemiring.toCommMonoidWithZero.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (IdemCommSemiring.toCommSemiring.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Ideal.instIdemCommSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (Ideal.uniqueFactorizationMonoid.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.17) I)) x) p) (Multiset.count.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Submodule.decidableEq.{u, u} R R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R (CommRing.toNonUnitalCommRing.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))))) (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (Subtype.val.{succ u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (fun (x : Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) => Membership.mem.{u, u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (SetLike.instMembership.{u, u} (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Finset.instSetLike.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))))) (Multiset.toFinset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Submodule.decidableEq.{u, u} R R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R (CommRing.toNonUnitalCommRing.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))))) (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (UniqueFactorizationMonoid.factors.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (CommSemiring.toCommMonoidWithZero.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (IdemCommSemiring.toCommSemiring.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Ideal.instIdemCommSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (Ideal.uniqueFactorizationMonoid.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.17) I)) x) p) (UniqueFactorizationMonoid.factors.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (CommSemiring.toCommMonoidWithZero.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (IdemCommSemiring.toCommSemiring.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3))) (Ideal.instIdemCommSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3)))) (Ideal.uniqueFactorizationMonoid.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Module.DedekindDomain.2481555069._hygCtx._hyg.17) I))))))","typeFull":"∀ {R : Type u} [inst : CommRing R] [IsDomain R] {M : Type v} [inst_2 : AddCommGroup M] [inst_3 : Module R M]\n  [inst_4 : IsDedekindDomain R] {I : Ideal R},\n  I ≠ ⊥ →\n    Module.IsTorsionBySet R M ↑I →\n      DirectSum.IsInternal fun p =>\n        Submodule.torsionBySet R M ↑(↑p ^ Multiset.count (↑p) (UniqueFactorizationMonoid.factors I))","typeReadable":"∀ {R : Type u} [inst : CommRing R] [IsDomain R] {M : Type v} [inst_2 : AddCommGroup M] [inst_3 : Module R M]\n  [inst_4 : IsDedekindDomain R] {I : Ideal R},\n  I ≠ ⊥ →\n    Module.IsTorsionBySet R M ↑I →\n      DirectSum.IsInternal fun p =>\n        Submodule.torsionBySet R M ↑(↑p ^ Multiset.count (↑p) (UniqueFactorizationMonoid.factors I))","typeReferences":[["UniqueFactorizationMonoid","factors"],["DirectSum","IsInternal"],["Finset","instSetLike"],["Multiset","count"],["Subtype"],["Finset"],["Module"],["Module","IsTorsionBySet"],["Membership","mem"],["IsDomain"],["Submodule","instPowNat"],["CommRing","toNonUnitalCommRing"],["Subtype","val"],["Algebra","id"],["Submodule"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["Submodule","torsionBySet"],["Ideal"],["Ideal","instIdemCommSemiring"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["IsScalarTower","right"],["IdemCommSemiring","toCommSemiring"],["Semiring","toModule"],["Submodule","addSubmonoidClass"],["instHPow"],["CommRing","toCommSemiring"],["SetLike","instMembership"],["Ideal","uniqueFactorizationMonoid"],["CommSemiring","toCommMonoidWithZero"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["IsDedekindDomain"],["CommSemiring","toSemiring"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["AddCommGroup"],["Multiset","toFinset"],["HPow","hPow"],["CommRing"],["Submodule","decidableEq"],["Bot","bot"],["Nat"],["SetLike","coe"],["Submodule","setLike"],["Submodule","instBot"],["AddCommGroup","toAddCommMonoid"],["Subtype","instDecidableEq"],["Ne"]],"valueReferences":[["UniqueFactorizationMonoid","factors"],["Multiset","count"],["Finset"],["MonoidWithZeroHom","funLike"],["Eq","trans"],["MonoidWithZero","toMulZeroOneClass"],["Ideal","instNoZeroDivisors"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Eq","symm"],["IsScalarTower","right"],["Eq","ndrec"],["Units"],["pow_ne_zero"],["IdemSemiring","toSemiring"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["Submodule","decidableEq"],["CommSemiring","toCommMonoid"],["Ideal","normalizationMonoid"],["SetLike","coe"],["Eq","refl"],["Submodule","instBot"],["HEq"],["AddCommGroup","toAddCommMonoid"],["Top","top"],["Eq","mpr"],["CompleteLattice","toBoundedOrder"],["pow_zero"],["AddMonoid","toAddZeroClass"],["MulOneClass","toMulOne"],["Submodule","torsionBySet_isInternal"],["IsDomain","to_noZeroDivisors"],["Submodule","idemSemiring"],["zero_min"],["Submodule","instPowNat"],["Multiset","replicate"],["Finset","prod_multiset_count"],["UniqueFactorizationMonoid","normalizedFactors"],["Ideal","uniqueUnits"],["isReduced_of_noZeroDivisors"],["instOfNatNat"],["iInf"],["Preorder","toLE"],["Eq"],["IdemCommSemiring","toCommSemiring"],["CompleteLattice","toConditionallyCompleteLattice"],["propext"],["Submodule","instOrderTop"],["associated_iff_eq"],["Ideal","one_eq_top"],["Nat","instAddMonoid"],["Set"],["Submodule","instTop"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Multiset","prod"],["HPow","hPow"],["OfNat","ofNat"],["Max","max"],["Associated"],["Prime","ne_zero"],["Submodule","completeLattice"],["Ne"],["CompleteLattice","toCompleteSemilatticeInf"],["PartialOrder","toPreorder"],["instMinNat"],["CommMonoidWithZero","toMonoidWithZero"],["Module","IsTorsionBySet"],["Membership","mem"],["Iff","mp"],["IdemSemiring","toSemilatticeSup"],["Algebra","id"],["irreducible_pow_sup"],["Finset","inf"],["Semiring","toNonAssocSemiring"],["Monoid","toPow"],["Ideal","instIdemCommSemiring"],["MonoidWithZeroHom"],["eq_of_heq"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Unique","instSubsingleton"],["Semiring","toModule"],["Ideal","isCancelMulZero"],["Nat","instLinearOrder"],["SemilatticeInf","toPartialOrder"],["UniqueFactorizationMonoid","normalizedFactors_of_irreducible_pow"],["instHPow"],["normalize_eq"],["SetLike","instMembership"],["MulOne","toOne"],["CompleteLattice","toLattice"],["CommSemiring","toCommMonoidWithZero"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["And","right"],["And"],["Multiset","mem_replicate"],["AddZeroClass","toAddZero"],["Bot","bot"],["Nat","instCanonicallyOrderedAdd"],["UniqueFactorizationMonoid","prime_of_factor"],["Nat"],["Multiset","count_eq_zero"],["id"],["ConditionallyCompleteLattice","toLattice"],["Prime","irreducible"],["AddZero","toZero"],["Submodule","one"],["Multiset","instMembership"],["Finset","instSetLike"],["SemilatticeInf","toMin"],["CommMonoid","toMonoid"],["Submodule","instInfSet"],["CommRing","toNonUnitalCommRing"],["DFunLike","coe"],["congrArg"],["instDistribLatticeOfLinearOrder"],["Ideal"],["BoundedOrder","toOrderTop"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["Zero","toOfNat0"],["Not"],["normalize"],["CommRing","toCommSemiring"],["Lattice","toSemilatticeInf"],["HEq","refl"],["Ideal","uniqueFactorizationMonoid"],["IsDedekindDomain","inf_pow_eq_prod_of_prime"],["CommSemiring","toSemiring"],["UniqueFactorizationMonoid","factors_prod"],["Semiring","toMonoidWithZero"],["Submodule","pointwiseZero"],["Multiset","toFinset"],["Eq","casesOn"],["Multiset"],["CompleteSemilatticeInf","toInfSet"],["Finset","prod"],["DistribLattice","toLattice"],["One","toOfNat1"],["Submodule","setLike"],["Min","min"],["SemilatticeSup","toMax"],["Multiset","mem_toFinset"],["Finset","inf_eq_iInf"],["Ideal","zero_eq_bot"]]},{"isProp":true,"kind":"theorem","name":["Submodule","exists_isInternal_prime_power_torsion"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3 : CommRing.{u} R] [inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.6 : IsDomain.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))] {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.10 : AddCommGroup.{v} M] [inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.13 : Module.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.10)] [inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.17 : IsDedekindDomain.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3] [inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.20 : Module.Finite.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.13], (Module.IsTorsion.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.13) -> (Exists.{succ u} (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)))) (fun (P : Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)))) => Exists.{succ u} (DecidableEq.{succ u} (Subtype.{succ u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (fun (x : Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) => Membership.mem.{u, u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)))) (SetLike.instMembership.{u, u} (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)))) (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (Finset.instSetLike.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))))) P x))) (fun (x._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.44 : DecidableEq.{succ u} (Subtype.{succ u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (fun (x : Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) => Membership.mem.{u, u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)))) (SetLike.instMembership.{u, u} (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)))) (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (Finset.instSetLike.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))))) P x))) => Exists.{0} (forall (p : Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))), (Membership.mem.{u, u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)))) (SetLike.instMembership.{u, u} (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)))) (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (Finset.instSetLike.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))))) P p) -> (Prime.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (CommSemiring.toCommMonoidWithZero.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (IdemCommSemiring.toCommSemiring.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (Ideal.instIdemCommSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)))) p)) (fun (x._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.52 : forall (p : Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))), (Membership.mem.{u, u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)))) (SetLike.instMembership.{u, u} (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)))) (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (Finset.instSetLike.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))))) P p) -> (Prime.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (CommSemiring.toCommMonoidWithZero.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (IdemCommSemiring.toCommSemiring.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (Ideal.instIdemCommSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)))) p)) => Exists.{succ u} ((Subtype.{succ u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (fun (x : Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) => Membership.mem.{u, u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)))) (SetLike.instMembership.{u, u} (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)))) (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (Finset.instSetLike.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))))) P x)) -> Nat) (fun (e : (Subtype.{succ u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (fun (x : Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) => Membership.mem.{u, u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)))) (SetLike.instMembership.{u, u} (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)))) (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (Finset.instSetLike.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))))) P x)) -> Nat) => DirectSum.IsInternal.{u, v, v} (Subtype.{succ u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (fun (x : Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) => Membership.mem.{u, u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)))) (SetLike.instMembership.{u, u} (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)))) (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (Finset.instSetLike.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))))) P x)) M (Submodule.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.13) x._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.44 (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.10) (Submodule.setLike.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.13) (Submodule.addSubmonoidClass.{u, v} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.13) (fun (p : Subtype.{succ u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (fun (x : Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) => Membership.mem.{u, u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)))) (SetLike.instMembership.{u, u} (Finset.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)))) (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (Finset.instSetLike.{u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))))) P x)) => Submodule.torsionBySet.{u, v} R M (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.13 (SetLike.coe.{u, u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) R (Submodule.setLike.{u, u} R R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))))) (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)))) (HPow.hPow.{u, 0, u} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) Nat (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (instHPow.{u, 0} (Ideal.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) Nat (Submodule.instPowNat.{u, u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)) R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)) (Semiring.toModule.{u} R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3))) (IsScalarTower.right.{u, u} R R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3) (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Module.DedekindDomain.1684941212._hygCtx._hyg.3)) (Algebra.id.{u} 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.Opposite.sym.json

@@ -0,0 +1 @@
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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Opposite.922663735._hygCtx._hyg.4 : Semiring.{u_1} R] (m : R), Eq.{succ u_1} R (HSMul.hSMul.{u_1, u_1, u_1} (MulOpposite.{u_1} R) R R (instHSMul.{u_1, u_1} (MulOpposite.{u_1} R) R (SemigroupAction.toSMul.{u_1, u_1} (MulOpposite.{u_1} R) R (Monoid.toSemigroup.{u_1} (MulOpposite.{u_1} R) (MonoidWithZero.toMonoid.{u_1} (MulOpposite.{u_1} R) (MulOpposite.instMonoidWithZero.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Opposite.922663735._hygCtx._hyg.4)))) (MulAction.toSemigroupAction.{u_1, u_1} (MulOpposite.{u_1} R) R (MonoidWithZero.toMonoid.{u_1} (MulOpposite.{u_1} R) (MulOpposite.instMonoidWithZero.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Opposite.922663735._hygCtx._hyg.4))) (MulActionWithZero.toMulAction.{u_1, u_1} (MulOpposite.{u_1} R) R (MulOpposite.instMonoidWithZero.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Opposite.922663735._hygCtx._hyg.4)) 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inst._@.Mathlib.Algebra.Module.Opposite.922663735._hygCtx._hyg.4))))))","typeFull":"∀ {R : Type u_1} [inst : Semiring R] (m : R), 0 • m = 0","typeReadable":"∀ {R : Type u_1} [inst : Semiring R] (m : R), 0 • m = 0","typeReferences":[["MonoidWithZero","toOppositeMulActionWithZero"],["SemigroupAction","toSMul"],["Semiring","toMonoidWithZero"],["MonoidWithZero","toMulZeroOneClass"],["OfNat","ofNat"],["MulOpposite","instMonoidWithZero"],["MulOpposite"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["HSMul","hSMul"],["MonoidWithZero","toMonoid"],["instHSMul"],["Zero","toOfNat0"],["Monoid","toSemigroup"],["Eq"],["MulAction","toSemigroupAction"],["Semiring"],["MulActionWithZero","toMulAction"]],"valueReferences":[["MulOpposite","instMonoidWithZero"],["MulOpposite"],["MonoidWithZero","toOppositeMulActionWithZero"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["MulActionWithZero","zero_smul"],["Semiring","toMonoidWithZero"],["MonoidWithZero","toMulZeroOneClass"]]},{"isProp":true,"kind":"theorem","name":["MulOpposite","instModule","_proof_1"],"typeFallback":"forall (R : Type.{u_2}) {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Opposite.955123615._hygCtx._hyg.4 : 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.Torsion.Prod.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Prod","moduleIsTorsionFree"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} {N : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Torsion.Prod.498984483._hygCtx._hyg.5 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Torsion.Prod.498984483._hygCtx._hyg.8 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Torsion.Prod.498984483._hygCtx._hyg.11 : AddCommMonoid.{u_3} N] [inst._@.Mathlib.Algebra.Module.Torsion.Prod.498984483._hygCtx._hyg.14 : Module.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Torsion.Prod.498984483._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Torsion.Prod.498984483._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Torsion.Prod.498984483._hygCtx._hyg.18 : Module.{u_1, u_3} R N inst._@.Mathlib.Algebra.Module.Torsion.Prod.498984483._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Torsion.Prod.498984483._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.Torsion.Prod.498984483._hygCtx._hyg.22 : Module.IsTorsionFree.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Torsion.Prod.498984483._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Torsion.Prod.498984483._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Torsion.Prod.498984483._hygCtx._hyg.14] [inst._@.Mathlib.Algebra.Module.Torsion.Prod.498984483._hygCtx._hyg.26 : Module.IsTorsionFree.{u_1, u_3} R N inst._@.Mathlib.Algebra.Module.Torsion.Prod.498984483._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Torsion.Prod.498984483._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Torsion.Prod.498984483._hygCtx._hyg.18], Module.IsTorsionFree.{u_1, max u_3 u_2} R (Prod.{u_2, u_3} M N) inst._@.Mathlib.Algebra.Module.Torsion.Prod.498984483._hygCtx._hyg.5 (Prod.instAddCommMonoid.{u_2, u_3} M N inst._@.Mathlib.Algebra.Module.Torsion.Prod.498984483._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Torsion.Prod.498984483._hygCtx._hyg.11) (Prod.instModule.{u_1, u_2, u_3} R M N inst._@.Mathlib.Algebra.Module.Torsion.Prod.498984483._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Torsion.Prod.498984483._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Torsion.Prod.498984483._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Torsion.Prod.498984483._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Torsion.Prod.498984483._hygCtx._hyg.18)","typeFull":"∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid N]\n  [inst_3 : Module R M] [inst_4 : Module R N] [Module.IsTorsionFree R M] [Module.IsTorsionFree R N],\n  Module.IsTorsionFree R (M × N)","typeReadable":"∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid N]\n  [inst_3 : Module R M] [inst_4 : Module R N] [Module.IsTorsionFree R M] [Module.IsTorsionFree R N],\n  Module.IsTorsionFree R (M × N)","typeReferences":[["Prod"],["Module","IsTorsionFree"],["AddCommMonoid"],["Prod","instModule"],["Module"],["Prod","instAddCommMonoid"],["Semiring"]],"valueReferences":[["Function","Injective","prodMap"],["DistribMulAction","toDistribSMul"],["SemigroupAction","toSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Prod","instAddCommMonoid"],["Module","IsTorsionFree","mk"],["Prod"],["IsRegular","isSMulRegular"],["Module","toDistribMulAction"],["Prod","instModule"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["DistribMulAction","toMulAction"],["instHSMul"],["Monoid","toSemigroup"],["MulAction","toSemigroupAction"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.MonoidAlgebra.Ideal.sym.json

@@ -0,0 +1 @@
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inst._@.Mathlib.Algebra.MonoidAlgebra.Ideal.1429574442._hygCtx._hyg.8 inst._@.Mathlib.Algebra.MonoidAlgebra.Ideal.1429574442._hygCtx._hyg.5) (Set.image.{u_2, max u_1 u_2} A (AddMonoidAlgebra.{u_1, u_2} k A inst._@.Mathlib.Algebra.MonoidAlgebra.Ideal.1429574442._hygCtx._hyg.8) (AddMonoidAlgebra.of'.{u_1, u_2} k A inst._@.Mathlib.Algebra.MonoidAlgebra.Ideal.1429574442._hygCtx._hyg.8) s)) x) (forall (m : A), (Membership.mem.{u_2, u_2} A (Finset.{u_2} A) (SetLike.instMembership.{u_2, u_2} (Finset.{u_2} A) A (Finset.instSetLike.{u_2} A)) (Finsupp.support.{u_2, u_1} A k (MulZeroClass.toZero.{u_1} k (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} k (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} k (Semiring.toNonAssocSemiring.{u_1} k inst._@.Mathlib.Algebra.MonoidAlgebra.Ideal.1429574442._hygCtx._hyg.8)))) x) m) -> (Exists.{succ u_2} A (fun (m' : A) => And (Membership.mem.{u_2, u_2} A (Set.{u_2} A) (Set.instMembership.{u_2} A) s m') (Exists.{succ u_2} A (fun (d : A) => Eq.{succ u_2} A m 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m'","typeReferences":[["Finset","instSetLike"],["Ideal","span"],["Finset"],["Membership","mem"],["Finsupp","support"],["Semiring","toNonAssocSemiring"],["Ideal"],["AddMonoidAlgebra","semiring"],["Eq"],["Semiring","toModule"],["AddSemigroup","toAdd"],["SetLike","instMembership"],["Exists"],["instHAdd"],["Set"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["And"],["AddMonoid"],["AddMonoidAlgebra","of'"],["Set","instMembership"],["HAdd","hAdd"],["Set","image"],["Submodule","setLike"],["MulZeroClass","toZero"],["AddMonoid","toAddSemigroup"],["Iff"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddMonoidAlgebra"],["Semiring"]],"valueReferences":[["MonoidAlgebra","mem_ideal_span_of_image"],["Multiplicative","monoid"],["Multiplicative"]]},{"isProp":true,"kind":"theorem","name":["MonoidAlgebra","mem_ideal_span_of_image"],"typeFallback":"forall {k : Type.{u_1}} {G : Type.{u_3}} 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Order.Field.Pi.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Order.Group.Int.Sum.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Order.Group.Opposite.sym.json

@@ -0,0 +1 @@
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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Order.GroupWithZero.Bounds.sym.json

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