diff --git "a/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.DirectSum.Module.sym.json" "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.DirectSum.Module.sym.json"
new file mode 100644--- /dev/null
+++ "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.DirectSum.Module.sym.json"
@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["DirectSum","lid","_proof_1"],"typeFallback":"forall (M : Type.{u_2}) (ι : Type.{u_1}) [inst._@.Mathlib.Algebra.DirectSum.Module.4176511332._hygCtx._hyg.33 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.DirectSum.Module.4176511332._hygCtx._hyg.40 : Unique.{succ u_1} ι] (x : DirectSum.{u_1, u_2} ι (fun (x._@.Mathlib.Algebra.DirectSum.Basic.1952539257._hygCtx._hyg.43 : ι) => M) (fun (i : ι) => inst._@.Mathlib.Algebra.DirectSum.Module.4176511332._hygCtx._hyg.33)) (y : DirectSum.{u_1, u_2} ι (fun (x._@.Mathlib.Algebra.DirectSum.Basic.1952539257._hygCtx._hyg.43 : ι) => M) (fun (i : ι) => inst._@.Mathlib.Algebra.DirectSum.Module.4176511332._hygCtx._hyg.33)), Eq.{succ u_2} M (Equiv.toFun.{succ (max u_2 u_1), succ u_2} (DirectSum.{u_1, u_2} ι (fun (x._@.Mathlib.Algebra.DirectSum.Basic.1952539257._hygCtx._hyg.43 : ι) => M) (fun (i : ι) => inst._@.Mathlib.Algebra.DirectSum.Module.4176511332._hygCtx._hyg.33)) M (AddEquiv.toEquiv.{max u_2 u_1, u_2} 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: ι) => inst._@.Mathlib.Algebra.DirectSum.Module.4176511332._hygCtx._hyg.33)) (instAddCommMonoidDirectSum.{u_1, u_2} ι (fun (x._@.Mathlib.Algebra.DirectSum.Basic.1952539257._hygCtx._hyg.43 : ι) => M) (fun (i : ι) => inst._@.Mathlib.Algebra.DirectSum.Module.4176511332._hygCtx._hyg.33)))))) x y)) (HAdd.hAdd.{u_2, u_2, u_2} M M M (instHAdd.{u_2} M (AddCommMagma.toAdd.{u_2} M (AddCommSemigroup.toAddCommMagma.{u_2} M (AddCommMonoid.toAddCommSemigroup.{u_2} M inst._@.Mathlib.Algebra.DirectSum.Module.4176511332._hygCtx._hyg.33)))) (Equiv.toFun.{succ (max u_2 u_1), succ u_2} (DirectSum.{u_1, u_2} ι (fun (x._@.Mathlib.Algebra.DirectSum.Basic.1952539257._hygCtx._hyg.43 : ι) => M) (fun (i : ι) => inst._@.Mathlib.Algebra.DirectSum.Module.4176511332._hygCtx._hyg.33)) M (AddEquiv.toEquiv.{max u_2 u_1, u_2} (DirectSum.{u_1, u_2} ι (fun (x._@.Mathlib.Algebra.DirectSum.Basic.1952539257._hygCtx._hyg.43 : ι) => M) (fun (i : ι) => inst._@.Mathlib.Algebra.DirectSum.Module.4176511332._hygCtx._hyg.33)) M 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(AddCommMonoid.toAddCommSemigroup.{u_2} M inst._@.Mathlib.Algebra.DirectSum.Module.4176511332._hygCtx._hyg.33))) (DirectSum.id.{u_2, u_1} M ι inst._@.Mathlib.Algebra.DirectSum.Module.4176511332._hygCtx._hyg.33 inst._@.Mathlib.Algebra.DirectSum.Module.4176511332._hygCtx._hyg.40)) x) (Equiv.toFun.{succ (max u_2 u_1), succ u_2} (DirectSum.{u_1, u_2} ι (fun (x._@.Mathlib.Algebra.DirectSum.Basic.1952539257._hygCtx._hyg.43 : ι) => M) (fun (i : ι) => inst._@.Mathlib.Algebra.DirectSum.Module.4176511332._hygCtx._hyg.33)) M (AddEquiv.toEquiv.{max u_2 u_1, u_2} (DirectSum.{u_1, u_2} ι (fun (x._@.Mathlib.Algebra.DirectSum.Basic.1952539257._hygCtx._hyg.43 : ι) => M) (fun (i : ι) => inst._@.Mathlib.Algebra.DirectSum.Module.4176511332._hygCtx._hyg.33)) M (AddCommMagma.toAdd.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (x._@.Mathlib.Algebra.DirectSum.Basic.1952539257._hygCtx._hyg.43 : ι) => M) (fun (i : ι) => inst._@.Mathlib.Algebra.DirectSum.Module.4176511332._hygCtx._hyg.33)) 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inst._@.Mathlib.Algebra.DirectSum.Module.4176511332._hygCtx._hyg.40)) y))","typeFull":"∀ (M : Type u_2) (ι : Type u_1) [inst : AddCommMonoid M] [inst_1 : Unique ι] (x y : DirectSum ι fun x => M),\n  (DirectSum.id M ι).toFun (x + y) = (DirectSum.id M ι).toFun x + (DirectSum.id M ι).toFun y","typeReadable":"∀ (M : Type u_2) (ι : Type u_1) [inst : AddCommMonoid M] [inst_1 : Unique ι] (x y : DirectSum ι fun x => M),\n  (DirectSum.id M ι).toFun (x + y) = (DirectSum.id M ι).toFun x + (DirectSum.id M ι).toFun y","typeReferences":[["Unique"],["instHAdd"],["AddEquiv","toEquiv"],["DirectSum"],["HAdd","hAdd"],["AddCommMonoid"],["AddCommMonoid","toAddCommSemigroup"],["instAddCommMonoidDirectSum"],["DirectSum","id"],["Equiv","toFun"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Eq"]],"valueReferences":[["DirectSum"],["DirectSum","id"],["instAddCommMonoidDirectSum"],["AddCommMonoid","toAddCommSemigroup"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["AddEquiv","map_add'"]]},{"isProp":true,"kind":"theorem","name":["DirectSum","mker_map"],"typeFallback":"forall {ι : Type.{v}} {M : ι -> Type.{w}} [inst._@.Mathlib.Algebra.DirectSum.Module.1126005069._hygCtx._hyg.10 : forall (i : ι), AddCommMonoid.{w} (M i)] {N : ι -> Type.{u_1}} [inst._@.Mathlib.Algebra.DirectSum.Module.1126005069._hygCtx._hyg.32 : forall (i : ι), AddCommMonoid.{u_1} (N i)] (f : forall (i : ι), AddMonoidHom.{w, u_1} (M i) (N i) (AddZeroClass.toAddZero.{w} (M i) (AddMonoid.toAddZeroClass.{w} (M i) 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inst._@.Mathlib.Algebra.DirectSum.Module.1126005069._hygCtx._hyg.10) (AddCommMonoid.toAddMonoid.{max v w} (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1126005069._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1126005069._hygCtx._hyg.10)))) (AddZeroClass.toAddZero.{max v u_1} (DirectSum.{v, u_1} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.1126005069._hygCtx._hyg.32) (AddMonoid.toAddZeroClass.{max v u_1} (DirectSum.{v, u_1} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.1126005069._hygCtx._hyg.32) (AddCommMonoid.toAddMonoid.{max v u_1} (DirectSum.{v, u_1} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.1126005069._hygCtx._hyg.32) (instAddCommMonoidDirectSum.{v, u_1} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.1126005069._hygCtx._hyg.32))))) (AddMonoidHom.instAddMonoidHomClass.{max v w, max u_1 v} (DirectSum.{v, 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inst._@.Mathlib.Algebra.DirectSum.Module.1364628504._hygCtx._hyg.3))))))))","typeFull":"∀ {R : Type u} [inst : Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] {A : ι → Submodule R M} (h : DirectSum.IsInternal A) {α : ι → Type u_2}\n  (v : (i : ι) → Module.Basis (α i) R ↥(A i)) {x : M} {i j : ι},\n  i ≠ j → ∀ {a : α j}, x ∈ A i → ((h.collectedBasis v).repr x) ⟨j, a⟩ = 0","typeReadable":"∀ {R : Type u} [inst : Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] {A : ι → Submodule R M} (h : DirectSum.IsInternal A) {α : ι → Type u_2}\n  (v : (i : ι) → Module.Basis (α i) R ↥(A i)) {x : M} {i j : ι},\n  i ≠ j → ∀ {a : α j}, x ∈ A i → ((h.collectedBasis v).repr x) ⟨j, a⟩ = 0","typeReferences":[["DFinsupp","instEquivLikeLinearEquiv"],["DirectSum","IsInternal"],["Subtype"],["Module"],["DecidableEq"],["Submodule","module"],["Membership","mem"],["LinearEquiv"],["DFunLike","coe"],["Submodule"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Sigma","mk"],["EquivLike","toFunLike"],["Sigma"],["Zero","toOfNat0"],["Module","Basis"],["Semiring","toModule"],["Eq"],["Submodule","addSubmonoidClass"],["Finsupp","module"],["Finsupp","instFunLike"],["SetLike","instMembership"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DirectSum","IsInternal","collectedBasis"],["OfNat","ofNat"],["Submodule","setLike"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["RingHomInvPair","ids"],["Module","Basis","repr"],["Ne"],["Finsupp","instAddCommMonoid"],["Submodule","addCommMonoid"],["Finsupp"],["Semiring"]],"valueReferences":[["RingHom"],["Eq","trans"],["Membership","mem"],["DistribMulActionSemiHomClass","toAddMonoidHomClass"],["Semiring","toNonAssocSemiring"],["instAddCommMonoidDirectSum"],["RingHom","id"],["Sigma"],["Semiring","toModule"],["Finsupp","instFunLike"],["SetLike","instMembership"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["LinearMap"],["AddZeroClass","toAddZero"],["DirectSum","coeLinearMap"],["DFinsupp","module"],["LinearEquiv","instSemilinearEquivClass"],["id"],["RingHomInvPair","ids"],["DirectSum","instModule"],["Module","Basis","repr"],["Finsupp","instAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["DFinsupp","instEquivLikeLinearEquiv"],["Subtype"],["LinearMap","instFunLike"],["DFinsupp"],["DFinsupp","addCommMonoid"],["Submodule","module"],["RingHom","instFunLike"],["AddCommMonoid","toAddMonoid"],["LinearEquiv"],["Sigma","fst"],["DFunLike","coe"],["congrArg"],["Submodule"],["DirectSum"],["SemilinearEquivClass","instSemilinearMapClass"],["Sigma","mk"],["EquivLike","toFunLike"],["MonoidWithZero","toMonoid"],["congrFun'"],["Zero","toOfNat0"],["DFinsupp","mapRange"],["LinearEquiv","instEquivLike"],["Eq"],["map_zero"],["Finsupp","module"],["True"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["AddMonoidHomClass","toZeroHomClass"],["Finsupp","instZero"],["Semiring","toMonoidWithZero"],["LinearEquiv","toLinearMap"],["DirectSum","IsInternal","collectedBasis"],["LinearEquiv","symm"],["OfNat","ofNat"],["DirectSum","IsInternal","ofBijective_coeLinearMap_of_mem_ne"],["eq_self"],["Module","toDistribMulAction"],["of_eq_true"],["MulZeroClass","toZero"],["Submodule","setLike"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Submodule","zero"],["DFinsupp","instDFunLike"],["sigmaFinsuppLequivDFinsupp"],["Submodule","addCommMonoid"],["SemilinearMapClass","distribMulActionSemiHomClass"],["Finsupp"],["LinearEquiv","ofBijective"]]},{"isProp":true,"kind":"theorem","name":["DirectSum","instModule","_proof_7"],"typeFallback":"forall {R : Type.{u_3}} [inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3 : Semiring.{u_3} R] {ι : Type.{u_1}} {M : ι -> Type.{u_2}} [inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10 : forall (i : ι), AddCommMonoid.{u_2} (M i)] [inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.19 : forall (i : ι), Module.{u_3, u_2} R (M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3 (inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10 i)] (r : R) (s : R) (x : DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10), Eq.{succ (max u_1 u_2)} (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) (HSMul.hSMul.{u_3, max u_1 u_2, max u_1 u_2} R (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) 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(MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10)) (DistribMulAction.mk.{u_3, max u_1 u_2} R (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10)) (MulAction.mk.{u_3, max u_1 u_2} R (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3)) (SemigroupAction.mk.{u_3, max u_1 u_2} R (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) (Monoid.toSemigroup.{u_3} R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3))) (SMul.mk.{u_3, max u_1 u_2} R (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) (DirectSum.instModule._aux_1.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3 ι M inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.19)) (DirectSum.instModule._proof_3.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3 ι M inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.19)) (DirectSum.instModule._proof_4.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3 ι M inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.19)) (DirectSum.instModule._proof_5.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3 ι M inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.19) (DirectSum.instModule._proof_6.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3 ι M inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.19)))))) (HAdd.hAdd.{u_3, u_3, u_3} R R R (instHAdd.{u_3} R (Distrib.toAdd.{u_3} R (NonUnitalNonAssocSemiring.toDistrib.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3))))) r s) x) (HAdd.hAdd.{max u_1 u_2, max u_1 u_2, max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) (instHAdd.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) (AddCommMagma.toAdd.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) 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inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3)) (SemigroupAction.mk.{u_3, max u_1 u_2} R (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) (Monoid.toSemigroup.{u_3} R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3))) (SMul.mk.{u_3, max u_1 u_2} R (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) (DirectSum.instModule._aux_1.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3 ι M inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.19)) (DirectSum.instModule._proof_3.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3 ι M inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.19)) (DirectSum.instModule._proof_4.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3 ι M inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.19)) (DirectSum.instModule._proof_5.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3 ι M inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.19) (DirectSum.instModule._proof_6.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3 ι M inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.19)))))) r x) (HSMul.hSMul.{u_3, max u_1 u_2, max u_1 u_2} R (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) 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inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.19)) (DirectSum.instModule._proof_3.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3 ι M inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.19)) (DirectSum.instModule._proof_4.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3 ι M inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.19)) (DirectSum.instModule._proof_5.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3 ι M inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.19) (DirectSum.instModule._proof_6.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3 ι M inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.19)))))) s x))","typeFull":"∀ {R : Type u_3} [inst : Semiring R] {ι : Type u_1} {M : ι → Type u_2} [inst_1 : (i : ι) → AddCommMonoid (M i)]\n  [inst_2 : (i : ι) → Module R (M i)] (r s : R) (x : DirectSum ι fun i => M i), (r + s) • x = r • x + s • x","typeReadable":"∀ {R : Type u_3} [inst : Semiring R] {ι : Type u_1} {M : ι → Type u_2} [inst_1 : (i : ι) → AddCommMonoid (M i)]\n  [inst_2 : (i : ι) → Module R (M i)] (r s : R) (x : DirectSum ι fun i => M i), (r + s) • x = r • x + s • 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dec_ι inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) (Submodule.addSubmonoidClass.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) A) {i : ι} {j : ι}, (Ne.{succ v} ι i j) -> (forall {x : M}, (Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13)) (A i) x) -> (Eq.{succ u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13)) (A j) x)) (DFunLike.coe.{max (succ v) (succ u_1), succ v, succ u_1} (DFinsupp.{v, u_1} ι (fun (i : ι) => (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13)) (A i) x)) i) (fun (i : ι) => AddZero.toZero.{u_1} ((fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13)) 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inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13)) (A i) x)) i) ((fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13 (A i)) i)))))) ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13)) (A i) x)) (DFinsupp.instDFunLike.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 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(Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13 (A i))) inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 (instAddCommMonoidDirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13 (A i))) inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13 (DirectSum.instModule.{u, v, u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13 (A i)) (fun (i : ι) => Submodule.module.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13 (A i))))) (LinearEquiv.symm.{u, u, max u_1 v, u_1} R R (DirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13 (A i))) M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 (instAddCommMonoidDirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13 (A i))) inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 (DirectSum.instModule.{u, v, u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13 (A i)) (fun (i : ι) => Submodule.module.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13 (A i))) inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3)) (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3)) (RingHomInvPair.ids.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3) (RingHomInvPair.ids.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3) (LinearEquiv.ofBijective.{u, u, max u_1 v, u_1} R R (DirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13 (A i))) M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 (instAddCommMonoidDirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13 (A i))) inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 (DirectSum.instModule.{u, v, u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13 (A i)) (fun (i : ι) => Submodule.module.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13 (A i))) inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3)) (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3)) (DirectSum.coeLinearMap.{u, v, u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 ι dec_ι M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13 A) (RingHomInvPair.ids.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3) (RingHomInvPair.ids.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3) h)) x) j) (OfNat.ofNat.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.13)) (A j) x)) 0 (Zero.toOfNat0.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2964844556._hygCtx._hyg.10 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inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13)) (A i) x)) (DirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13 (A i))) (Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13 (A i)) (instAddCommMonoidDirectSum.{v, 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inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13 (A i)) (fun (i : ι) => Submodule.module.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13 (A i)))) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13)) (A i) x)) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13)) (A i) x)) => DirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13 (A i))) (LinearMap.instFunLike.{u, u, u_1, max u_1 v} R R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13)) (A i) x)) (DirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13 (A i))) inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 (Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13 (A i)) (instAddCommMonoidDirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13 (A i))) (Submodule.module.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13 (A i)) (DirectSum.instModule.{u, v, u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13 (A i)) (fun (i : ι) => Submodule.module.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13 (A i))) (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3))) (DirectSum.lof.{u, v, u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13 (A i)) (fun (i : ι) => Submodule.module.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13 (A i)) dec_ι i) x)) (Subtype.val.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1567020609._hygCtx._hyg.13)) (A i) x) x)","typeFull":"∀ {R : Type u} [inst : Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] (A : ι → Submodule R M) (i : ι) (x : ↥(A i)),\n  (DirectSum.coeLinearMap A) ((DirectSum.lof R ι (fun i => ↥(A i)) i) x) = ↑x","typeReadable":"∀ {R : Type u} [inst : Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] (A : ι → Submodule R M) (i : ι) (x : ↥(A i)),\n  (DirectSum.coeLinearMap A) ((DirectSum.lof R ι (fun i => ↥(A i)) i) x) = ↑x","typeReferences":[["Subtype"],["SetLike","instMembership"],["LinearMap","instFunLike"],["Module"],["Submodule","module"],["DecidableEq"],["Membership","mem"],["LinearMap"],["Subtype","val"],["DFunLike","coe"],["DirectSum","coeLinearMap"],["Submodule"],["DirectSum"],["AddCommMonoid"],["DirectSum","lof"],["Semiring","toNonAssocSemiring"],["instAddCommMonoidDirectSum"],["Submodule","setLike"],["RingHom","id"],["DirectSum","instModule"],["Submodule","addCommMonoid"],["Eq"],["Semiring"]],"valueReferences":[["DirectSum","coeLinearMap_of"]]},{"isProp":true,"kind":"theorem","name":["DirectSum","sigmaLcurry","_proof_1"],"typeFallback":"forall (R : Type.{u_4}) [inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.3 : Semiring.{u_4} R] {ι : Type.{u_2}} {α : ι -> Type.{u_1}} {δ : forall (i : ι), (α i) -> Type.{u_3}} [inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.39 : DecidableEq.{succ u_2} ι] [inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 : forall (i : ι) (j : α i), AddCommMonoid.{u_3} (δ i j)] [inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.53 : forall (i : ι) (j : α i), Module.{u_4, u_3} R (δ i j) inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.3 (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i j)] (r : R) (x : DirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => δ (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun 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(_i : ι) => α _i) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_2, u_1} ι (fun (_i : ι) => α _i) i))) (AddCommMonoid.toAddMonoid.{max (max u_1 u_2) u_3} (DirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) (fun (i : Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) => δ (Sigma.fst.{u_2, u_1} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_2, u_1} ι (fun (_i : ι) => α _i) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_2, u_1} ι (fun (_i : ι) => α _i) i))) (instAddCommMonoidDirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) (fun (i : Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) => δ (Sigma.fst.{u_2, u_1} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_2, u_1} ι (fun (_i : ι) => α _i) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_2, u_1} ι (fun (_i : ι) => α _i) i)))))) (AddZeroClass.toAddZero.{max (max u_1 u_2) u_3} (DirectSum.{u_2, max u_3 u_1} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))) (AddMonoid.toAddZeroClass.{max (max u_1 u_2) u_3} (DirectSum.{u_2, max u_3 u_1} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))) (AddCommMonoid.toAddMonoid.{max (max u_1 u_2) u_3} (DirectSum.{u_2, max u_3 u_1} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))) (instAddCommMonoidDirectSum.{u_2, max u_1 u_3} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)))))) (DirectSum.sigmaCurry.{u_1, u_2, u_3} ι inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.39 α δ inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42)) (HSMul.hSMul.{u_4, max (max u_1 u_2) u_3, max (max u_1 u_2) u_3} R (DirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => δ (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i))) (DirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => δ (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i))) (instHSMul.{u_4, max (max u_1 u_2) u_3} R (DirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => δ (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i))) (SMulZeroClass.toSMul.{u_4, max (max u_1 u_2) u_3} R (DirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => δ (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i))) (AddZero.toZero.{max (max u_1 u_2) u_3} (DirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => δ (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i))) (AddZeroClass.toAddZero.{max (max u_1 u_2) u_3} (DirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => δ (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i))) (AddMonoid.toAddZeroClass.{max (max u_1 u_2) u_3} (DirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => δ (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i))) (AddCommMonoid.toAddMonoid.{max (max u_1 u_2) u_3} (DirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => δ (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i))) (instAddCommMonoidDirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => δ (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i))))))) (DistribSMul.toSMulZeroClass.{u_4, max (max u_1 u_2) u_3} R (DirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => δ (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i))) (AddMonoid.toAddZeroClass.{max (max u_1 u_2) u_3} (DirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => δ (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i))) (AddCommMonoid.toAddMonoid.{max (max u_1 u_2) u_3} (DirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => δ (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i))) (instAddCommMonoidDirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => δ (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i))))) (DistribMulAction.toDistribSMul.{u_4, max (max u_1 u_2) u_3} R (DirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => δ (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i))) (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{max (max u_1 u_2) u_3} (DirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => δ (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i))) (instAddCommMonoidDirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => δ (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)))) (Module.toDistribMulAction.{u_4, max (max u_1 u_2) u_3} R (DirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => δ (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i))) inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.3 (instAddCommMonoidDirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => δ (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i))) (DirectSum.instModule.{u_4, max u_1 u_2, u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.3 (Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => δ (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.53 (Sigma.fst.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{u_2, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)))))))) r x)) (HSMul.hSMul.{u_4, max (max u_1 u_2) u_3, max (max u_1 u_2) u_3} R (DirectSum.{u_2, max u_3 u_1} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))) (DirectSum.{u_2, max u_3 u_1} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))) (instHSMul.{u_4, max (max u_1 u_2) u_3} R (DirectSum.{u_2, max u_3 u_1} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))) (SMulZeroClass.toSMul.{u_4, max (max u_1 u_2) u_3} R (DirectSum.{u_2, max u_3 u_1} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))) (AddZero.toZero.{max (max u_1 u_2) u_3} (DirectSum.{u_2, max u_3 u_1} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))) (AddZeroClass.toAddZero.{max (max u_1 u_2) u_3} (DirectSum.{u_2, max u_3 u_1} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))) (AddMonoid.toAddZeroClass.{max (max u_1 u_2) u_3} (DirectSum.{u_2, max u_3 u_1} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))) (AddCommMonoid.toAddMonoid.{max (max u_1 u_2) u_3} (DirectSum.{u_2, max u_3 u_1} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))) (instAddCommMonoidDirectSum.{u_2, max u_1 u_3} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))))))) (DistribSMul.toSMulZeroClass.{u_4, max (max u_1 u_2) u_3} R (DirectSum.{u_2, max u_3 u_1} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))) (AddMonoid.toAddZeroClass.{max (max u_1 u_2) u_3} (DirectSum.{u_2, max u_3 u_1} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))) (AddCommMonoid.toAddMonoid.{max (max u_1 u_2) u_3} (DirectSum.{u_2, max u_3 u_1} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))) (instAddCommMonoidDirectSum.{u_2, max u_1 u_3} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))))) (DistribMulAction.toDistribSMul.{u_4, max (max u_1 u_2) u_3} R (DirectSum.{u_2, max u_3 u_1} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))) (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{max (max u_1 u_2) u_3} (DirectSum.{u_2, max u_3 u_1} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))) (instAddCommMonoidDirectSum.{u_2, max u_1 u_3} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)))) (Module.toDistribMulAction.{u_4, max (max u_1 u_2) u_3} R (DirectSum.{u_2, max u_3 u_1} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))) inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.3 (instAddCommMonoidDirectSum.{u_2, max u_1 u_3} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))) (DirectSum.instModule.{u_4, u_2, max u_1 u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.3 ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => DirectSum.instModule.{u_4, u_1, u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.3 (α i) (δ i) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.53 i)))))))) (DFunLike.coe.{succ u_4, succ u_4, succ u_4} (RingHom.{u_4, u_4} R R (Semiring.toNonAssocSemiring.{u_4} R inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.3) (Semiring.toNonAssocSemiring.{u_4} R inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.3)) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => R) (RingHom.instFunLike.{u_4, u_4} R R (Semiring.toNonAssocSemiring.{u_4} R inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.3) (Semiring.toNonAssocSemiring.{u_4} R inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.3)) (RingHom.id.{u_4} R (Semiring.toNonAssocSemiring.{u_4} R inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.3)) r) (ZeroHom.toFun.{max u_3 u_1 u_2, max 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DecidableEq ι] [inst_2 : (i : ι) → (j : α i) → AddCommMonoid (δ i j)]\n  [inst_3 : (i : ι) → (j : α i) → Module R (δ i j)] (r : R) (x : DirectSum ((x : ι) × α x) fun i => δ i.fst i.snd),\n  (↑DirectSum.sigmaCurry).toFun (r • x) = (RingHom.id R) r • (↑DirectSum.sigmaCurry).toFun x","typeReadable":"∀ (R : Type u_4) [inst : Semiring R] {ι : Type u_2} {α : ι → Type u_1} {δ : (i : ι) → α i → Type u_3}\n  [inst_1 : DecidableEq ι] [inst_2 : (i : ι) → (j : α i) → AddCommMonoid (δ i j)]\n  [inst_3 : (i : ι) → (j : α i) → Module R (δ i j)] (r : R) (x : DirectSum ((x : ι) × α x) fun i => δ i.fst i.snd),\n  (↑DirectSum.sigmaCurry).toFun (r • x) = (RingHom.id R) r • (↑DirectSum.sigmaCurry).toFun 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{R : Type.{u}} [inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.3 : Ring.{u} R] {ι : Type.{v}} [dec_ι : DecidableEq.{succ v} ι] {M : Type.{u_1}} [inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.10 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.13 : Module.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.10)] (A : ι -> (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.13)), Iff (DirectSum.IsInternal.{v, u_1, u_1} ι M (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.3) 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R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [inst_1 : AddCommGroup M]\n  [inst_2 : Module R M] {A : ι → Submodule R M}, (DirectSum.IsInternal fun i => A ↑i) = DirectSum.IsInternal A","typeReadable":"∀ {R : Type u} [inst : Ring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [inst_1 : AddCommGroup M]\n  [inst_2 : Module R M] {A : ι → Submodule R M}, (DirectSum.IsInternal fun i => A ↑i) = DirectSum.IsInternal 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(AddEquiv.toEquiv.{max u_1 u_2, max u_1 u_2} (DirectSum.{u_2, u_1} (Option.{u_2} ι) (fun (i : Option.{u_2} ι) => α i) inst._@.Mathlib.Algebra.DirectSum.Module.2478368259._hygCtx._hyg.33) (Prod.{u_1, max u_1 u_2} (α (Option.none.{u_2} ι)) (DirectSum.{u_2, u_1} ι (fun (i : ι) => α (Option.some.{u_2} ι i)) (fun (i : ι) => inst._@.Mathlib.Algebra.DirectSum.Module.2478368259._hygCtx._hyg.33 (Option.some.{u_2} ι i)))) (AddCommMagma.toAdd.{max u_2 u_1} (DirectSum.{u_2, u_1} (Option.{u_2} ι) (fun (i : Option.{u_2} ι) => α i) inst._@.Mathlib.Algebra.DirectSum.Module.2478368259._hygCtx._hyg.33) (AddCommSemigroup.toAddCommMagma.{max u_2 u_1} (DirectSum.{u_2, u_1} (Option.{u_2} ι) (fun (i : Option.{u_2} ι) => α i) inst._@.Mathlib.Algebra.DirectSum.Module.2478368259._hygCtx._hyg.33) (AddCommMonoid.toAddCommSemigroup.{max u_2 u_1} (DirectSum.{u_2, u_1} (Option.{u_2} ι) (fun (i : Option.{u_2} ι) => α i) inst._@.Mathlib.Algebra.DirectSum.Module.2478368259._hygCtx._hyg.33) (instAddCommMonoidDirectSum.{u_2, u_1} (Option.{u_2} ι) (fun (i : Option.{u_2} ι) => α i) inst._@.Mathlib.Algebra.DirectSum.Module.2478368259._hygCtx._hyg.33)))) (Prod.instAdd.{u_1, max u_2 u_1} (α (Option.none.{u_2} ι)) (DirectSum.{u_2, u_1} ι (fun (i : ι) => α (Option.some.{u_2} ι i)) (fun (i : ι) => inst._@.Mathlib.Algebra.DirectSum.Module.2478368259._hygCtx._hyg.33 (Option.some.{u_2} ι i))) (AddCommMagma.toAdd.{u_1} (α (Option.none.{u_2} ι)) (AddCommSemigroup.toAddCommMagma.{u_1} (α (Option.none.{u_2} ι)) (AddCommMonoid.toAddCommSemigroup.{u_1} (α (Option.none.{u_2} ι)) (inst._@.Mathlib.Algebra.DirectSum.Module.2478368259._hygCtx._hyg.33 (Option.none.{u_2} ι))))) (AddCommMagma.toAdd.{max u_2 u_1} (DirectSum.{u_2, u_1} ι (fun (i : ι) => α (Option.some.{u_2} ι i)) (fun (i : ι) => inst._@.Mathlib.Algebra.DirectSum.Module.2478368259._hygCtx._hyg.33 (Option.some.{u_2} ι i))) (AddCommSemigroup.toAddCommMagma.{max u_2 u_1} (DirectSum.{u_2, u_1} ι (fun (i : ι) => α (Option.some.{u_2} ι i)) (fun (i : ι) => inst._@.Mathlib.Algebra.DirectSum.Module.2478368259._hygCtx._hyg.33 (Option.some.{u_2} ι i))) (AddCommMonoid.toAddCommSemigroup.{max u_2 u_1} (DirectSum.{u_2, u_1} ι (fun (i : ι) => α (Option.some.{u_2} ι i)) (fun (i : ι) => inst._@.Mathlib.Algebra.DirectSum.Module.2478368259._hygCtx._hyg.33 (Option.some.{u_2} ι i))) (instAddCommMonoidDirectSum.{u_2, u_1} ι (fun (i : ι) => α (Option.some.{u_2} ι i)) (fun (i : ι) => inst._@.Mathlib.Algebra.DirectSum.Module.2478368259._hygCtx._hyg.33 (Option.some.{u_2} ι i))))))) (DirectSum.addEquivProdDirectSum.{u_2, u_1} ι α inst._@.Mathlib.Algebra.DirectSum.Module.2478368259._hygCtx._hyg.33)))","typeFull":"∀ {ι : Type u_2} {α : Option ι → Type u_1} [inst : (i : Option ι) → AddCommMonoid (α i)],\n  Function.LeftInverse DirectSum.addEquivProdDirectSum.invFun DirectSum.addEquivProdDirectSum.toFun","typeReadable":"∀ {ι : Type u_2} {α : Option ι → Type u_1} [inst : (i : Option ι) → AddCommMonoid (α i)],\n  Function.LeftInverse DirectSum.addEquivProdDirectSum.invFun DirectSum.addEquivProdDirectSum.toFun","typeReferences":[["DirectSum","addEquivProdDirectSum"],["Option","some"],["AddEquiv","toEquiv"],["Prod","instAdd"],["DirectSum"],["Prod"],["AddCommMonoid"],["Option","none"],["AddCommMonoid","toAddCommSemigroup"],["instAddCommMonoidDirectSum"],["Equiv","invFun"],["Option"],["Function","LeftInverse"],["Equiv","toFun"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"]],"valueReferences":[["Equiv","left_inv"],["DirectSum","addEquivProdDirectSum"],["Option","some"],["AddEquiv","toEquiv"],["Prod","instAdd"],["DirectSum"],["Prod"],["Option","none"],["AddCommMonoid","toAddCommSemigroup"],["instAddCommMonoidDirectSum"],["Option"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"]]},{"isProp":false,"kind":"definition","name":["DirectSum","instModule","_aux_1"],"typeFallback":"forall {R : Type.{u_3}} [inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3 : Semiring.{u_3} R] {ι : Type.{u_1}} {M : ι -> Type.{u_2}} [inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10 : forall (i : ι), AddCommMonoid.{u_2} (M i)] [inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.19 : forall (i : ι), Module.{u_3, u_2} R (M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3 (inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10 i)], R -> (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) -> (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10)","typeFull":"{R : Type u_3} →\n  [inst : Semiring R] →\n    {ι : Type u_1} →\n      {M : ι → Type u_2} →\n        [inst_1 : (i : ι) → AddCommMonoid (M i)] →\n          [(i : ι) → Module R (M i)] → R → (DirectSum ι fun i => M i) → DirectSum ι fun i => M i","typeReadable":"{R : Type u_3} →\n  [inst : Semiring R] →\n    {ι : Type u_1} →\n      {M : ι → Type u_2} →\n        [inst_1 : (i : ι) → AddCommMonoid (M i)] →\n          [(i : ι) → Module R (M i)] → R → (DirectSum ι fun i => M i) → DirectSum ι fun i => M i","typeReferences":[["DirectSum"],["AddCommMonoid"],["Module"],["Semiring"]],"valueReferences":[["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DFinsupp","instSMulZeroClass","_proof_1"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["HSMul","hSMul"],["MonoidWithZero","toMonoid"],["instHSMul"],["DFinsupp","mapRange"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["DirectSum","isInternal_biSup_submodule_of_iSupIndep"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3 : Ring.{u} R] {ι : Type.{v}} [dec_ι : DecidableEq.{succ v} ι] {M : Type.{u_1}} [inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13 : Module.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10)] {A : ι -> (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13)} (s : Set.{v} ι), (iSupIndep.{succ v, u_1} (Set.Elem.{v} ι s) (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (fun (i : Set.Elem.{v} ι s) => A (Subtype.val.{succ v} ι (fun (x : ι) => Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s x) i))) -> (DirectSum.IsInternal.{v, u_1, u_1} (Set.Elem.{v} ι s) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13)) (iSup.{u_1, succ v} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (i : ι) => iSup.{u_1, 0} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (h._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.54 : Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) => A i))) x)) (Submodule.{u, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13)) (iSup.{u_1, succ v} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (i : ι) => iSup.{u_1, 0} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (h._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.54 : Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) => A i))) x)) (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (Submodule.addCommMonoid.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13 (iSup.{u_1, succ v} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (i : ι) => iSup.{u_1, 0} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (h._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.54 : Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) => A i)))) (Submodule.module.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13 (iSup.{u_1, succ v} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (i : ι) => iSup.{u_1, 0} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (h._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.54 : Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) => A i))))) (fun (a : Set.Elem.{v} ι s) (b : Set.Elem.{v} ι s) => Subtype.instDecidableEq.{succ v} ι (Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s) dec_ι a b) (Submodule.addCommMonoid.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13 (iSup.{u_1, succ v} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (i : ι) => iSup.{u_1, 0} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (h._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.54 : Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) => A i)))) (Submodule.setLike.{u, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13)) (iSup.{u_1, succ v} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (i : ι) => iSup.{u_1, 0} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (h._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.54 : Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) => A i))) x)) (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (Submodule.addCommMonoid.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13 (iSup.{u_1, succ v} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (i : ι) => iSup.{u_1, 0} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (h._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.54 : Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) => A i)))) (Submodule.module.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13 (iSup.{u_1, succ v} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (i : ι) => iSup.{u_1, 0} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (h._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.54 : Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) => A i))))) (Submodule.addSubmonoidClass.{u, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13)) (iSup.{u_1, succ v} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (i : ι) => iSup.{u_1, 0} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (h._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.54 : Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) => A i))) x)) (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (Submodule.addCommMonoid.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13 (iSup.{u_1, succ v} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (i : ι) => iSup.{u_1, 0} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (h._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.54 : Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) => A i)))) (Submodule.module.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13 (iSup.{u_1, succ v} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (i : ι) => iSup.{u_1, 0} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (h._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.54 : Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) => A i))))) (fun (i : Set.Elem.{v} ι s) => Submodule.comap.{u, u, u_1, u_1} R R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13)) (iSup.{u_1, succ v} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (i : ι) => iSup.{u_1, 0} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (h._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.54 : Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) => A i))) x)) M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (Submodule.addCommMonoid.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13 (iSup.{u_1, succ v} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (i : ι) => iSup.{u_1, 0} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (h._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.54 : Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) => A i)))) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) (Submodule.module.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13 (iSup.{u_1, succ v} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (i : ι) => iSup.{u_1, 0} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (h._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.54 : Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) => A i)))) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3))) (Submodule.subtype.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13 (iSup.{u_1, succ v} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (i : ι) => iSup.{u_1, 0} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.13))))) (fun (h._@.Mathlib.Algebra.DirectSum.Module.3020627411._hygCtx._hyg.54 : Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s i) => A i)))) (A (Subtype.val.{succ v} ι (fun (x : ι) => Membership.mem.{v, v} ι (Set.{v} ι) (Set.instMembership.{v} ι) s x) i))))","typeFull":"∀ {R : Type u} [inst : Ring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [inst_1 : AddCommGroup M]\n  [inst_2 : Module R M] {A : ι → Submodule R M} (s : Set ι),\n  (iSupIndep fun i => A ↑i) → DirectSum.IsInternal fun i => Submodule.comap (⨆ i ∈ s, A i).subtype (A ↑i)","typeReadable":"∀ {R : Type u} [inst : Ring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [inst_1 : AddCommGroup M]\n  [inst_2 : Module R M] {A : ι → Submodule R M} (s : Set ι),\n  (iSupIndep fun i => A ↑i) → DirectSum.IsInternal fun i => Submodule.comap (⨆ i ∈ s, A i).subtype (A ↑i)","typeReferences":[["DirectSum","IsInternal"],["Subtype"],["Module"],["DecidableEq"],["Submodule","module"],["Membership","mem"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["Subtype","val"],["Set","Elem"],["Submodule","subtype"],["Submodule"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderSup"],["Semiring","toNonAssocSemiring"],["iSup"],["RingHom","id"],["ConditionallyCompletePartialOrderSup","toSupSet"],["iSupIndep"],["Submodule","addSubmonoidClass"],["CompleteLattice","toConditionallyCompleteLattice"],["SetLike","instMembership"],["Set"],["AddCommGroup"],["Ring","toSemiring"],["Set","instMembership"],["Submodule","comap"],["Submodule","setLike"],["Submodule","completeLattice"],["AddCommGroup","toAddCommMonoid"],["Subtype","instDecidableEq"],["Submodule","addCommMonoid"],["Ring"]],"valueReferences":[["Submodule","ext"],["PartialOrder","toPreorder"],["Eq","trans"],["Subtype","instLE"],["Submodule","map"],["Membership","mem"],["OrderIso"],["le_biSup"],["Subtype","val"],["Set","Elem"],["Submodule","subtype"],["And","intro"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderSup"],["iSup"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["ConditionallyCompletePartialOrderSup","toSupSet"],["funext"],["iSupIndep","of_coe_Iic_comp"],["Eq","symm"],["Submodule","biSup_comap_subtype_eq_top"],["iSupIndep"],["Submodule","addSubmonoidClass"],["SetLike","instMembership"],["Submodule","addCommGroup"],["And"],["Subtype","ext"],["Set","Iic"],["Set","instMembership"],["Ring","toSemiring"],["iSupIndep_map_orderIso_iff"],["Iff","of_eq"],["Submodule","comap"],["Eq","refl"],["Iff"],["Iff","mpr"],["id"],["AddCommGroup","toAddCommMonoid"],["Top","top"],["Eq","mpr"],["ConditionallyCompleteLattice","toLattice"],["Submodule","instMin"],["DirectSum","IsInternal"],["SemilatticeInf","toMin"],["Subtype"],["Submodule","mapIic"],["Submodule","module"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["CompleteSemilatticeSup","toSupSet"],["CompleteLattice","toCompleteSemilatticeSup"],["DFunLike","coe"],["congrArg"],["Submodule"],["ConditionallyCompletePartialOrderSup","toPartialOrder"],["Submodule","map_comap_subtype"],["Subtype","property"],["DirectSum","isInternal_submodule_iff_iSupIndep_and_iSup_eq_top"],["iSup_subtype"],["congrFun'"],["Eq"],["Preorder","toLE"],["propext"],["CompleteLattice","toConditionallyCompleteLattice"],["Lattice","toSemilatticeInf"],["True"],["Set"],["Submodule","instTop"],["Submodule","instPartialOrder"],["inf_of_le_right"],["Iff","rfl"],["Function","comp"],["iSup_congr_Prop"],["instFunLikeOrderIso"],["RingHomSurjective","ids"],["Set","Iic","instCompleteLattice"],["eq_self"],["of_eq_true"],["Min","min"],["Submodule","setLike"],["Submodule","completeLattice"],["Subtype","instDecidableEq"],["Subtype","mk"],["Submodule","addCommMonoid"]]},{"isProp":true,"kind":"theorem","name":["DirectSum","IsInternal","ofBijective_coeLinearMap_same"],"typeFallback":"forall 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dec_ι inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13) (Submodule.addSubmonoidClass.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13) A) {i : ι} (x : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13)) (A i) x)), Eq.{succ u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13)) (A i) x)) (DFunLike.coe.{max (succ v) (succ u_1), succ v, succ u_1} (DFinsupp.{v, u_1} ι (fun (i : ι) => (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 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inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13 (A i)) (fun (i : ι) => Submodule.module.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13 (A i)))) M (DirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M 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inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13 (A i))) inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13 (DirectSum.instModule.{u, v, u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13 (A i)) (fun (i : ι) => Submodule.module.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13 (A i))))) (LinearEquiv.symm.{u, u, max u_1 v, u_1} R R (DirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13 (A i))) M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 (instAddCommMonoidDirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13 (A i))) inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 (DirectSum.instModule.{u, v, u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13 (A i)) (fun (i : ι) => Submodule.module.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13 (A i))) inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3)) (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3)) (RingHomInvPair.ids.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3) (RingHomInvPair.ids.{u} R 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(Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13 (A i))) inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 (DirectSum.instModule.{u, v, u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13) 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inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13 (A i))) inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3)) (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3)) (DirectSum.coeLinearMap.{u, v, u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 ι dec_ι M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.13 A) (RingHomInvPair.ids.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3) (RingHomInvPair.ids.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3) h)) (Subtype.val.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2018255034._hygCtx._hyg.3 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(R : Type.{u}) [inst._@.Mathlib.Algebra.DirectSum.Module.1468946332._hygCtx._hyg.3 : Semiring.{u} R] (ι : Type.{v}) (M : ι -> Type.{w}) [inst._@.Mathlib.Algebra.DirectSum.Module.1468946332._hygCtx._hyg.10 : forall (i : ι), AddCommMonoid.{w} (M i)] [inst._@.Mathlib.Algebra.DirectSum.Module.1468946332._hygCtx._hyg.19 : forall (i : ι), Module.{u, w} R (M i) inst._@.Mathlib.Algebra.DirectSum.Module.1468946332._hygCtx._hyg.3 (inst._@.Mathlib.Algebra.DirectSum.Module.1468946332._hygCtx._hyg.10 i)] [inst._@.Mathlib.Algebra.DirectSum.Module.1468946332._hygCtx._hyg.29 : DecidableEq.{succ v} ι] (s : Finset.{v} ι), LinearMap.{u, u, max v w, max w v} R R inst._@.Mathlib.Algebra.DirectSum.Module.1468946332._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1468946332._hygCtx._hyg.3 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1468946332._hygCtx._hyg.3)) (forall (i : Set.Elem.{v} ι (SetLike.coe.{v, v} (Finset.{v} ι) ι (Finset.instSetLike.{v} ι) 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(fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1193137464._hygCtx._hyg.10) (AddMonoid.toAddZeroClass.{max v w} (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1193137464._hygCtx._hyg.10) (AddCommMonoid.toAddMonoid.{max v w} (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1193137464._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1193137464._hygCtx._hyg.10))))) (DistribSMul.toSMulZeroClass.{u_1, max v w} S (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1193137464._hygCtx._hyg.10) (AddMonoid.toAddZeroClass.{max v w} (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1193137464._hygCtx._hyg.10) (AddCommMonoid.toAddMonoid.{max v w} (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1193137464._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{v, w} ι 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i)","typeReferences":[["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["DirectSum"],["SMulCommClass"],["Module","toDistribMulAction"],["AddCommMonoid"],["instAddCommMonoidDirectSum"],["MonoidWithZero","toMonoid"],["DirectSum","instModule"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["_private","Mathlib","Algebra","DirectSum","Module",0,"DirectSum","instSMulCommClass","_proof_1"]]},{"isProp":true,"kind":"theorem","name":["DirectSum","instIsCentralScalar"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.3 : Semiring.{u} R] {ι : Type.{v}} {M : ι -> Type.{w}} [inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10 : forall (i : ι), AddCommMonoid.{w} (M i)] [inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.19 : forall (i : ι), Module.{u, w} R (M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.3 (inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10 i)] [inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.29 : forall (i : ι), Module.{u, w} (MulOpposite.{u} R) (M i) (MulOpposite.instSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.3) (inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10 i)] [inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.42 : forall (i : ι), IsCentralScalar.{u, w} R (M i) (SMulZeroClass.toSMul.{u, w} R (M i) (AddZero.toZero.{w} (M i) (AddZeroClass.toAddZero.{w} (M i) (AddMonoid.toAddZeroClass.{w} (M i) (AddCommMonoid.toAddMonoid.{w} (M i) (inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10 i))))) (DistribSMul.toSMulZeroClass.{u, w} R (M i) (AddMonoid.toAddZeroClass.{w} (M i) (AddCommMonoid.toAddMonoid.{w} (M i) 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(AddCommMonoid.toAddMonoid.{w} (M i) (inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10 i))) (DistribMulAction.toDistribSMul.{u, w} (MulOpposite.{u} R) (M i) (MulOpposite.instMonoid.{u} R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{w} (M i) (inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10 i)) (Module.toDistribMulAction.{u, w} (MulOpposite.{u} R) (M i) (MulOpposite.instSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.3) (inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10 i) (inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.29 i)))))], IsCentralScalar.{u, max w v} R (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (SMulZeroClass.toSMul.{u, max v w} R (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (AddZero.toZero.{max v w} (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (AddZeroClass.toAddZero.{max v w} (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (AddMonoid.toAddZeroClass.{max v w} (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (AddCommMonoid.toAddMonoid.{max v w} (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10))))) (DistribSMul.toSMulZeroClass.{u, max v w} R (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (AddMonoid.toAddZeroClass.{max v w} (DirectSum.{v, w} ι (fun (i : ι) => M 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inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.3 (instAddCommMonoidDirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (DirectSum.instModule.{u, v, w} R inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.3 ι M inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.19))))) (SMulZeroClass.toSMul.{u, max v w} (MulOpposite.{u} R) (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (AddZero.toZero.{max v w} (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (AddZeroClass.toAddZero.{max v w} (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (AddMonoid.toAddZeroClass.{max v w} (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (AddCommMonoid.toAddMonoid.{max v w} (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10))))) (DistribSMul.toSMulZeroClass.{u, max v w} (MulOpposite.{u} R) (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (AddMonoid.toAddZeroClass.{max v w} (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (AddCommMonoid.toAddMonoid.{max v w} (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10))) (DistribMulAction.toDistribSMul.{u, max v w} (MulOpposite.{u} R) (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (MulOpposite.instMonoid.{u} R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{max v w} (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10)) (Module.toDistribMulAction.{u, max v w} (MulOpposite.{u} R) (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (MulOpposite.instSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.3) (instAddCommMonoidDirectSum.{v, w} ι (fun (i : ι) => M i) 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inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13 (A i))) => M) (LinearMap.instFunLike.{u, u, max u_1 v, u_1} R R (DirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13 (A i))) M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 (instAddCommMonoidDirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13 (A i))) inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 (DirectSum.instModule.{u, v, u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13 (A i)) (fun (i : ι) => Submodule.module.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13 (A i))) inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3))) (DirectSum.coeLinearMap.{u, v, u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 ι dec_ι M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13 A) x) (DFinsupp.sum.{v, u_1, u_1} ι M (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) 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inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13)) (A i) x)) i) (AddCommMonoid.toAddMonoid.{u_1} ((fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13)) (A 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inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13)) (A i) x)) i) (AddZeroClass.toAddZero.{u_1} ((fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13)) (A i) x)) i) (AddMonoid.toAddZeroClass.{u_1} ((fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13)) (A i) x)) i) (AddCommMonoid.toAddMonoid.{u_1} ((fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13)) (A i) x)) i) ((fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13 (A i)) i)))))))) (Subtype.instDecidableEq.{succ u_1} M (Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13)) (A i)) inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.22 x (OfNat.ofNat.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13)) (A i) x)) 0 (Zero.toOfNat0.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13)) (A i) x)) (AddZero.toZero.{u_1} ((fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13)) (A i) x)) i) (AddZeroClass.toAddZero.{u_1} ((fun (i : ι) => Subtype.{succ u_1} M (fun 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inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13)) (A i) x)) i) (AddCommMonoid.toAddMonoid.{u_1} ((fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13)) (A i) x)) i) ((fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13 (A i)) i))))))))) inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 x (fun (i : ι) (x : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13)) (A i) x)) => Subtype.val.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1563678033._hygCtx._hyg.13)) (A i) x) x))","typeFull":"∀ {R : Type u} [inst : Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] (A : ι → Submodule R M) [inst_3 : DecidableEq M] (x : DirectSum ι fun i => ↥(A i)),\n  (DirectSum.coeLinearMap A) x = DFinsupp.sum x fun i x => ↑x","typeReadable":"∀ {R : Type u} [inst : Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [inst_1 : 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{R : Type.{u}} [inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.3 : Semiring.{u} R] {ι : Type.{v}} {M : ι -> Type.{w}} [inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10 : forall (i : ι), AddCommMonoid.{w} (M i)] [inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.19 : forall (i : ι), Module.{u, w} R (M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.3 (inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10 i)] {S : Type.{u_1}} [inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.30 : Semiring.{u_1} S] [inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.33 : SMul.{u, u_1} R S] [inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.37 : forall (i : ι), Module.{u_1, w} S (M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.30 (inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10 i)] [inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.47 : forall (i : ι), IsScalarTower.{u, u_1, w} R S (M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.33 (SMulZeroClass.toSMul.{u_1, w} S (M i) (AddZero.toZero.{w} (M i) (AddZeroClass.toAddZero.{w} (M i) (AddMonoid.toAddZeroClass.{w} (M i) (AddCommMonoid.toAddMonoid.{w} (M i) (inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10 i))))) (DistribSMul.toSMulZeroClass.{u_1, w} S (M i) (AddMonoid.toAddZeroClass.{w} (M i) (AddCommMonoid.toAddMonoid.{w} (M i) (inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10 i))) (DistribMulAction.toDistribSMul.{u_1, w} S (M i) (MonoidWithZero.toMonoid.{u_1} S (Semiring.toMonoidWithZero.{u_1} S inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.30)) (AddCommMonoid.toAddMonoid.{w} (M i) (inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10 i)) (Module.toDistribMulAction.{u_1, w} S (M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.30 (inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10 i) (inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.37 i))))) (SMulZeroClass.toSMul.{u, w} R (M i) (AddZero.toZero.{w} (M i) (AddZeroClass.toAddZero.{w} (M i) (AddMonoid.toAddZeroClass.{w} (M i) (AddCommMonoid.toAddMonoid.{w} (M i) (inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10 i))))) (DistribSMul.toSMulZeroClass.{u, w} R (M i) (AddMonoid.toAddZeroClass.{w} (M i) (AddCommMonoid.toAddMonoid.{w} (M i) (inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10 i))) (DistribMulAction.toDistribSMul.{u, w} R (M i) (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{w} (M i) (inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10 i)) (Module.toDistribMulAction.{u, w} R (M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.3 (inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10 i) (inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.19 i)))))], IsScalarTower.{u, u_1, max w v} R S (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.33 (SMulZeroClass.toSMul.{u_1, max v w} S (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (AddZero.toZero.{max v w} (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (AddZeroClass.toAddZero.{max v w} (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (AddMonoid.toAddZeroClass.{max v w} (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (AddCommMonoid.toAddMonoid.{max v w} (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10))))) (DistribSMul.toSMulZeroClass.{u_1, max v w} S (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (AddMonoid.toAddZeroClass.{max v w} (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (AddCommMonoid.toAddMonoid.{max v w} (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10))) (DistribMulAction.toDistribSMul.{u_1, max v w} S (DirectSum.{v, w} ι (fun (i : ι) => M i) 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inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.37))))) (SMulZeroClass.toSMul.{u, max v w} R (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (AddZero.toZero.{max v w} (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (AddZeroClass.toAddZero.{max v w} (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (AddMonoid.toAddZeroClass.{max v w} (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (AddCommMonoid.toAddMonoid.{max v w} (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10))))) (DistribSMul.toSMulZeroClass.{u, max v w} R (DirectSum.{v, w} ι (fun (i : ι) => M 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i)","typeReferences":[["DirectSum","lmap"],["LinearMap","instFunLike"],["Module"],["DFinsupp"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["DirectSum"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["instAddCommMonoidDirectSum"],["RingHom","id"],["DirectSum","instModule"],["DFinsupp","instDFunLike"],["Eq"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["rfl"],["DirectSum","lmap"],["LinearMap","instFunLike"],["DFinsupp"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["LinearMap"],["DFunLike","coe"],["DirectSum"],["Semiring","toNonAssocSemiring"],["instAddCommMonoidDirectSum"],["RingHom","id"],["DirectSum","instModule"],["DFinsupp","instDFunLike"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["DirectSum","isInternal_submodule_of_iSupIndep_of_iSup_eq_top"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.3 : Ring.{u} R] {ι : Type.{v}} [dec_ι : DecidableEq.{succ v} ι] {M : Type.{u_1}} [inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.10 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.13 : Module.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.10)] {A : ι -> (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.13)}, (iSupIndep.{succ v, u_1} ι (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.3) 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inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.13) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.13))))) A) (Top.top.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.13) (Submodule.instTop.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.13))) -> (DirectSum.IsInternal.{v, u_1, u_1} ι M (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.13) dec_ι (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.1265894544._hygCtx._hyg.10) 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Submodule R M}, iSupIndep A → iSup A = ⊤ → DirectSum.IsInternal 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{ι : Type.{u_1}} (S : Set.{u_1} ι) (T : Set.{u_1} ι), (HasSubset.Subset.{u_1} (Set.{u_1} ι) (Set.instHasSubset.{u_1} ι) S T) -> (forall (i : Set.Elem.{u_1} ι S), Membership.mem.{u_1, u_1} ι (Set.{u_1} ι) (Set.instMembership.{u_1} ι) T (Subtype.val.{succ u_1} ι (fun (x : ι) => Membership.mem.{u_1, u_1} ι (Set.{u_1} ι) (Set.instMembership.{u_1} ι) S x) i))","typeFull":"∀ {ι : Type u_1} (S T : Set ι), S ⊆ T → ∀ (i : ↑S), ↑i ∈ T","typeReadable":"∀ {ι : Type u_1} (S T : Set ι), S ⊆ T → ∀ (i : ↑S), ↑i ∈ T","typeReferences":[["Set","instHasSubset"],["HasSubset","Subset"],["Set"],["Membership","mem"],["Subtype","val"],["Set","Elem"],["Set","instMembership"]],"valueReferences":[["Subtype","prop"],["Set"],["Membership","mem"],["Subtype","val"],["Set","instMembership"]]},{"isProp":false,"kind":"definition","name":["DirectSum","lmap"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.3 : Semiring.{u} R] {ι : Type.{v}} {M : ι -> Type.{w}} [inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.10 : forall (i : ι), AddCommMonoid.{w} (M i)] [inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.19 : forall (i : ι), Module.{u, w} R (M i) inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.3 (inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.10 i)] {N : ι -> Type.{u_1}} [inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.32 : forall (i : ι), AddCommMonoid.{u_1} (N i)] [inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.41 : forall (i : ι), Module.{u, u_1} R (N i) inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.3 (inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.32 i)], (forall (i : ι), LinearMap.{u, u, w, u_1} R R inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.3 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.3)) (M i) (N i) (inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.10 i) (inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.32 i) (inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.19 i) (inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.41 i)) -> (LinearMap.{u, u, max w v, max u_1 v} R R inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.3 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.3)) (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.10) (DirectSum.{v, u_1} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.32) (instAddCommMonoidDirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{v, u_1} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.32) (DirectSum.instModule.{u, v, w} R inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.3 ι M inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.19) (DirectSum.instModule.{u, v, u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.3 ι N inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.32 inst._@.Mathlib.Algebra.DirectSum.Module.1921081898._hygCtx._hyg.41))","typeFull":"{R : Type u} →\n  [inst : Semiring R] →\n    {ι : Type v} →\n      {M : ι → Type w} →\n        [inst_1 : (i : ι) → AddCommMonoid (M i)] →\n          [inst_2 : (i : ι) → Module R (M i)] →\n            {N : ι → Type u_1} →\n              [inst_3 : (i : ι) → AddCommMonoid (N i)] →\n                [inst_4 : (i : ι) → Module R (N i)] →\n                  ((i : ι) → M i →ₗ[R] N i) → (DirectSum ι fun i => M i) →ₗ[R] DirectSum ι fun i => N i","typeReadable":"{R : Type u} →\n  [inst : Semiring R] →\n    {ι : Type v} →\n      {M : ι → Type w} →\n        [inst_1 : (i : ι) → AddCommMonoid (M i)] →\n          [inst_2 : (i : ι) → Module R (M i)] →\n            {N : ι → Type u_1} →\n              [inst_3 : (i : ι) → AddCommMonoid (N i)] →\n                [inst_4 : (i : ι) → Module R (N i)] →\n                  ((i : ι) → M i →ₗ[R] N i) → (DirectSum ι fun i => M i) →ₗ[R] DirectSum ι fun i => N i","typeReferences":[["DirectSum"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["instAddCommMonoidDirectSum"],["RingHom","id"],["Module"],["DirectSum","instModule"],["LinearMap"],["Semiring"]],"valueReferences":[["DFinsupp","mapRange","linearMap"]]},{"isProp":true,"kind":"theorem","name":["DirectSum","IsInternal","addSubgroup_iSupIndep"],"typeFallback":"forall {ι : Type.{v}} [dec_ι : DecidableEq.{succ v} ι] {G : Type.{u_2}} [inst._@.Mathlib.Algebra.DirectSum.Module.1775035287._hygCtx._hyg.18 : AddCommGroup.{u_2} G] {A : ι -> (AddSubgroup.{u_2} G (AddCommGroup.toAddGroup.{u_2} G inst._@.Mathlib.Algebra.DirectSum.Module.1775035287._hygCtx._hyg.18))}, (DirectSum.IsInternal.{v, u_2, u_2} ι G (AddSubgroup.{u_2} G (AddCommGroup.toAddGroup.{u_2} G inst._@.Mathlib.Algebra.DirectSum.Module.1775035287._hygCtx._hyg.18)) dec_ι (AddCommGroup.toAddCommMonoid.{u_2} G inst._@.Mathlib.Algebra.DirectSum.Module.1775035287._hygCtx._hyg.18) (AddSubgroup.instSetLike.{u_2} G (AddCommGroup.toAddGroup.{u_2} G inst._@.Mathlib.Algebra.DirectSum.Module.1775035287._hygCtx._hyg.18)) (AddSubgroupClass.toAddSubmonoidClass.{u_2, u_2} (AddSubgroup.{u_2} G (AddCommGroup.toAddGroup.{u_2} G inst._@.Mathlib.Algebra.DirectSum.Module.1775035287._hygCtx._hyg.18)) G (AddGroup.toSubNegMonoid.{u_2} G (AddCommGroup.toAddGroup.{u_2} G inst._@.Mathlib.Algebra.DirectSum.Module.1775035287._hygCtx._hyg.18)) (AddSubgroup.instSetLike.{u_2} G (AddCommGroup.toAddGroup.{u_2} G inst._@.Mathlib.Algebra.DirectSum.Module.1775035287._hygCtx._hyg.18)) (AddSubgroup.instAddSubgroupClass.{u_2} G (AddCommGroup.toAddGroup.{u_2} G inst._@.Mathlib.Algebra.DirectSum.Module.1775035287._hygCtx._hyg.18))) A) -> (iSupIndep.{succ v, u_2} ι (AddSubgroup.{u_2} G (AddCommGroup.toAddGroup.{u_2} G inst._@.Mathlib.Algebra.DirectSum.Module.1775035287._hygCtx._hyg.18)) (AddSubgroup.instCompleteLattice.{u_2} G (AddCommGroup.toAddGroup.{u_2} G inst._@.Mathlib.Algebra.DirectSum.Module.1775035287._hygCtx._hyg.18)) A)","typeFull":"∀ {ι : Type v} [dec_ι : DecidableEq ι] {G : Type u_2} [inst : AddCommGroup G] {A : ι → AddSubgroup G},\n  DirectSum.IsInternal A → iSupIndep A","typeReadable":"∀ {ι : Type v} [dec_ι : DecidableEq ι] {G : Type u_2} [inst : AddCommGroup G] {A : ι → AddSubgroup G},\n  DirectSum.IsInternal A → iSupIndep A","typeReferences":[["DirectSum","IsInternal"],["AddSubgroup"],["AddSubgroup","instAddSubgroupClass"],["AddCommGroup","toAddGroup"],["DecidableEq"],["AddSubgroup","instSetLike"],["AddSubgroupClass","toAddSubmonoidClass"],["AddCommGroup","toAddCommMonoid"],["AddCommGroup"],["AddGroup","toSubNegMonoid"],["iSupIndep"],["AddSubgroup","instCompleteLattice"]],"valueReferences":[["Subtype"],["SetLike","instMembership"],["AddCommGroup","toAddGroup"],["Membership","mem"],["AddSubgroup","instSetLike"],["AddSubgroupClass","toAddSubmonoidClass"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["AddSubmonoidClass","toAddCommMonoid"],["DirectSum"],["AddSubgroup"],["Function","Bijective","injective"],["instAddCommMonoidDirectSum"],["AddSubgroup","instAddSubgroupClass"],["AddMonoidHom"],["AddCommGroup","toAddCommMonoid"],["AddMonoidHom","instFunLike"],["AddGroup","toSubNegMonoid"],["DirectSum","coeAddMonoidHom"],["iSupIndep_of_dfinsuppSumAddHom_injective'"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","DirectSum","Module",0,"DirectSum","instIsCentralScalar","_proof_1"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.3 : Semiring.{u_1} R] {ι : Type.{u_3}} {M : ι -> Type.{u_2}} [inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10 : forall (i : ι), AddCommMonoid.{u_2} (M i)] [inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.19 : forall (i : ι), Module.{u_1, u_2} R (M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.3 (inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10 i)] [inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.29 : forall (i : ι), Module.{u_1, u_2} (MulOpposite.{u_1} R) (M i) (MulOpposite.instSemiring.{u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.3) (inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10 i)] [inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.42 : forall (i : ι), IsCentralScalar.{u_1, u_2} R (M i) (SMulZeroClass.toSMul.{u_1, u_2} R (M i) (AddZero.toZero.{u_2} (M i) (AddZeroClass.toAddZero.{u_2} (M i) (AddMonoid.toAddZeroClass.{u_2} (M i) (AddCommMonoid.toAddMonoid.{u_2} (M i) (inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10 i))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R (M i) (AddMonoid.toAddZeroClass.{u_2} (M i) (AddCommMonoid.toAddMonoid.{u_2} (M i) (inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10 i))) (DistribMulAction.toDistribSMul.{u_1, u_2} R (M i) (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_2} (M i) (inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10 i)) (Module.toDistribMulAction.{u_1, u_2} R (M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.3 (inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10 i) (inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.19 i))))) (SMulZeroClass.toSMul.{u_1, u_2} (MulOpposite.{u_1} R) (M i) (AddZero.toZero.{u_2} (M i) (AddZeroClass.toAddZero.{u_2} (M i) (AddMonoid.toAddZeroClass.{u_2} (M i) (AddCommMonoid.toAddMonoid.{u_2} (M i) (inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10 i))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} (MulOpposite.{u_1} R) (M i) (AddMonoid.toAddZeroClass.{u_2} (M i) (AddCommMonoid.toAddMonoid.{u_2} (M i) (inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10 i))) (DistribMulAction.toDistribSMul.{u_1, u_2} (MulOpposite.{u_1} R) (M i) (MulOpposite.instMonoid.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.3))) (AddCommMonoid.toAddMonoid.{u_2} (M i) (inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10 i)) (Module.toDistribMulAction.{u_1, u_2} (MulOpposite.{u_1} R) (M i) (MulOpposite.instSemiring.{u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.3) (inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10 i) (inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.29 i)))))], IsCentralScalar.{u_1, max u_2 u_3} R (DirectSum.{u_3, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (SMulZeroClass.toSMul.{u_1, max u_3 u_2} R (DirectSum.{u_3, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (AddZero.toZero.{max u_3 u_2} (DirectSum.{u_3, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (AddZeroClass.toAddZero.{max u_3 u_2} (DirectSum.{u_3, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (AddMonoid.toAddZeroClass.{max u_3 u_2} (DirectSum.{u_3, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (AddCommMonoid.toAddMonoid.{max u_3 u_2} (DirectSum.{u_3, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{u_3, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10))))) (DistribSMul.toSMulZeroClass.{u_1, max u_3 u_2} R (DirectSum.{u_3, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (AddMonoid.toAddZeroClass.{max u_3 u_2} (DirectSum.{u_3, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (AddCommMonoid.toAddMonoid.{max u_3 u_2} (DirectSum.{u_3, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{u_3, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10))) (DistribMulAction.toDistribSMul.{u_1, max u_3 u_2} R (DirectSum.{u_3, u_2} ι (fun (i : ι) => M i) 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inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.19))))) (SMulZeroClass.toSMul.{u_1, max u_3 u_2} (MulOpposite.{u_1} R) (DirectSum.{u_3, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (AddZero.toZero.{max u_3 u_2} (DirectSum.{u_3, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (AddZeroClass.toAddZero.{max u_3 u_2} (DirectSum.{u_3, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (AddMonoid.toAddZeroClass.{max u_3 u_2} (DirectSum.{u_3, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (AddCommMonoid.toAddMonoid.{max u_3 u_2} (DirectSum.{u_3, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1836316326._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{u_3, u_2} ι (fun (i : ι) => M i) 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: ι) => α _i) i))) (instAddCommMonoidDirectSum.{max u_3 u_1, u_2} (Sigma.{u_1, u_3} ι (fun (_i : ι) => α _i)) (fun (i : Sigma.{u_1, u_3} ι (fun (_i : ι) => α _i)) => δ (Sigma.fst.{u_1, u_3} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_1, u_3} ι (fun (_i : ι) => α _i) i)) (fun (i : Sigma.{u_1, u_3} ι (fun (_i : ι) => α _i)) => inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.42 (Sigma.fst.{u_1, u_3} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_1, u_3} ι (fun (_i : ι) => α _i) i))))))) (AddMonoidHom.toZeroHom.{max (max u_2 u_3) u_1, max u_2 u_3 u_1} (DirectSum.{u_1, max u_2 u_3} ι (fun (i : ι) => DirectSum.{u_3, u_2} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_3, u_2} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.42 i))) (DirectSum.{max u_3 u_1, u_2} (Sigma.{u_1, u_3} ι (fun (_i : ι) => α _i)) (fun (i : Sigma.{u_1, u_3} ι (fun (_i : ι) => α _i)) => δ (Sigma.fst.{u_1, u_3} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_1, u_3} ι (fun (_i : ι) => α _i) i)) (fun (i : Sigma.{u_1, u_3} ι (fun (_i : ι) => α _i)) => inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.42 (Sigma.fst.{u_1, u_3} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_1, u_3} ι (fun (_i : ι) => α _i) i))) (AddZeroClass.toAddZero.{max (max u_3 u_1) u_2} (DirectSum.{u_1, max u_2 u_3} ι (fun (i : ι) => DirectSum.{u_3, u_2} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_3, u_2} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.42 i))) (AddMonoid.toAddZeroClass.{max (max u_3 u_1) u_2} (DirectSum.{u_1, max u_2 u_3} ι (fun (i : ι) => DirectSum.{u_3, u_2} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_3, u_2} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.42 i))) (AddCommMonoid.toAddMonoid.{max (max u_3 u_1) u_2} (DirectSum.{u_1, max u_2 u_3} ι (fun (i : ι) => DirectSum.{u_3, u_2} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_3, u_2} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.42 i))) (instAddCommMonoidDirectSum.{u_1, max u_3 u_2} ι (fun (i : ι) => DirectSum.{u_3, u_2} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_3, u_2} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.42 i)))))) (AddZeroClass.toAddZero.{max (max u_3 u_1) u_2} (DirectSum.{max u_3 u_1, u_2} (Sigma.{u_1, u_3} ι (fun (_i : ι) => α _i)) 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inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.42)) x) (ZeroHom.toFun.{max (max u_2 u_3) u_1, max u_2 u_3 u_1} (DirectSum.{u_1, max u_2 u_3} ι (fun (i : ι) => DirectSum.{u_3, u_2} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_3, u_2} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.42 i))) (DirectSum.{max u_3 u_1, u_2} (Sigma.{u_1, u_3} ι (fun (_i : ι) => α _i)) (fun (i : Sigma.{u_1, u_3} ι (fun (_i : ι) => α _i)) => δ (Sigma.fst.{u_1, u_3} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_1, u_3} ι (fun (_i : ι) => α _i) i)) (fun (i : Sigma.{u_1, u_3} ι (fun (_i : ι) => α _i)) => inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.42 (Sigma.fst.{u_1, u_3} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_1, u_3} ι (fun (_i : ι) => α _i) i))) (AddZero.toZero.{max (max u_2 u_3) u_1} (DirectSum.{u_1, max u_2 u_3} ι (fun (i 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=> instAddCommMonoidDirectSum.{u_1, w} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.2416241005._hygCtx._hyg.42 i))) (instAddCommMonoidDirectSum.{max u_1 v, w} (Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.118 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.118)) (fun (i : Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.118 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.118)) => δ (Sigma.fst.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.118 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.118) i) (Sigma.snd.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.118 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.118) i)) (fun (i : Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.118 : ι) => α 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_i) i))) (AddMonoid.toAddZeroClass.{max (max u_1 u_2) u_3} (DirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) (fun (i : Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) => δ (Sigma.fst.{u_2, u_1} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_2, u_1} ι (fun (_i : ι) => α _i) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_2, u_1} ι (fun (_i : ι) => α _i) i))) (AddCommMonoid.toAddMonoid.{max (max u_1 u_2) u_3} (DirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) (fun (i : Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) => δ (Sigma.fst.{u_2, u_1} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_2, u_1} ι (fun (_i : ι) => α _i) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_2, u_1} ι (fun (_i : ι) => α _i) i))) (instAddCommMonoidDirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) (fun (i : Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) => δ (Sigma.fst.{u_2, u_1} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_2, u_1} ι (fun (_i : ι) => α _i) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_2, u_1} ι (fun (_i : ι) => α _i) i)))))) (AddZeroClass.toAddZero.{max (max u_1 u_2) u_3} (DirectSum.{u_2, max u_3 u_1} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))) (AddMonoid.toAddZeroClass.{max (max u_1 u_2) u_3} (DirectSum.{u_2, max u_3 u_1} ι (fun (i : ι) => DirectSum.{u_1, 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i)))))) (DirectSum.sigmaCurry.{u_1, u_2, u_3} ι inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.39 α δ inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42)) (HAdd.hAdd.{max u_3 u_1 u_2, max u_3 u_1 u_2, max u_3 u_1 u_2} (DirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) (fun (i : Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) => δ (Sigma.fst.{u_2, u_1} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_2, u_1} ι (fun (_i : ι) => α _i) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_2, u_1} ι (fun (_i : ι) => α _i) i))) (DirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) (fun (i : Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) => δ (Sigma.fst.{u_2, u_1} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_2, u_1} ι (fun (_i : ι) => α _i) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) => 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(inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)))))))) (ZeroHom.toFun.{max u_3 u_1 u_2, max (max u_3 u_1) u_2} (DirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) (fun (i : Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) => δ (Sigma.fst.{u_2, u_1} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_2, u_1} ι (fun (_i : ι) => α _i) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_2, u_1} ι (fun (_i : ι) => α _i) i))) (DirectSum.{u_2, max u_3 u_1} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))) (AddZero.toZero.{max u_3 u_1 u_2} (DirectSum.{max u_1 u_2, u_3} 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(instAddCommMonoidDirectSum.{u_2, max u_1 u_3} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))))))) (AddMonoidHom.toZeroHom.{max u_3 u_1 u_2, max (max u_3 u_1) u_2} (DirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) (fun (i : Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) => δ (Sigma.fst.{u_2, u_1} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_2, u_1} ι (fun (_i : ι) => α _i) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_2, u_1} ι (fun (_i : ι) => α _i) i))) (DirectSum.{u_2, max u_3 u_1} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) 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_i) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_2, u_1} ι (fun (_i : ι) => α _i) i)))))) (AddZeroClass.toAddZero.{max (max u_1 u_2) u_3} (DirectSum.{u_2, max u_3 u_1} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))) (AddMonoid.toAddZeroClass.{max (max u_1 u_2) u_3} (DirectSum.{u_2, max u_3 u_1} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))) 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u_3} (Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) (fun (i : Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) => δ (Sigma.fst.{u_2, u_1} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_2, u_1} ι (fun (_i : ι) => α _i) i)) (fun (i : Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{u_2, u_1} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_2, u_1} ι (fun (_i : ι) => α _i) i))) (DirectSum.{u_2, max u_3 u_1} ι (fun (i : ι) => DirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, u_3} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))) (AddZero.toZero.{max u_3 u_1 u_2} (DirectSum.{max u_1 u_2, u_3} (Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) (fun (i : Sigma.{u_2, u_1} ι (fun (_i : ι) => α _i)) => δ (Sigma.fst.{u_2, u_1} ι (fun (_i : ι) => α _i) i) (Sigma.snd.{u_2, 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(AddMonoid.toAddZeroClass.{max v u_1} (DirectSum.{v, u_1} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.755558530._hygCtx._hyg.32) (AddCommMonoid.toAddMonoid.{max v u_1} (DirectSum.{v, u_1} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.755558530._hygCtx._hyg.32) (instAddCommMonoidDirectSum.{v, u_1} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.755558530._hygCtx._hyg.32))))) (DirectSum.map.{v, w, u_1} ι M N inst._@.Mathlib.Algebra.DirectSum.Module.755558530._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.755558530._hygCtx._hyg.32 (fun (i : ι) => LinearMap.toAddMonoidHom.{u, u, w, u_1} R R (M i) (N i) inst._@.Mathlib.Algebra.DirectSum.Module.755558530._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.755558530._hygCtx._hyg.3 (inst._@.Mathlib.Algebra.DirectSum.Module.755558530._hygCtx._hyg.10 i) (inst._@.Mathlib.Algebra.DirectSum.Module.755558530._hygCtx._hyg.32 i) (inst._@.Mathlib.Algebra.DirectSum.Module.755558530._hygCtx._hyg.19 i) (inst._@.Mathlib.Algebra.DirectSum.Module.755558530._hygCtx._hyg.41 i) (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.755558530._hygCtx._hyg.3)) (f i))) x)","typeFull":"∀ {R : Type u} [inst : Semiring R] {ι : Type v} {M : ι → Type w} [inst_1 : (i : ι) → AddCommMonoid (M i)]\n  [inst_2 : (i : ι) → Module R (M i)] {N : ι → Type u_1} [inst_3 : (i : ι) → AddCommMonoid (N i)]\n  [inst_4 : (i : ι) → Module R (N i)] (f : (i : ι) → M i →ₗ[R] N i) (x : DirectSum ι fun i => M i),\n  (DirectSum.lmap f) x = (DirectSum.map fun i => (f i).toAddMonoidHom) x","typeReadable":"∀ {R : Type u} [inst : Semiring R] {ι : Type v} {M : ι → Type w} [inst_1 : (i : ι) → AddCommMonoid (M i)]\n  [inst_2 : (i : ι) → Module R (M i)] {N : ι → Type u_1} [inst_3 : (i : ι) → AddCommMonoid (N i)]\n  [inst_4 : (i : ι) → Module R (N i)] (f : (i : ι) → M i →ₗ[R] N i) (x : DirectSum ι fun i => M i),\n  (DirectSum.lmap f) x = (DirectSum.map fun i => (f i).toAddMonoidHom) x","typeReferences":[["LinearMap","toAddMonoidHom"],["DirectSum","map"],["LinearMap","instFunLike"],["DirectSum","lmap"],["Module"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["LinearMap"],["DFunLike","coe"],["DirectSum"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["instAddCommMonoidDirectSum"],["RingHom","id"],["AddMonoidHom"],["AddMonoidHom","instFunLike"],["DirectSum","instModule"],["Eq"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["DirectSum"],["rfl"],["Semiring","toNonAssocSemiring"],["DirectSum","lmap"],["LinearMap","instFunLike"],["instAddCommMonoidDirectSum"],["RingHom","id"],["DirectSum","instModule"],["LinearMap"],["DFunLike","coe"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","DirectSum","Module",0,"DirectSum","isInternal_submodule_iff_isCompl","match_1_3"],"typeFallback":"forall {R : Type.{u_2}} [inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3 : Ring.{u_2} R] {ι : Type.{u_3}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13 : Module.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10)] (A : ι -> (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13)) {i : ι} {j : ι} (motive : (IsCompl.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.instPartialOrder.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (CompleteLattice.toBoundedOrder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.completeLattice.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13)) (A i) (A j)) -> Prop) (x._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx.214.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.222 : IsCompl.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.instPartialOrder.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (CompleteLattice.toBoundedOrder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.completeLattice.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13)) (A i) (A j)), (forall (hd : Disjoint.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.instPartialOrder.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (BoundedOrder.toOrderBot.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Preorder.toLE.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (PartialOrder.toPreorder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.instPartialOrder.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13))) (CompleteLattice.toBoundedOrder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.completeLattice.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13))) (A i) (A j)) (ht : Codisjoint.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.instPartialOrder.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (BoundedOrder.toOrderTop.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Preorder.toLE.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (PartialOrder.toPreorder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.instPartialOrder.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13))) (CompleteLattice.toBoundedOrder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.completeLattice.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13))) (A i) (A j)), motive (IsCompl.mk.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.instPartialOrder.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (CompleteLattice.toBoundedOrder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.completeLattice.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13)) (A i) (A j) hd ht)) -> (motive x._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx.214.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.222)","typeFull":"∀ {R : Type u_2} [inst : Ring R] {ι : Type u_3} {M : Type u_1} [inst_1 : AddCommGroup M] [inst_2 : Module R M]\n  (A : ι → Submodule R M) {i j : ι} (motive : IsCompl (A i) (A j) → Prop) (x : IsCompl (A i) (A j)),\n  (∀ (hd : Disjoint (A i) (A j)) (ht : Codisjoint (A i) (A j)), motive ⋯) → motive x","typeReadable":"∀ {R : Type u_2} [inst : Ring R] {ι : Type u_3} {M : Type u_1} [inst_1 : AddCommGroup M] [inst_2 : Module R M]\n  (A : ι → Submodule R M) {i j : ι} (motive : IsCompl (A i) (A j) → Prop) (x : IsCompl (A i) (A j)),\n  (∀ (hd : Disjoint (A i) (A j)) (ht : Codisjoint (A i) (A j)), motive ⋯) → motive x","typeReferences":[["IsCompl"],["PartialOrder","toPreorder"],["Module"],["Submodule","instPartialOrder"],["AddCommGroup"],["IsCompl","mk"],["Ring","toSemiring"],["Submodule"],["BoundedOrder","toOrderBot"],["Codisjoint"],["BoundedOrder","toOrderTop"],["Disjoint"],["AddCommGroup","toAddCommMonoid"],["Submodule","completeLattice"],["Preorder","toLE"],["CompleteLattice","toBoundedOrder"],["Ring"]],"valueReferences":[["IsCompl","casesOn"],["Submodule","instPartialOrder"],["Submodule","completeLattice"],["AddCommGroup","toAddCommMonoid"],["CompleteLattice","toBoundedOrder"],["Ring","toSemiring"],["Submodule"]]},{"isProp":false,"kind":"definition","name":["DirectSum","component"],"typeFallback":"forall (R : Type.{u}) [inst._@.Mathlib.Algebra.DirectSum.Module.2042137856._hygCtx._hyg.3 : Semiring.{u} R] (ι : Type.{v}) (M : ι -> Type.{w}) [inst._@.Mathlib.Algebra.DirectSum.Module.2042137856._hygCtx._hyg.10 : forall (i : ι), AddCommMonoid.{w} (M i)] 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inst._@.Mathlib.Algebra.DirectSum.Module.2042137856._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2042137856._hygCtx._hyg.19) (inst._@.Mathlib.Algebra.DirectSum.Module.2042137856._hygCtx._hyg.19 i)","typeFull":"(R : Type u) →\n  [inst : Semiring R] →\n    (ι : Type v) →\n      (M : ι → Type w) →\n        [inst_1 : (i : ι) → AddCommMonoid (M i)] →\n          [inst_2 : (i : ι) → Module R (M i)] → (i : ι) → (DirectSum ι fun i => M i) →ₗ[R] M i","typeReadable":"(R : Type u) →\n  [inst : Semiring R] →\n    (ι : Type v) →\n      (M : ι → Type w) →\n        [inst_1 : (i : ι) → AddCommMonoid (M i)] →\n          [inst_2 : (i : ι) → Module R (M i)] → (i : ι) → (DirectSum ι fun i => M i) →ₗ[R] M 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(i : ι) → AddCommMonoid (M i)] →\n          [inst_2 : (i : ι) → Module R (M i)] → (DirectSum ι fun i => M i) →ₗ[R] (i : ι) → M i","typeReferences":[["DirectSum"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["instAddCommMonoidDirectSum"],["RingHom","id"],["Module"],["Pi","addCommMonoid"],["DirectSum","instModule"],["LinearMap"],["Pi","module"],["Semiring"]],"valueReferences":[["DFinsupp","coeFnLinearMap"]]},{"isProp":true,"kind":"theorem","name":["DirectSum","IsInternal","ofBijective_coeLinearMap_of_mem"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 : Semiring.{u} R] {ι : Type.{v}} [dec_ι : DecidableEq.{succ v} ι] {M : Type.{u_1}} [inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13 : Module.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10] {A : ι -> (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13)} (h : DirectSum.IsInternal.{v, u_1, u_1} ι M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13) dec_ι inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13) (Submodule.addSubmonoidClass.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13) A) {i : ι} {x : M} (hx : Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13)) (A i) x), Eq.{succ u_1} (Subtype.{succ u_1} M (fun (x : M) => 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inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13)) (A i) x)) (DFinsupp.instDFunLike.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M 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inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13 (A i))) inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13 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inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13 (A i)) (fun (i : ι) => Submodule.module.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13 (A i)))) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => DirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13 (A i))) (EquivLike.toFunLike.{max (succ u_1) (succ v), succ u_1, max (succ u_1) (succ v)} (LinearEquiv.{u, u, u_1, max u_1 v} R R inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R 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u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13 (A i))) inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 (instAddCommMonoidDirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13 (A i))) inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13 (DirectSum.instModule.{u, v, u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13 (A i)) (fun (i : ι) => Submodule.module.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13 (A i)))) M (DirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13 (A i))) (DFinsupp.instEquivLikeLinearEquiv.{u, u, u_1, max u_1 v} R R inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3)) (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3)) (RingHomInvPair.ids.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3) (RingHomInvPair.ids.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3) M (DirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13 (A i))) inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 (instAddCommMonoidDirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13 (A i))) inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13 (DirectSum.instModule.{u, v, u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13 (A i)) (fun (i : ι) => Submodule.module.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1727633501._hygCtx._hyg.13 (A i))))) (LinearEquiv.symm.{u, u, 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(Monoid.toSemigroup.{u_3} R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3))) (MulAction.toSemigroupAction.{u_3, max u_1 u_2} R (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3)) (DistribMulAction.toMulAction.{u_3, max u_1 u_2} R (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10)) (DistribMulAction.mk.{u_3, max u_1 u_2} R (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10)) (MulAction.mk.{u_3, max u_1 u_2} R (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3)) (SemigroupAction.mk.{u_3, max u_1 u_2} R (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) (Monoid.toSemigroup.{u_3} R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3))) (SMul.mk.{u_3, max u_1 u_2} R (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) (DirectSum.instModule._aux_1.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3 ι M inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.19)) (DirectSum.instModule._proof_3.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3 ι M inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.19)) (DirectSum.instModule._proof_4.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3 ι M inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.19)) (DirectSum.instModule._proof_5.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3 ι M inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.19) (DirectSum.instModule._proof_6.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3 ι M inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.19)))))) (OfNat.ofNat.{u_3} R 0 (Zero.toOfNat0.{u_3} R (MulZeroClass.toZero.{u_3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.3)))))) x) (OfNat.ofNat.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) 0 (Zero.toOfNat0.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) (AddZero.toZero.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) (AddZeroClass.toAddZero.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) (AddMonoid.toAddZeroClass.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) (AddCommMonoid.toAddMonoid.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{u_1, u_2} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.955123615._hygCtx._hyg.10)))))))","typeFull":"∀ {R : Type u_3} [inst : Semiring R] {ι : Type u_1} {M : ι → Type u_2} [inst_1 : (i : ι) → AddCommMonoid (M i)]\n  [inst_2 : (i : ι) → Module R (M i)] (x : DirectSum ι fun i => M i), 0 • x = 0","typeReadable":"∀ {R : Type u_3} [inst : Semiring R] {ι : Type u_1} {M : ι → Type u_2} [inst_1 : (i : ι) → AddCommMonoid (M i)]\n  [inst_2 : (i : ι) → Module R (M i)] (x : DirectSum ι fun i => M i), 0 • x = 0","typeReferences":[["Module"],["SemigroupAction","mk"],["DirectSum","instModule","_aux_1"],["SemigroupAction","toSMul"],["AddCommMonoid","toAddMonoid"],["DirectSum","instModule","_proof_5"],["DirectSum"],["DirectSum","instModule","_proof_4"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["instAddCommMonoidDirectSum"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["Zero","toOfNat0"],["Monoid","toSemigroup"],["Eq"],["DirectSum","instModule","_proof_3"],["DirectSum","instModule","_proof_6"],["SMul","mk"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","mk"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["HSMul","hSMul"],["AddZero","toZero"],["MulAction","toSemigroupAction"],["AddMonoid","toAddZeroClass"],["Semiring"],["MulAction","mk"]],"valueReferences":[["DFinsupp","module","_proof_4"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","DirectSum","Module",0,"DirectSum","isInternal_submodule_iff_isCompl","match_1_1"],"typeFallback":"forall {R : Type.{u_2}} [inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3 : Ring.{u_2} R] {ι : Type.{u_3}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13 : Module.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10)] (A : ι -> (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13)) {i : ι} {j : ι} (motive : (And (Disjoint.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (ConditionallyCompletePartialOrderSup.toPartialOrder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.completeLattice.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13))))) (BoundedOrder.toOrderBot.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Preorder.toLE.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (PartialOrder.toPreorder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (ConditionallyCompletePartialOrderSup.toPartialOrder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.completeLattice.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13))))))) (CompleteLattice.toBoundedOrder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.completeLattice.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13))) (A i) (A j)) (Eq.{succ u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Max.max.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (SemilatticeSup.toMax.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Lattice.toSemilatticeSup.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (CompleteLattice.toLattice.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.completeLattice.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13)))) (A i) (A j)) (Top.top.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.instTop.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13)))) -> Prop) (x._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx.188.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.196 : And (Disjoint.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (ConditionallyCompletePartialOrderSup.toPartialOrder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.completeLattice.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13))))) (BoundedOrder.toOrderBot.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Preorder.toLE.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (PartialOrder.toPreorder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (ConditionallyCompletePartialOrderSup.toPartialOrder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.completeLattice.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13))))))) (CompleteLattice.toBoundedOrder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.completeLattice.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13))) (A i) (A j)) (Eq.{succ u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Max.max.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (SemilatticeSup.toMax.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Lattice.toSemilatticeSup.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (CompleteLattice.toLattice.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.completeLattice.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13)))) (A i) (A j)) (Top.top.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.instTop.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13)))), (forall (hd : Disjoint.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (ConditionallyCompletePartialOrderSup.toPartialOrder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.completeLattice.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13))))) (BoundedOrder.toOrderBot.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Preorder.toLE.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (PartialOrder.toPreorder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (ConditionallyCompletePartialOrderSup.toPartialOrder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.completeLattice.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13))))))) (CompleteLattice.toBoundedOrder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.completeLattice.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13))) (A i) (A j)) (ht : Eq.{succ u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Max.max.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (SemilatticeSup.toMax.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Lattice.toSemilatticeSup.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (CompleteLattice.toLattice.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.completeLattice.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13)))) (A i) (A j)) (Top.top.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.instTop.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13))), motive (And.intro (Disjoint.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (ConditionallyCompletePartialOrderSup.toPartialOrder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.completeLattice.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13))))) (BoundedOrder.toOrderBot.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Preorder.toLE.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (PartialOrder.toPreorder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (ConditionallyCompletePartialOrderSup.toPartialOrder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.completeLattice.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13))))))) (CompleteLattice.toBoundedOrder.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.completeLattice.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13))) (A i) (A j)) (Eq.{succ u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Max.max.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (SemilatticeSup.toMax.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Lattice.toSemilatticeSup.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (CompleteLattice.toLattice.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.completeLattice.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13)))) (A i) (A j)) (Top.top.{u_1} (Submodule.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13) (Submodule.instTop.{u_2, u_1} R M (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.13))) hd ht)) -> (motive x._@.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx.188.Mathlib.Algebra.DirectSum.Module.529924417._hygCtx._hyg.196)","typeFull":"∀ {R : Type u_2} [inst : Ring R] {ι : Type u_3} {M : Type u_1} [inst_1 : AddCommGroup M] [inst_2 : Module R M]\n  (A : ι → Submodule R M) {i j : ι} (motive : Disjoint (A i) (A j) ∧ A i ⊔ A j = ⊤ → Prop)\n  (x : Disjoint (A i) (A j) ∧ A i ⊔ A j = ⊤), (∀ (hd : Disjoint (A i) (A j)) (ht : A i ⊔ A j = ⊤), motive ⋯) → motive x","typeReadable":"∀ {R : Type u_2} [inst : Ring R] {ι : Type u_3} {M : Type u_1} [inst_1 : AddCommGroup M] [inst_2 : Module R M]\n  (A : ι → Submodule R M) {i j : ι} (motive : Disjoint (A i) (A j) ∧ A i ⊔ A j = ⊤ → Prop)\n  (x : Disjoint (A i) (A j) ∧ A i ⊔ A j = ⊤), (∀ (hd : Disjoint (A i) (A j)) (ht : A i ⊔ A j = ⊤), motive ⋯) → motive x","typeReferences":[["Lattice","toSemilatticeSup"],["PartialOrder","toPreorder"],["Module"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["Submodule"],["ConditionallyCompletePartialOrderSup","toPartialOrder"],["And","intro"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderSup"],["Disjoint"],["Preorder","toLE"],["Eq"],["CompleteLattice","toConditionallyCompleteLattice"],["CompleteLattice","toLattice"],["Submodule","instTop"],["And"],["AddCommGroup"],["Ring","toSemiring"],["Max","max"],["BoundedOrder","toOrderBot"],["SemilatticeSup","toMax"],["Submodule","completeLattice"],["AddCommGroup","toAddCommMonoid"],["Top","top"],["CompleteLattice","toBoundedOrder"],["Ring"]],"valueReferences":[["Lattice","toSemilatticeSup"],["PartialOrder","toPreorder"],["CompleteLattice","toLattice"],["Submodule","instTop"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["Submodule"],["Ring","toSemiring"],["ConditionallyCompletePartialOrderSup","toPartialOrder"],["Max","max"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderSup"],["BoundedOrder","toOrderBot"],["Disjoint"],["SemilatticeSup","toMax"],["Top","top"],["AddCommGroup","toAddCommMonoid"],["Submodule","completeLattice"],["Eq"],["Preorder","toLE"],["CompleteLattice","toBoundedOrder"],["CompleteLattice","toConditionallyCompleteLattice"],["And","casesOn"]]},{"isProp":false,"kind":"definition","name":["DirectSum","sigmaLcurry"],"typeFallback":"forall (R : Type.{u}) [inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.3 : Semiring.{u} R] {ι : Type.{v}} {α : ι -> Type.{u_1}} {δ : forall (i : ι), (α i) -> Type.{w}} [inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.39 : DecidableEq.{succ v} ι] [inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 : forall (i : ι) (j : α i), AddCommMonoid.{w} (δ i j)] [inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.53 : forall (i : ι) (j : α i), Module.{u, w} R (δ i j) inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.3 (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i j)], LinearMap.{u, u, max w u_1 v, max (max w u_1) v} R R inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.3 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.3)) (DirectSum.{max u_1 v, w} (Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) (fun (i : Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => δ (Sigma.fst.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)) (fun (i : Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i))) (DirectSum.{v, max w u_1} ι (fun (i : ι) => DirectSum.{u_1, w} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, w} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))) (instAddCommMonoidDirectSum.{max u_1 v, w} (Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) (fun (i : Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => δ (Sigma.fst.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)) (fun (i : Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i))) (instAddCommMonoidDirectSum.{v, max u_1 w} ι (fun (i : ι) => DirectSum.{u_1, w} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, w} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i))) (DirectSum.instModule.{u, max u_1 v, w} R inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.3 (Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) (fun (i : Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => δ (Sigma.fst.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)) (fun (i : Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 (Sigma.fst.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)) (fun (i : Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.53 (Sigma.fst.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i))) (DirectSum.instModule.{u, v, max u_1 w} R inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.3 ι (fun (i : ι) => DirectSum.{u_1, w} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, w} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i)) (fun (i : ι) => DirectSum.instModule.{u, u_1, w} R inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.3 (α i) (δ i) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.42 i) (inst._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.53 i)))","typeFull":"(R : Type u) →\n  [inst : Semiring R] →\n    {ι : Type v} →\n      {α : ι → Type u_1} →\n        {δ : (i : ι) → α i → Type w} →\n          [DecidableEq ι] →\n            [inst_2 : (i : ι) → (j : α i) → AddCommMonoid (δ i j)] →\n              [inst_3 : (i : ι) → (j : α i) → Module R (δ i j)] →\n                (DirectSum ((x : ι) × α x) fun i => δ i.fst i.snd) →ₗ[R]\n                  DirectSum ι fun i => DirectSum (α i) fun j => δ i j","typeReadable":"(R : Type u) →\n  [inst : Semiring R] →\n    {ι : Type v} →\n      {α : ι → Type u_1} →\n        {δ : (i : ι) → α i → Type w} →\n          [DecidableEq ι] →\n            [inst_2 : (i : ι) → (j : α i) → AddCommMonoid (δ i j)] →\n              [inst_3 : (i : ι) → (j : α i) → Module R (δ i j)] →\n                (DirectSum ((x : ι) × α x) fun i => δ i.fst i.snd) →ₗ[R]\n                  DirectSum ι fun i => DirectSum (α i) fun j => δ i 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inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13 (A i))) M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 (instAddCommMonoidDirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 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inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13 (A i)) (fun (i : ι) => Submodule.module.{u, u_1} R M 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inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13 (A i))) (AddZeroClass.toAddZero.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13)) (A i) x)) (AddMonoid.toAddZeroClass.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13)) (A i) x)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13)) (A i) x)) (Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13 (A i))))) (AddZeroClass.toAddZero.{max v u_1} (DirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13 (A i))) (AddMonoid.toAddZeroClass.{max v u_1} (DirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13 (A i))) (AddCommMonoid.toAddMonoid.{max v u_1} (DirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13 (A i))) (instAddCommMonoidDirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13 (A i))))))) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13)) (A i) x)) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13)) (A i) x)) => DirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13 (A i))) (AddMonoidHom.instFunLike.{u_1, max u_1 v} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13)) (A i) x)) (DirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13 (A i))) (AddZeroClass.toAddZero.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13)) (A i) x)) (AddMonoid.toAddZeroClass.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13)) (A i) x)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13)) (A i) x)) (Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3597860278._hygCtx._hyg.13 (A i))))) (AddZeroClass.toAddZero.{max v u_1} (DirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun 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b","typeReferences":[["MulOneClass","toMulOne"],["SMul","mk"],["MulOne","toOne"],["Module"],["SemigroupAction","mk"],["DirectSum","instModule","_aux_1"],["SemigroupAction","toSMul"],["Semiring","toMonoidWithZero"],["OfNat","ofNat"],["DirectSum"],["AddCommMonoid"],["One","toOfNat1"],["Monoid","toMulOneClass"],["HSMul","hSMul"],["MonoidWithZero","toMonoid"],["instHSMul"],["Monoid","toSemigroup"],["Eq"],["DirectSum","instModule","_proof_3"],["Semiring"]],"valueReferences":[["Function","Injective","mulAction"],["DFinsupp"],["Pi","addMonoid"],["SMulZeroClass","toSMul"],["DFinsupp","distribMulAction","_proof_2"],["AddCommMonoid","toAddMonoid"],["DFinsupp","instAddMonoid"],["DFunLike","coe"],["DFinsupp","coeFnAddMonoidHom"],["AddMonoidHom"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["AddMonoidHom","instFunLike"],["DistribSMul","toSMulZeroClass"],["Pi","distribMulAction"],["DFinsupp","instSMulZeroClass"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["MulAction","one_smul"],["DFinsupp","distribMulAction","_proof_1"],["DFinsupp","instZero"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","DirectSum","Module",0,"DirectSum","isInternal_ne_bot_iff","_simp_1_1"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.3 : Ring.{u} R] {ι : Type.{v}} [dec_ι : DecidableEq.{succ v} ι] {M : Type.{u_1}} [inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.10 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.13 : Module.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.10)] (A : ι -> (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.13)), Eq.{1} Prop (DirectSum.IsInternal.{v, u_1, u_1} ι M (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.13) dec_ι (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.10) (Submodule.setLike.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.13) (Submodule.addSubmonoidClass.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.13) A) (And (iSupIndep.{succ v, u_1} ι (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.13) A) (Eq.{succ u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.13) (iSup.{u_1, succ v} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.13) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.13))))) A) (Top.top.{u_1} (Submodule.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.13) (Submodule.instTop.{u, u_1} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.749824039._hygCtx._hyg.13))))","typeFull":"∀ {R : Type u} [inst : Ring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [inst_1 : AddCommGroup M]\n  [inst_2 : Module R M] (A : ι → Submodule R M), DirectSum.IsInternal A = (iSupIndep A ∧ iSup A = ⊤)","typeReadable":"∀ {R : Type u} [inst : Ring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [inst_1 : AddCommGroup M]\n  [inst_2 : Module R M] (A : ι → Submodule R M), DirectSum.IsInternal A = (iSupIndep A ∧ iSup A = 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{ι : Type.{u_1}} {N : ι -> Type.{u_2}} [inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10 : forall (i : ι), AddCommMonoid.{u_2} (N i)] {P : ι -> Type.{u_3}} [inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30 : forall (i : ι), AddCommMonoid.{u_3} (P i)] (u : forall (i : ι), AddEquiv.{u_2, u_3} (N i) (P i) (AddCommMagma.toAdd.{u_2} (N i) (AddCommSemigroup.toAddCommMagma.{u_2} (N i) (AddCommMonoid.toAddCommSemigroup.{u_2} (N i) (inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10 i)))) (AddCommMagma.toAdd.{u_3} (P i) (AddCommSemigroup.toAddCommMagma.{u_3} (P i) (AddCommMonoid.toAddCommSemigroup.{u_3} (P i) (inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30 i))))) (x : DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (y : DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10), Eq.{succ (max u_1 u_3)} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddHom.toFun.{max u_1 u_2, max u_1 u_3} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddCommMagma.toAdd.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (AddCommSemigroup.toAddCommMagma.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (AddCommMonoid.toAddCommSemigroup.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10)))) 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inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (AddMonoid.toAddZeroClass.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (AddCommMonoid.toAddMonoid.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10)))) (AddZeroClass.toAddZero.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddMonoid.toAddZeroClass.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddCommMonoid.toAddMonoid.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (instAddCommMonoidDirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30))))) (DirectSum.congrAddEquiv._proof_1.{u_1, u_2, u_3} ι N inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10 P inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (DirectSum.map.{u_1, u_2, u_3} ι N P inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30 (fun (i : ι) => AddEquiv.toAddMonoidHom.{u_2, u_3} (N i) (P i) (AddMonoid.toAddZeroClass.{u_2} (N i) (AddCommMonoid.toAddMonoid.{u_2} (N i) (inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10 i))) (AddMonoid.toAddZeroClass.{u_3} (P i) (AddCommMonoid.toAddMonoid.{u_3} (P i) (inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30 i))) (u i)))) (HAdd.hAdd.{max u_1 u_2, max u_1 u_2, max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (instHAdd.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (AddCommMagma.toAdd.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (AddCommSemigroup.toAddCommMagma.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (AddCommMonoid.toAddCommSemigroup.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10))))) x y)) (HAdd.hAdd.{max u_1 u_3, max u_1 u_3, max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (instHAdd.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddCommMagma.toAdd.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddCommSemigroup.toAddCommMagma.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddCommMonoid.toAddCommSemigroup.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (instAddCommMonoidDirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30))))) (AddHom.toFun.{max u_1 u_2, max u_1 u_3} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddCommMagma.toAdd.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (AddCommSemigroup.toAddCommMagma.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (AddCommMonoid.toAddCommSemigroup.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10)))) (AddCommMagma.toAdd.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddCommSemigroup.toAddCommMagma.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddCommMonoid.toAddCommSemigroup.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (instAddCommMonoidDirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30)))) (AddHomClass.toAddHom.{max u_1 u_2, max u_1 u_3, max (max u_1 u_2) u_3} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddMonoidHom.{max u_2 u_1, max u_3 u_1} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddZeroClass.toAddZero.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (AddMonoid.toAddZeroClass.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (AddCommMonoid.toAddMonoid.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10)))) (AddZeroClass.toAddZero.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddMonoid.toAddZeroClass.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddCommMonoid.toAddMonoid.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (instAddCommMonoidDirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30))))) (AddCommMagma.toAdd.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (AddCommSemigroup.toAddCommMagma.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (AddCommMonoid.toAddCommSemigroup.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10)))) (AddCommMagma.toAdd.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddCommSemigroup.toAddCommMagma.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddCommMonoid.toAddCommSemigroup.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (instAddCommMonoidDirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30)))) (AddMonoidHom.instFunLike.{max u_1 u_2, max u_1 u_3} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddZeroClass.toAddZero.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (AddMonoid.toAddZeroClass.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (AddCommMonoid.toAddMonoid.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10)))) (AddZeroClass.toAddZero.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddMonoid.toAddZeroClass.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddCommMonoid.toAddMonoid.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (instAddCommMonoidDirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30))))) (DirectSum.congrAddEquiv._proof_1.{u_1, u_2, u_3} ι N inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10 P inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (DirectSum.map.{u_1, u_2, u_3} ι N P inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30 (fun (i : ι) => AddEquiv.toAddMonoidHom.{u_2, u_3} (N i) (P i) (AddMonoid.toAddZeroClass.{u_2} (N i) (AddCommMonoid.toAddMonoid.{u_2} (N i) (inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10 i))) (AddMonoid.toAddZeroClass.{u_3} (P i) (AddCommMonoid.toAddMonoid.{u_3} (P i) (inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30 i))) (u i)))) x) (AddHom.toFun.{max u_1 u_2, max u_1 u_3} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddCommMagma.toAdd.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (AddCommSemigroup.toAddCommMagma.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (AddCommMonoid.toAddCommSemigroup.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10)))) (AddCommMagma.toAdd.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddCommSemigroup.toAddCommMagma.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddCommMonoid.toAddCommSemigroup.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (instAddCommMonoidDirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30)))) (AddHomClass.toAddHom.{max u_1 u_2, max u_1 u_3, max (max u_1 u_2) u_3} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddMonoidHom.{max u_2 u_1, max u_3 u_1} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddZeroClass.toAddZero.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (AddMonoid.toAddZeroClass.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (AddCommMonoid.toAddMonoid.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10)))) (AddZeroClass.toAddZero.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddMonoid.toAddZeroClass.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddCommMonoid.toAddMonoid.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (instAddCommMonoidDirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30))))) (AddCommMagma.toAdd.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (AddCommSemigroup.toAddCommMagma.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (AddCommMonoid.toAddCommSemigroup.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10)))) (AddCommMagma.toAdd.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddCommSemigroup.toAddCommMagma.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddCommMonoid.toAddCommSemigroup.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (instAddCommMonoidDirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30)))) (AddMonoidHom.instFunLike.{max u_1 u_2, max u_1 u_3} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddZeroClass.toAddZero.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (AddMonoid.toAddZeroClass.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (AddCommMonoid.toAddMonoid.{max u_1 u_2} (DirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{u_1, u_2} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10)))) (AddZeroClass.toAddZero.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddMonoid.toAddZeroClass.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (AddCommMonoid.toAddMonoid.{max u_1 u_3} (DirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (instAddCommMonoidDirectSum.{u_1, u_3} ι (fun (i : ι) => P i) inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30))))) (DirectSum.congrAddEquiv._proof_1.{u_1, u_2, u_3} ι N inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10 P inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30) (DirectSum.map.{u_1, u_2, u_3} ι N P inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30 (fun (i : ι) => AddEquiv.toAddMonoidHom.{u_2, u_3} (N i) (P i) (AddMonoid.toAddZeroClass.{u_2} (N i) (AddCommMonoid.toAddMonoid.{u_2} (N i) (inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.10 i))) (AddMonoid.toAddZeroClass.{u_3} (P i) (AddCommMonoid.toAddMonoid.{u_3} (P i) (inst._@.Mathlib.Algebra.DirectSum.Module.3390033309._hygCtx._hyg.30 i))) (u i)))) y))","typeFull":"∀ {ι : Type u_1} {N : ι → Type u_2} [inst : (i : ι) → AddCommMonoid (N i)] {P : ι → Type u_3}\n  [inst_1 : (i : ι) → AddCommMonoid (P i)] (u : (i : ι) → N i ≃+ P i) (x y : DirectSum ι fun i => N i),\n  (↑(DirectSum.map fun i => (u i).toAddMonoidHom)).toFun (x + y) =\n    (↑(DirectSum.map fun i => (u i).toAddMonoidHom)).toFun x + (↑(DirectSum.map fun i => (u i).toAddMonoidHom)).toFun y","typeReadable":"∀ {ι : Type u_1} {N : ι → Type u_2} [inst : (i : ι) → AddCommMonoid (N i)] {P : ι → Type u_3}\n  [inst_1 : (i : ι) → AddCommMonoid (P i)] (u : (i : ι) → N i ≃+ P i) (x y : DirectSum ι fun i => N i),\n  (↑(DirectSum.map fun i => (u i).toAddMonoidHom)).toFun (x + y) =\n    (↑(DirectSum.map fun i => (u i).toAddMonoidHom)).toFun x + (↑(DirectSum.map fun i => (u i).toAddMonoidHom)).toFun y","typeReferences":[["DirectSum","map"],["instHAdd"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["DirectSum","congrAddEquiv","_proof_1"],["AddEquiv"],["HAdd","hAdd"],["DirectSum"],["AddCommMonoid"],["AddHomClass","toAddHom"],["AddCommMonoid","toAddCommSemigroup"],["instAddCommMonoidDirectSum"],["AddEquiv","toAddMonoidHom"],["AddMonoidHom"],["AddMonoidHom","instFunLike"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["AddHom","toFun"],["Eq"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["DirectSum","map"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["DirectSum","congrAddEquiv","_proof_1"],["DirectSum"],["AddHom","map_add'"],["AddHomClass","toAddHom"],["AddCommMonoid","toAddCommSemigroup"],["instAddCommMonoidDirectSum"],["AddEquiv","toAddMonoidHom"],["AddMonoidHom"],["AddMonoidHom","instFunLike"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["DirectSum","lequivProdDirectSum","_proof_1"],"typeFallback":"forall 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inst._@.Mathlib.Algebra.DirectSum.Module.3212809767._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3212809767._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3212809767._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.3212809767._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3212809767._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.3212809767._hygCtx._hyg.13)) (A i) x)) (Finsupp.{u_2, u} (α i) R (MulZeroClass.toZero.{u} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.3212809767._hygCtx._hyg.3))))) (DFinsupp.instEquivLikeLinearEquiv.{u, u, u_1, max u u_2} R R inst._@.Mathlib.Algebra.DirectSum.Module.3212809767._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.3212809767._hygCtx._hyg.3 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R 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[inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] {A : ι → Submodule R M} (h : DirectSum.IsInternal A) {α : ι → Type u_2}\n  (v : (i : ι) → Module.Basis (α i) R ↥(A i)) {x : M} {i : ι} {a : α i} (hx : x ∈ A i),\n  ((h.collectedBasis v).repr x) ⟨i, a⟩ = ((v i).repr ⟨x, hx⟩) a","typeReadable":"∀ {R : Type u} [inst : Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] {A : ι → Submodule R M} (h : DirectSum.IsInternal A) {α : ι → Type u_2}\n  (v : (i : ι) → Module.Basis (α i) R ↥(A i)) {x : M} {i : ι} {a : α i} (hx : x ∈ A i),\n  ((h.collectedBasis v).repr x) ⟨i, a⟩ = ((v i).repr ⟨x, hx⟩) 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{ι : Type.{v}} {M : ι -> Type.{w}} [inst._@.Mathlib.Algebra.DirectSum.Module.3681857551._hygCtx._hyg.10 : forall (i : ι), AddCommMonoid.{w} (M i)] {N : ι -> Type.{u_1}} [inst._@.Mathlib.Algebra.DirectSum.Module.3681857551._hygCtx._hyg.32 : forall (i : ι), AddCommMonoid.{u_1} (N i)] (f : forall (i : ι), AddMonoidHom.{w, u_1} (M i) (N i) (AddZeroClass.toAddZero.{w} (M i) (AddMonoid.toAddZeroClass.{w} (M i) (AddCommMonoid.toAddMonoid.{w} (M i) (inst._@.Mathlib.Algebra.DirectSum.Module.3681857551._hygCtx._hyg.10 i)))) (AddZeroClass.toAddZero.{u_1} (N i) (AddMonoid.toAddZeroClass.{u_1} (N i) (AddCommMonoid.toAddMonoid.{u_1} (N i) (inst._@.Mathlib.Algebra.DirectSum.Module.3681857551._hygCtx._hyg.32 i))))), Eq.{max (succ u_1) (succ v)} (AddSubmonoid.{max u_1 v} (DirectSum.{v, u_1} ι (fun (i : ι) => N i) inst._@.Mathlib.Algebra.DirectSum.Module.3681857551._hygCtx._hyg.32) (AddMonoid.toAddZeroClass.{max u_1 v} (DirectSum.{v, u_1} ι (fun (i : ι) => N i) 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AddMonoidHom.mrange (DirectSum.map f) =\n    AddSubmonoid.comap (DirectSum.coeFnAddMonoidHom N) (AddSubmonoid.pi Set.univ fun i => AddMonoidHom.mrange (f i))","typeReadable":"∀ {ι : Type v} {M : ι → Type w} [inst : (i : ι) → AddCommMonoid (M i)] {N : ι → Type u_1}\n  [inst_1 : (i : ι) → AddCommMonoid (N i)] (f : (i : ι) → M i →+ N i),\n  AddMonoidHom.mrange (DirectSum.map f) =\n    AddSubmonoid.comap (DirectSum.coeFnAddMonoidHom N) (AddSubmonoid.pi Set.univ fun i => AddMonoidHom.mrange (f 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.3 : Semiring.{u_1} R] {ι : Type.{u_4}} {M : ι -> Type.{u_3}} [inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10 : forall (i : ι), AddCommMonoid.{u_3} (M i)] [inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.19 : forall (i : ι), Module.{u_1, u_3} R (M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.3 (inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10 i)] {S : Type.{u_2}} [inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.30 : Semiring.{u_2} S] [inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.33 : SMul.{u_1, u_2} R S] [inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.37 : forall (i : ι), Module.{u_2, u_3} S (M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.30 (inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10 i)] [inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.47 : forall (i : ι), IsScalarTower.{u_1, u_2, u_3} R S (M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.33 (SMulZeroClass.toSMul.{u_2, u_3} S (M i) (AddZero.toZero.{u_3} (M i) (AddZeroClass.toAddZero.{u_3} (M i) (AddMonoid.toAddZeroClass.{u_3} (M i) (AddCommMonoid.toAddMonoid.{u_3} (M i) (inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10 i))))) (DistribSMul.toSMulZeroClass.{u_2, u_3} S (M i) (AddMonoid.toAddZeroClass.{u_3} (M i) (AddCommMonoid.toAddMonoid.{u_3} (M i) (inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10 i))) (DistribMulAction.toDistribSMul.{u_2, u_3} S (M i) (MonoidWithZero.toMonoid.{u_2} S (Semiring.toMonoidWithZero.{u_2} S inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.30)) (AddCommMonoid.toAddMonoid.{u_3} (M i) (inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10 i)) (Module.toDistribMulAction.{u_2, u_3} S (M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.30 (inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10 i) (inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.37 i))))) (SMulZeroClass.toSMul.{u_1, u_3} R (M i) (AddZero.toZero.{u_3} (M i) (AddZeroClass.toAddZero.{u_3} (M i) (AddMonoid.toAddZeroClass.{u_3} (M i) (AddCommMonoid.toAddMonoid.{u_3} (M i) (inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10 i))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R (M i) (AddMonoid.toAddZeroClass.{u_3} (M i) (AddCommMonoid.toAddMonoid.{u_3} (M i) (inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10 i))) (DistribMulAction.toDistribSMul.{u_1, u_3} R (M i) (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_3} (M i) (inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10 i)) (Module.toDistribMulAction.{u_1, u_3} R (M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.3 (inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10 i) (inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.19 i)))))], IsScalarTower.{u_1, u_2, max u_3 u_4} R S (DirectSum.{u_4, u_3} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.33 (SMulZeroClass.toSMul.{u_2, max u_4 u_3} S (DirectSum.{u_4, u_3} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (AddZero.toZero.{max u_4 u_3} (DirectSum.{u_4, u_3} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (AddZeroClass.toAddZero.{max u_4 u_3} (DirectSum.{u_4, u_3} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (AddMonoid.toAddZeroClass.{max u_4 u_3} (DirectSum.{u_4, u_3} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (AddCommMonoid.toAddMonoid.{max u_4 u_3} (DirectSum.{u_4, u_3} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{u_4, u_3} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10))))) (DistribSMul.toSMulZeroClass.{u_2, max u_4 u_3} S (DirectSum.{u_4, u_3} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (AddMonoid.toAddZeroClass.{max u_4 u_3} (DirectSum.{u_4, u_3} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (AddCommMonoid.toAddMonoid.{max u_4 u_3} (DirectSum.{u_4, u_3} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{u_4, u_3} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10))) (DistribMulAction.toDistribSMul.{u_2, max u_4 u_3} S (DirectSum.{u_4, u_3} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (MonoidWithZero.toMonoid.{u_2} S (Semiring.toMonoidWithZero.{u_2} S inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.30)) (AddCommMonoid.toAddMonoid.{max u_4 u_3} (DirectSum.{u_4, u_3} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{u_4, u_3} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10)) (Module.toDistribMulAction.{u_2, max u_4 u_3} S (DirectSum.{u_4, u_3} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.30 (instAddCommMonoidDirectSum.{u_4, u_3} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (DirectSum.instModule.{u_2, u_4, u_3} S inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.30 ι M inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.37))))) (SMulZeroClass.toSMul.{u_1, max u_4 u_3} R (DirectSum.{u_4, u_3} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (AddZero.toZero.{max u_4 u_3} (DirectSum.{u_4, u_3} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (AddZeroClass.toAddZero.{max u_4 u_3} (DirectSum.{u_4, u_3} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (AddMonoid.toAddZeroClass.{max u_4 u_3} (DirectSum.{u_4, u_3} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (AddCommMonoid.toAddMonoid.{max u_4 u_3} (DirectSum.{u_4, u_3} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.772748180._hygCtx._hyg.10) (instAddCommMonoidDirectSum.{u_4, 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Semiring R] {ι : Type u_4} {M : ι → Type u_3} [inst_1 : (i : ι) → AddCommMonoid (M i)]\n  [inst_2 : (i : ι) → Module R (M i)] {S : Type u_2} [inst_3 : Semiring S] [inst_4 : SMul R S]\n  [inst_5 : (i : ι) → Module S (M i)] [∀ (i : ι), IsScalarTower R S (M i)], IsScalarTower R S (DirectSum ι fun i => M i)","typeReadable":"∀ {R : Type u_1} [inst : Semiring R] {ι : Type u_4} {M : ι → Type u_3} [inst_1 : (i : ι) → AddCommMonoid (M i)]\n  [inst_2 : (i : ι) → Module R (M i)] {S : Type u_2} [inst_3 : Semiring S] [inst_4 : SMul R S]\n  [inst_5 : (i : ι) → Module S (M i)] [∀ (i : ι), IsScalarTower R S (M i)], IsScalarTower R S (DirectSum ι fun i => M i)","typeReferences":[["Module"],["IsScalarTower"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["DirectSum"],["Module","toDistribMulAction"],["AddCommMonoid"],["SMul"],["instAddCommMonoidDirectSum"],["MonoidWithZero","toMonoid"],["DirectSum","instModule"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["Module","toDistribMulAction"],["MonoidWithZero","toMonoid"],["DistribMulAction","toDistribSMul"],["DFinsupp","isScalarTower"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["DirectSum","IsInternal","isCompl"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.3 : Semiring.{u} R] {ι : Type.{v}} [dec_ι : DecidableEq.{succ v} ι] {M : Type.{u_1}} [inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.13 : Module.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.10] {A : ι -> (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.13)} {i : ι} {j : ι}, (Ne.{succ v} ι i j) -> (Eq.{succ v} (Set.{v} ι) (Set.univ.{v} ι) (Insert.insert.{v, v} ι (Set.{v} ι) (Set.instInsert.{v} ι) i (Singleton.singleton.{v, v} ι (Set.{v} ι) (Set.instSingletonSet.{v} ι) j))) -> (DirectSum.IsInternal.{v, u_1, u_1} ι M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.13) dec_ι inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.10 (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.13) (Submodule.addSubmonoidClass.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.13) A) -> (IsCompl.{u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.13) (Submodule.instPartialOrder.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.13) (CompleteLattice.toBoundedOrder.{u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1865590668._hygCtx._hyg.13)) (A i) (A j))","typeFull":"∀ {R : Type u} [inst : Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] {A : ι → Submodule R M} {i j : ι},\n  i ≠ j → Set.univ = {i, j} → DirectSum.IsInternal A → IsCompl (A i) (A j)","typeReadable":"∀ {R : Type u} [inst : Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] {A : ι → Submodule R M} {i j : ι},\n  i ≠ j → Set.univ = {i, j} → DirectSum.IsInternal A → IsCompl (A i) (A j)","typeReferences":[["DirectSum","IsInternal"],["IsCompl"],["Module"],["Set"],["DecidableEq"],["Singleton","singleton"],["Submodule","instPartialOrder"],["Insert","insert"],["Set","instInsert"],["Set","instSingletonSet"],["Submodule"],["Set","univ"],["AddCommMonoid"],["Submodule","setLike"],["Submodule","completeLattice"],["Ne"],["Eq"],["CompleteLattice","toBoundedOrder"],["Submodule","addSubmonoidClass"],["Semiring"]],"valueReferences":[["Lattice","toSemilatticeSup"],["Set","image_insert_eq"],["PartialOrder","toPreorder"],["Eq","trans"],["Singleton","singleton"],["Set","instInsert"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderSup"],["iSup"],["ConditionallyCompletePartialOrderSup","toSupSet"],["SupSet","sSup"],["Eq","symm"],["sSup_pair"],["CompleteLattice","toLattice"],["Insert","insert"],["IsCompl","mk"],["Set","image_univ"],["Set","image"],["Eq","refl"],["Iff","mpr"],["Top","top"],["id"],["DirectSum","IsInternal","submodule_iSup_eq_top"],["Eq","mpr"],["ConditionallyCompleteLattice","toLattice"],["CompleteLattice","toBoundedOrder"],["Set","image_singleton"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["SemilatticeSup","toPartialOrder"],["iSupIndep","pairwiseDisjoint"],["CompleteSemilatticeSup","toSupSet"],["CompleteLattice","toCompleteSemilatticeSup"],["Set","range"],["Submodule"],["congrArg"],["Codisjoint"],["BoundedOrder","toOrderTop"],["Eq"],["Preorder","toLE"],["CompleteLattice","toConditionallyCompleteLattice"],["Set"],["codisjoint_iff"],["Submodule","instTop"],["Submodule","instPartialOrder"],["Set","instSingletonSet"],["iSup","eq_1"],["Set","univ"],["Max","max"],["SemilatticeSup","toMax"],["Submodule","completeLattice"],["DirectSum","IsInternal","submodule_iSupIndep"],["OrderTop","toTop"]]},{"isProp":true,"kind":"theorem","name":["DirectSum","IsInternal","ofBijective_coeLinearMap_of_ne"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 : Semiring.{u} R] {ι : Type.{v}} [dec_ι : DecidableEq.{succ v} ι] {M : Type.{u_1}} [inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13 : Module.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10] {A : ι -> (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13)} (h : DirectSum.IsInternal.{v, u_1, u_1} ι M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13) dec_ι inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13) (Submodule.addSubmonoidClass.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13) A) {i : ι} {j : ι}, (Ne.{succ v} ι i j) -> (forall (x : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13)) (A i) x)), Eq.{succ u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13)) (A j) x)) (DFunLike.coe.{max (succ v) (succ u_1), succ v, succ u_1} (DFinsupp.{v, u_1} ι (fun (i : ι) => (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13)) (A i) x)) i) (fun (i : ι) => AddZero.toZero.{u_1} ((fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13)) (A i) x)) i) (AddZeroClass.toAddZero.{u_1} ((fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13)) (A i) x)) i) (AddMonoid.toAddZeroClass.{u_1} ((fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13)) (A i) x)) i) (AddCommMonoid.toAddMonoid.{u_1} ((fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13)) (A i) x)) i) ((fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13 (A i)) i)))))) ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13)) (A i) x)) (DFinsupp.instDFunLike.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => AddZero.toZero.{u_1} ((fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13)) (A i) x)) i) (AddZeroClass.toAddZero.{u_1} ((fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 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(Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13)) (A i) x)) i) (AddCommMonoid.toAddMonoid.{u_1} ((fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.750442154._hygCtx._hyg.3 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(inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10 i) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.33 (inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.19 i) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.36 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3)))) (LinearMap.addCommMonoid.{u, u, max v w, u₁} R R (DFinsupp.{v, w} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{w} (M i) (AddZeroClass.toAddZero.{w} (M i) (AddMonoid.toAddZeroClass.{w} (M i) (AddCommMonoid.toAddMonoid.{w} (M i) (inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10 i)))))) N inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 (DFinsupp.addCommMonoid.{v, w} ι M inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.33 (DFinsupp.module.{v, w, u} ι R M inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.19) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.36 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3))) (Pi.module.{v, max w u₁, 0} ι (fun (i : ι) => LinearMap.{u, u, w, u₁} R R inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3)) (M i) N (inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10 i) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.33 (inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.19 i) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.36) Nat Nat.instSemiring (fun (i : ι) => LinearMap.addCommMonoid.{u, u, w, u₁} R R (M i) N inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 (inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10 i) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.33 (inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.19 i) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.36 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3))) (fun (i : ι) => LinearMap.module.{u, u, 0, w, u₁} R R Nat (M i) N inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 (inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10 i) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.33 (inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.19 i) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.36 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3)) Nat.instSemiring (AddCommMonoid.toNatModule.{u₁} N inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.33) (DirectSum.toModule._proof_1.{u, u₁} R inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 N inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.33 inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.36))) (LinearMap.module.{u, u, 0, max v w, u₁} R R Nat (DFinsupp.{v, w} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{w} (M i) (AddZeroClass.toAddZero.{w} (M i) (AddMonoid.toAddZeroClass.{w} (M i) (AddCommMonoid.toAddMonoid.{w} (M i) (inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10 i)))))) N inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 (DFinsupp.addCommMonoid.{v, w} ι M inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.33 (DFinsupp.module.{v, w, u} ι R M inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.19) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.36 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3)) Nat.instSemiring (AddCommMonoid.toNatModule.{u₁} N inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.33) (DirectSum.toModule._proof_1.{u, u₁} R inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 N inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.33 inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.36))) (forall (i : ι), LinearMap.{u, u, w, u₁} R R inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3)) (M i) N (inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10 i) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.33 (inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.19 i) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.36) (LinearMap.{u, u, max w v, u₁} R R inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3)) (DFinsupp.{v, w} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{w} (M i) (AddZeroClass.toAddZero.{w} (M i) (AddMonoid.toAddZeroClass.{w} (M i) (AddCommMonoid.toAddMonoid.{w} (M i) (inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10 i)))))) N (DFinsupp.addCommMonoid.{v, w} ι M inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.33 (DFinsupp.module.{v, w, u} ι R M inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.19) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.36) (DFinsupp.instEquivLikeLinearEquiv.{0, 0, max (max u₁ w) v, max (max u₁ w) v} Nat Nat Nat.instSemiring Nat.instSemiring (RingHom.id.{0} Nat 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(Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3)) (DFinsupp.{v, w} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{w} (M i) (AddZeroClass.toAddZero.{w} (M i) (AddMonoid.toAddZeroClass.{w} (M i) (AddCommMonoid.toAddMonoid.{w} (M i) (inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10 i)))))) N (DFinsupp.addCommMonoid.{v, w} ι M inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.33 (DFinsupp.module.{v, w, u} ι R M inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.19) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.36) (Pi.addCommMonoid.{v, max w u₁} ι (fun (i : ι) => LinearMap.{u, u, w, u₁} R R inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3)) (M i) N (inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10 i) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.33 (inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.19 i) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.36) (fun (i : ι) => LinearMap.addCommMonoid.{u, u, w, u₁} R R (M i) N inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 (inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10 i) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.33 (inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.19 i) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.36 (RingHom.id.{u} R 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inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.36 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3))) (Pi.module.{v, max w u₁, 0} ι (fun (i : ι) => LinearMap.{u, u, w, u₁} R R inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3)) (M i) N (inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10 i) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.33 (inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.19 i) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.36) Nat Nat.instSemiring (fun (i : ι) => LinearMap.addCommMonoid.{u, u, w, u₁} R R (M i) N inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 (inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10 i) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.33 (inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.19 i) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.36 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3))) (fun (i : ι) => LinearMap.module.{u, u, 0, w, u₁} R R Nat (M i) N inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 (inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10 i) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.33 (inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.19 i) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.36 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3)) Nat.instSemiring (AddCommMonoid.toNatModule.{u₁} N inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.33) (DirectSum.toModule._proof_1.{u, u₁} R inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 N inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.33 inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.36))) (LinearMap.module.{u, u, 0, max v w, u₁} R R Nat (DFinsupp.{v, w} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{w} (M i) (AddZeroClass.toAddZero.{w} (M i) (AddMonoid.toAddZeroClass.{w} (M i) (AddCommMonoid.toAddMonoid.{w} (M i) (inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10 i)))))) N inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 (DFinsupp.addCommMonoid.{v, w} ι M inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10) 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{R : Type.{u}} [inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.3 : Semiring.{u} R] {ι : Type.{v}} [dec_ι : DecidableEq.{succ v} ι] {M : Type.{u_1}} [inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.13 : Module.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.10] {A : ι -> (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.13)}, Eq.{succ u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.13) (LinearMap.range.{u, u, max u_1 v, u_1} R 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inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.13 (A i)) (fun (i : ι) => Submodule.module.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.13 (A i))) inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.13 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.3)) (RingHomSurjective.ids.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.3) (DirectSum.coeLinearMap.{u, v, u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.3 ι dec_ι M inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.13 A)) (iSup.{u_1, succ v} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.13) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.10 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inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.129696875._hygCtx._hyg.13))))) (fun (i : ι) => A i))","typeFull":"∀ {R : Type u} [inst : Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] {A : ι → Submodule R M}, (DirectSum.coeLinearMap A).range = ⨆ i, A i","typeReadable":"∀ {R : Type u} [inst : Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] {A : ι → Submodule R M}, (DirectSum.coeLinearMap A).range = ⨆ i, A 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{R : Type.{u}} [inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 : Semiring.{u} R] {ι : Type.{v}} [dec_ι : DecidableEq.{succ v} ι] {M : Type.{u_1}} [inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13 : Module.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10] {A : ι -> (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13)} (h : DirectSum.IsInternal.{v, u_1, u_1} ι M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13) dec_ι 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inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13 (A i))) M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 (instAddCommMonoidDirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13 (A i))) inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 (DirectSum.instModule.{u, v, u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13 (A i)) (fun (i : ι) => Submodule.module.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13 (A i))) inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3)) (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3)) (RingHomInvPair.ids.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3) (RingHomInvPair.ids.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3) (LinearEquiv.ofBijective.{u, u, max u_1 v, u_1} R R (DirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13 (A i))) M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 (instAddCommMonoidDirectSum.{v, u_1} ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13 (A i))) inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 (DirectSum.instModule.{u, v, u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 ι (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13 (A i)) (fun (i : ι) => Submodule.module.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13 (A i))) inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3)) (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3)) (DirectSum.coeLinearMap.{u, v, u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 ι dec_ι M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13 A) (RingHomInvPair.ids.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3) (RingHomInvPair.ids.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3) h)) (DFinsupp.mapRange.linearEquiv.{v, u, u_1, max u u_2} ι R inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 (fun (i : ι) => Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13)) (A i) x)) (fun (i : ι) => Finsupp.{u_2, u} (α i) R (MulZeroClass.toZero.{u} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3))))) (fun (i : ι) => 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inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13)) (A i) x)) inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 (Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13 (A i)) (Submodule.module.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13 (A i)) (v i)))) (LinearEquiv.symm.{u, u, max (max u u_2) v, max (max u u_2) v} R R (Finsupp.{max u_2 v, u} (Sigma.{v, u_2} ι (fun (i : ι) => α i)) R (AddZero.toZero.{u} R (AddZeroClass.toAddZero.{u} R (AddMonoid.toAddZeroClass.{u} R (AddCommMonoid.toAddMonoid.{u} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R 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[dec_ι : DecidableEq ι] {M : Type u_1} [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] {A : ι → Submodule R M} (h : DirectSum.IsInternal A) {α : ι → Type u_2}\n  (v : (i : ι) → Module.Basis (α i) R ↥(A i)),\n  h.collectedBasis v =\n    {\n      repr :=\n        ((LinearEquiv.ofBijective (DirectSum.coeLinearMap A) h).symm ≪≫ₗ\n            DFinsupp.mapRange.linearEquiv fun i => (v i).repr) ≪≫ₗ\n          (sigmaFinsuppLequivDFinsupp R).symm }","typeReadable":"∀ {R : Type u} [inst : Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] {A : ι → Submodule R M} (h : DirectSum.IsInternal A) {α : ι → Type u_2}\n  (v : (i : ι) → Module.Basis (α i) R ↥(A i)),\n  h.collectedBasis v =\n    {\n      repr :=\n        ((LinearEquiv.ofBijective (DirectSum.coeLinearMap A) h).symm ≪≫ₗ\n            DFinsupp.mapRange.linearEquiv fun i => (v i).repr) ≪≫ₗ\n          (sigmaFinsuppLequivDFinsupp R).symm 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AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13 : Module.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10] {A : ι -> (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13)}, (DirectSum.IsInternal.{v, u_1, u_1} ι M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13) dec_ι inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13) (Submodule.addSubmonoidClass.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13) A) -> (forall {α : ι -> Type.{u_2}}, (forall (i : ι), Module.Basis.{u_2, u, u_1} (α i) R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13) (SetLike.instMembership.{u_1, u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13) M (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13)) (A i) x)) inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 (Submodule.addCommMonoid.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13 (A i)) (Submodule.module.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13 (A i))) -> (Module.Basis.{max u_2 v, u, u_1} (Sigma.{v, u_2} ι (fun (i : ι) => α i)) R M inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.13))","typeFull":"{R : Type u} →\n  [inst : Semiring R] →\n    {ι : Type v} →\n      [dec_ι : DecidableEq ι] →\n        {M : Type u_1} →\n          [inst_1 : AddCommMonoid M] →\n            [inst_2 : Module R M] →\n              {A : ι → Submodule R M} →\n                DirectSum.IsInternal A →\n                  {α : ι → Type u_2} → ((i : ι) → Module.Basis (α i) R ↥(A i)) → Module.Basis ((i : ι) × α i) R M","typeReadable":"{R : Type u} →\n  [inst : Semiring R] →\n    {ι : Type v} →\n      [dec_ι : DecidableEq ι] →\n        {M : Type u_1} →\n          [inst_1 : AddCommMonoid M] →\n            [inst_2 : Module R M] →\n              {A : ι → Submodule R M} →\n                DirectSum.IsInternal A →\n                  {α : ι → Type u_2} → ((i : ι) → Module.Basis (α i) R ↥(A i)) → Module.Basis ((i : ι) × α i) R M","typeReferences":[["DirectSum","IsInternal"],["Subtype"],["SetLike","instMembership"],["Module"],["Submodule","module"],["DecidableEq"],["Membership","mem"],["Submodule"],["AddCommMonoid"],["Submodule","setLike"],["Sigma"],["Module","Basis"],["Submodule","addCommMonoid"],["Submodule","addSubmonoidClass"],["Semiring"]],"valueReferences":[["Subtype"],["DFinsupp"],["Submodule","module"],["DFinsupp","addCommMonoid"],["Membership","mem"],["AddCommMonoid","toAddMonoid"],["Submodule"],["DirectSum"],["Semiring","toNonAssocSemiring"],["instAddCommMonoidDirectSum"],["RingHom","id"],["Sigma"],["DirectSum","IsInternal","collectedBasis","_proof_1"],["Semiring","toModule"],["Finsupp","module"],["SetLike","instMembership"],["Module","Basis","ofRepr"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["AddZeroClass","toAddZero"],["LinearEquiv","symm"],["DirectSum","coeLinearMap"],["DFinsupp","module"],["LinearEquiv","trans"],["Submodule","setLike"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["RingHomInvPair","ids"],["Module","Basis","repr"],["DFinsupp","mapRange","linearEquiv"],["DirectSum","instModule"],["sigmaFinsuppLequivDFinsupp"],["Submodule","addCommMonoid"],["Finsupp","instAddCommMonoid"],["AddZero","toZero"],["Finsupp"],["LinearEquiv","ofBijective"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["DirectSum","IsInternal","collectedBasis","_proof_1"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 : Semiring.{u_1} R], RingHomCompTriple.{u_1, u_1, u_1} R R R inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3)) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3)) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.593871674._hygCtx._hyg.3))","typeFull":"∀ {R : Type u_1} [inst : Semiring R], RingHomCompTriple (RingHom.id R) (RingHom.id R) (RingHom.id R)","typeReadable":"∀ {R : Type u_1} [inst : Semiring R], RingHomCompTriple (RingHom.id R) (RingHom.id R) (RingHom.id 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inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.13)}, (DirectSum.IsInternal.{v, u_1, u_1} ι M (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.13) dec_ι inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.10 (Submodule.setLike.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.13) (Submodule.addSubmonoidClass.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.13) A) -> (Eq.{succ u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.13) (iSup.{u_1, succ v} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.13) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.13) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.13) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.13) (CompleteLattice.toConditionallyCompleteLattice.{u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.13) (Submodule.completeLattice.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.13))))) A) (Top.top.{u_1} (Submodule.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.13) (Submodule.instTop.{u, u_1} R M inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.2580992357._hygCtx._hyg.13)))","typeFull":"∀ {R : Type u} [inst : Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] {A : ι → Submodule R M}, DirectSum.IsInternal A → iSup A = ⊤","typeReadable":"∀ {R : Type u} [inst : Semiring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [inst_1 : AddCommMonoid M]\n  [inst_2 : Module R M] {A : ι → Submodule R M}, DirectSum.IsInternal A → iSup A = ⊤","typeReferences":[["DirectSum","IsInternal"],["Module"],["DecidableEq"],["Submodule","instTop"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["Submodule"],["AddCommMonoid"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderSup"],["iSup"],["Submodule","setLike"],["ConditionallyCompletePartialOrderSup","toSupSet"],["Top","top"],["Submodule","completeLattice"],["Eq"],["CompleteLattice","toConditionallyCompleteLattice"],["Submodule","addSubmonoidClass"],["Semiring"]],"valueReferences":[["Membership","mem"],["Pi","addCommMonoid"],["Pi","module"],["Submodule","subtype"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderSup"],["iSup"],["Semiring","toNonAssocSemiring"],["ConditionallyCompletePartialOrderSup","toSupSet"],["RingHom","id"],["SetLike","instMembership"],["AddMonoid","nat_smulCommClass'"],["LinearMap","module"],["DistribMulAction","toDistribSMul"],["AddZeroClass","toAddZero"],["LinearMap"],["DFinsupp","module"],["Nat"],["Top","top"],["RingHomInvPair","ids"],["id"],["Eq","mpr"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["DFinsupp","instEquivLikeLinearEquiv"],["LinearMap","range_eq_top"],["Subtype"],["LinearMap","instFunLike"],["DFinsupp"],["Submodule","module"],["DFinsupp","addCommMonoid"],["LinearMap","range"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["AddCommMonoid","toAddMonoid"],["LinearEquiv"],["DFunLike","coe"],["congrArg"],["Submodule"],["Nat","instSemiring"],["MonoidWithZero","toMonoid"],["EquivLike","toFunLike"],["AddCommMonoid","toNatModule"],["Eq"],["propext"],["CompleteLattice","toConditionallyCompleteLattice"],["Function","Bijective","surjective"],["Submodule","instTop"],["Semiring","toMonoidWithZero"],["Function","Surjective"],["RingHomSurjective","ids"],["LinearMap","addCommMonoid"],["Module","toDistribMulAction"],["Submodule","setLike"],["Submodule","completeLattice"],["DFinsupp","lsum"],["Submodule","iSup_eq_range_dfinsupp_lsum"],["Submodule","addCommMonoid"]]},{"isProp":false,"kind":"definition","name":["DirectSum","toModule"],"typeFallback":"forall (R : Type.{u}) [inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 : Semiring.{u} R] (ι : Type.{v}) {M : ι -> Type.{w}} [inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10 : forall (i : ι), AddCommMonoid.{w} (M i)] [inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.19 : forall (i : ι), Module.{u, w} R (M i) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 (inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10 i)] [inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.29 : DecidableEq.{succ v} ι] (N : Type.{u₁}) [inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.33 : AddCommMonoid.{u₁} N] [inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.36 : Module.{u, u₁} R N inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.33], (forall (i : ι), LinearMap.{u, u, w, u₁} R R inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3)) (M i) N (inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10 i) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.33 (inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.19 i) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.36) -> (LinearMap.{u, u, max w v, u₁} R R inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.3)) (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1426575160._hygCtx._hyg.10) N (instAddCommMonoidDirectSum.{v, w} ι (fun (i : ι) 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AddCommMonoid (M i)] →\n          [inst_2 : (i : ι) → Module R (M i)] →\n            [DecidableEq ι] →\n              (N : Type u₁) →\n                [inst_4 : AddCommMonoid N] →\n                  [inst_5 : Module R N] → ((i : ι) → M i →ₗ[R] N) → (DirectSum ι fun i => M i) →ₗ[R] 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(x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i))) (DirectSum.{v, max w u_1} ι (fun (i : ι) => DirectSum.{u_1, w} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, w} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.42 i))) inst._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.3 (instAddCommMonoidDirectSum.{max u_1 v, w} (Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) (fun (i : Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => δ (Sigma.fst.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)) (fun (i : Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => inst._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.42 (Sigma.fst.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i))) (instAddCommMonoidDirectSum.{v, max u_1 w} ι (fun (i : ι) => DirectSum.{u_1, w} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, w} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.42 i))) (DirectSum.instModule.{u, max u_1 v, w} R inst._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.3 (Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) (fun (i : Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => δ (Sigma.fst.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)) (fun (i : Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => inst._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.42 (Sigma.fst.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i)) (fun (i : Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92)) => inst._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.53 (Sigma.fst.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i) (Sigma.snd.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.4028603747._hygCtx._hyg.92) i))) (DirectSum.instModule.{u, v, max u_1 w} R inst._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.3 ι (fun (i : ι) => DirectSum.{u_1, w} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.42 i)) (fun (i : ι) => instAddCommMonoidDirectSum.{u_1, w} (α i) (fun (j : α i) => δ i j) (inst._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.42 i)) (fun (i : ι) => DirectSum.instModule.{u, u_1, w} R inst._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.3 (α i) (δ i) (inst._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.42 i) (inst._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.53 i))) (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.3))) (DirectSum.sigmaLcurry.{u, v, w, u_1} R inst._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.3 ι α δ inst._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.39 inst._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.42 inst._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.53) f) i) j) (DFunLike.coe.{max (succ (max u_1 v)) (succ w), succ (max u_1 v), succ w} (DFinsupp.{max u_1 v, w} (Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83)) (fun (i : Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83)) => (fun (i : Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83)) => δ (Sigma.fst.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83) i) (Sigma.snd.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83) i)) i) (fun (i : Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83)) => AddZero.toZero.{w} ((fun (i : Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α 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=> α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83) i)) i) ((fun (i : Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83)) => inst._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.42 (Sigma.fst.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83) i) (Sigma.snd.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83) i)) i)))))) (Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83)) (fun (i : Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83)) => δ (Sigma.fst.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83) i) (Sigma.snd.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83) i)) (DFinsupp.instDFunLike.{max u_1 v, w} (Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83)) (fun (i : Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83)) => δ (Sigma.fst.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83) i) (Sigma.snd.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83) i)) (fun (i : Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83)) => AddZero.toZero.{w} ((fun (i : Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83)) => δ (Sigma.fst.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83) i) (Sigma.snd.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83) i)) i) (AddZeroClass.toAddZero.{w} ((fun (i : Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83)) => δ (Sigma.fst.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83) i) (Sigma.snd.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83) i)) i) (AddMonoid.toAddZeroClass.{w} ((fun (i : Sigma.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83)) => δ (Sigma.fst.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83) i) (Sigma.snd.{v, u_1} ι (fun (x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83 : ι) => α x._@.Mathlib.Algebra.DirectSum.Module.920101297._hygCtx._hyg.83) i)) i) (AddCommMonoid.toAddMonoid.{w} 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inst._@.Mathlib.Algebra.DirectSum.Module.1257460159._hygCtx._hyg.3 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1257460159._hygCtx._hyg.3))) ψ' (DirectSum.lof.{u, v, w} R inst._@.Mathlib.Algebra.DirectSum.Module.1257460159._hygCtx._hyg.3 ι M inst._@.Mathlib.Algebra.DirectSum.Module.1257460159._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1257460159._hygCtx._hyg.19 inst._@.Mathlib.Algebra.DirectSum.Module.1257460159._hygCtx._hyg.29 i))) -> (Eq.{max (max (succ u₁) (succ v)) (succ w)} (LinearMap.{u, u, max w v, u₁} R R inst._@.Mathlib.Algebra.DirectSum.Module.1257460159._hygCtx._hyg.3 inst._@.Mathlib.Algebra.DirectSum.Module.1257460159._hygCtx._hyg.3 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.DirectSum.Module.1257460159._hygCtx._hyg.3)) (DirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1257460159._hygCtx._hyg.10) N (instAddCommMonoidDirectSum.{v, w} ι (fun (i : ι) => M i) inst._@.Mathlib.Algebra.DirectSum.Module.1257460159._hygCtx._hyg.10) inst._@.Mathlib.Algebra.DirectSum.Module.1257460159._hygCtx._hyg.33 (DirectSum.instModule.{u, v, w} R inst._@.Mathlib.Algebra.DirectSum.Module.1257460159._hygCtx._hyg.3 ι M inst._@.Mathlib.Algebra.DirectSum.Module.1257460159._hygCtx._hyg.10 inst._@.Mathlib.Algebra.DirectSum.Module.1257460159._hygCtx._hyg.19) inst._@.Mathlib.Algebra.DirectSum.Module.1257460159._hygCtx._hyg.36) ψ ψ')","typeFull":"∀ (R : Type u) [inst : Semiring R] {ι : Type v} {M : ι → Type w} [inst_1 : (i : ι) → AddCommMonoid (M i)]\n  [inst_2 : (i : ι) → Module R (M i)] [inst_3 : DecidableEq ι] {N : Type u₁} [inst_4 : AddCommMonoid N]\n  [inst_5 : Module R N] ⦃ψ ψ' : (DirectSum ι fun i => M i) →ₗ[R] N⦄,\n  (∀ (i : ι), ψ ∘ₗ DirectSum.lof R ι M i = ψ' ∘ₗ DirectSum.lof R ι M i) → ψ = ψ'","typeReadable":"∀ (R : Type u) [inst : Semiring R] {ι : Type v} {M : ι → Type w} [inst_1 : (i : ι) → AddCommMonoid (M i)]\n  [inst_2 : (i : ι) → Module R (M i)] [inst_3 : DecidableEq ι] {N : Type u₁} [inst_4 : AddCommMonoid N]\n  [inst_5 : Module R N] ⦃ψ ψ' : (DirectSum ι fun i => M i) →ₗ[R] N⦄,\n  (∀ (i : ι), ψ ∘ₗ DirectSum.lof R ι M i = ψ' ∘ₗ DirectSum.lof R ι M i) → ψ = ψ'","typeReferences":[["Module"],["RingHomCompTriple","ids"],["DecidableEq"],["LinearMap","comp"],["LinearMap"],["DirectSum"],["AddCommMonoid"],["DirectSum","lof"],["Semiring","toNonAssocSemiring"],["instAddCommMonoidDirectSum"],["RingHom","id"],["DirectSum","instModule"],["Eq"],["Semiring"]],"valueReferences":[["DFinsupp","lhom_ext'"]]},{"isProp":true,"kind":"theorem","name":["DirectSum","lid","_proof_3"],"typeFallback":"forall (M : Type.{u_1}) (ι : Type.{u_2}) [inst._@.Mathlib.Algebra.DirectSum.Module.4176511332._hygCtx._hyg.33 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.DirectSum.Module.4176511332._hygCtx._hyg.40 : Unique.{succ u_2} ι], Function.LeftInverse.{succ (max u_1 u_2), succ u_1} (DirectSum.{u_2, u_1} ι (fun 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k)","typeReferences":[["Equiv","instEquivLike"],["Module"],["LinearEquiv"],["DFunLike","coe"],["Equiv"],["DirectSum"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["instAddCommMonoidDirectSum"],["RingHom","id"],["EquivLike","toFunLike"],["RingHomInvPair","ids"],["Equiv","symm"],["DirectSum","instModule"],["Semiring"]],"valueReferences":[["DFinsupp","domLCongr"]]},{"isProp":true,"kind":"theorem","name":["DirectSum","sigmaLuncurry","_proof_2"],"typeFallback":"forall (R : Type.{u_4}) [inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.3 : Semiring.{u_4} R] {ι : Type.{u_1}} {α : ι -> Type.{u_3}} {δ : forall (i : ι), (α i) -> Type.{u_2}} [inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.39 : DecidableEq.{succ u_1} ι] [inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.42 : forall (i : ι) (j : α i), AddCommMonoid.{u_2} (δ i j)] [inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.53 : forall (i : ι) (j : α i), Module.{u_4, u_2} 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(AddZeroClass.toAddZero.{u_2} (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (AddMonoid.toAddZeroClass.{u_2} (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (AddCommMonoid.toAddMonoid.{u_2} (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.42 (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i))))))) (DFinsupp.{max u_3 u_1, u_2} (Sigma.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79)) (fun (i : Sigma.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79)) => δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (fun (i : Sigma.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79)) => AddZero.toZero.{u_2} (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (AddZeroClass.toAddZero.{u_2} (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (AddMonoid.toAddZeroClass.{u_2} (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (AddCommMonoid.toAddMonoid.{u_2} (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.42 (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i))))))) (instHSMul.{u_4, max (max u_1 u_3) u_2} R (DFinsupp.{max u_3 u_1, u_2} (Sigma.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79)) (fun (i : Sigma.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79)) => δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => �� x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (fun (i : Sigma.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79)) => AddZero.toZero.{u_2} (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (AddZeroClass.toAddZero.{u_2} (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (AddMonoid.toAddZeroClass.{u_2} (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (AddCommMonoid.toAddMonoid.{u_2} (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.42 (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i))))))) (SMulZeroClass.toSMul.{u_4, max (max u_1 u_3) u_2} R (DFinsupp.{max u_3 u_1, u_2} (Sigma.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79)) (fun (i : Sigma.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79)) => δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (fun (i : Sigma.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79)) => AddZero.toZero.{u_2} (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (AddZeroClass.toAddZero.{u_2} (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (AddMonoid.toAddZeroClass.{u_2} (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (AddCommMonoid.toAddMonoid.{u_2} (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.42 (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i))))))) (DFinsupp.instZero.{max u_1 u_3, u_2} (Sigma.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79)) (fun (i : Sigma.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79)) => δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (fun (i : Sigma.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79)) => AddZero.toZero.{u_2} (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (AddZeroClass.toAddZero.{u_2} (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (AddMonoid.toAddZeroClass.{u_2} (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (AddCommMonoid.toAddMonoid.{u_2} (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.42 (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i))))))) (DFinsupp.instSMulZeroClass.{max u_1 u_3, u_2, u_4} (Sigma.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79)) R (fun (i : Sigma.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79)) => δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (fun (i : Sigma.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79)) => AddZero.toZero.{u_2} (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (AddZeroClass.toAddZero.{u_2} (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (AddMonoid.toAddZeroClass.{u_2} (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (AddCommMonoid.toAddMonoid.{u_2} (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.42 (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)))))) (fun (i : Sigma.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79)) => DistribSMul.toSMulZeroClass.{u_4, u_2} R (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (AddMonoid.toAddZeroClass.{u_2} (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (AddCommMonoid.toAddMonoid.{u_2} (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.42 (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)))) (DistribMulAction.toDistribSMul.{u_4, u_2} R (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.3)) (AddCommMonoid.toAddMonoid.{u_2} (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.42 (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i))) (Module.toDistribMulAction.{u_4, u_2} R (δ (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.3 (inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.42 (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)) (inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.53 (Sigma.fst.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i) (Sigma.snd.{u_1, u_3} ι (fun (x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79 : ι) => α x._@.Mathlib.Data.DFinsupp.Sigma.1085907084._hygCtx._hyg.79) i)))))))) r (DFinsupp.sigmaUncurry.{u_1, u_2, u_3} ι α (fun (i : ι) (j : α i) => δ i j) inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.39 (fun (i : ι) (i_1 : α i) => AddZero.toZero.{u_2} (δ i i_1) (AddZeroClass.toAddZero.{u_2} (δ i i_1) (AddMonoid.toAddZeroClass.{u_2} (δ i i_1) (AddCommMonoid.toAddMonoid.{u_2} (δ i i_1) (inst._@.Mathlib.Algebra.DirectSum.Module.3472913571._hygCtx._hyg.42 i i_1))))) f))","typeFull":"∀ (R : Type u_4) [inst : Semiring R] {ι : Type u_1} {α : ι → Type u_3} {δ : (i : ι) → α i → Type u_2}\n  [inst_1 : DecidableEq ι] [inst_2 : (i : ι) → (j : α i) → AddCommMonoid (δ i j)]\n  [inst_3 : (i : ι) → (j : α i) → Module R (δ i j)] (r : R) (f : Π₀ (i : ι) (j : α i), δ i j),\n  (r • f).sigmaUncurry = r • f.sigmaUncurry","typeReadable":"∀ (R : Type u_4) [inst : Semiring R] {ι : Type u_1} {α : ι → Type u_3} {δ : (i : ι) → α i → Type u_2}\n  [inst_1 : DecidableEq ι] [inst_2 : (i : ι) → (j : α i) → AddCommMonoid (δ i j)]\n  [inst_3 : (i : ι) → (j : α i) → Module R (δ i j)] (r : R) (f : Π₀ (i : ι) (j : α i), δ i j),\n  (r • f).sigmaUncurry = r • f.sigmaUncurry","typeReferences":[["DFinsupp","instSMulZeroClass"],["Module"],["DFinsupp"],["DecidableEq"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Sigma","fst"],["Sigma","snd"],["Module","toDistribMulAction"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["instHSMul"],["Sigma"],["DFinsupp","instZero"],["Eq"],["DFinsupp","sigmaUncurry"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Module","toDistribMulAction"],["MonoidWithZero","toMonoid"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"],["DFinsupp","sigmaUncurry_smul"]]},{"isProp":true,"kind":"theorem","name":["DFinsupp","lsum","eq_1"],"typeFallback":"forall {ι : Type.{u_1}} {R : Type.{u_3}} (S : Type.{u_4}) {M : ι -> Type.{u_5}} {N : Type.{u_6}} [inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 : Semiring.{u_3} R] [inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 : forall (i : ι), AddCommMonoid.{u_5} (M i)] [inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 : forall (i : ι), Module.{u_3, u_5} R (M i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)] [inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 : AddCommMonoid.{u_6} N] [inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 : Module.{u_3, u_6} R N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32] [inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.39 : DecidableEq.{succ u_1} ι] [inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42 : Semiring.{u_4} S] [inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.45 : Module.{u_4, u_6} S N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32] [inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.49 : SMulCommClass.{u_3, u_4, u_6} R S N (SMulZeroClass.toSMul.{u_3, u_6} R N (AddZero.toZero.{u_6} N (AddZeroClass.toAddZero.{u_6} N (AddMonoid.toAddZeroClass.{u_6} N (AddCommMonoid.toAddMonoid.{u_6} N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32)))) (DistribSMul.toSMulZeroClass.{u_3, u_6} R N (AddMonoid.toAddZeroClass.{u_6} N (AddCommMonoid.toAddMonoid.{u_6} N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32)) (DistribMulAction.toDistribSMul.{u_3, u_6} R N (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (AddCommMonoid.toAddMonoid.{u_6} N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32) (Module.toDistribMulAction.{u_3, u_6} R N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35)))) (SMulZeroClass.toSMul.{u_4, u_6} S N (AddZero.toZero.{u_6} N (AddZeroClass.toAddZero.{u_6} N (AddMonoid.toAddZeroClass.{u_6} N (AddCommMonoid.toAddMonoid.{u_6} N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32)))) (DistribSMul.toSMulZeroClass.{u_4, u_6} S N (AddMonoid.toAddZeroClass.{u_6} N (AddCommMonoid.toAddMonoid.{u_6} N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32)) (DistribMulAction.toDistribSMul.{u_4, u_6} S N (MonoidWithZero.toMonoid.{u_4} S (Semiring.toMonoidWithZero.{u_4} S inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42)) (AddCommMonoid.toAddMonoid.{u_6} N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32) (Module.toDistribMulAction.{u_4, u_6} S N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.45))))], Eq.{max (succ (max (max u_1 u_5) u_6)) (succ (max u_6 u_5 u_1))} (LinearEquiv.{u_4, u_4, max (max u_1 u_5) u_6, max u_6 u_5 u_1} S S inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42 (RingHom.id.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42)) (RingHom.id.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42)) (RingHomInvPair.ids.{u_4} S inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42) (RingHomInvPair.ids.{u_4} S inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42) (forall (i : ι), LinearMap.{u_3, u_3, u_5, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (M i) N (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) (LinearMap.{u_3, u_3, max u_5 u_1, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) N (DFinsupp.addCommMonoid.{u_1, u_5} ι M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (DFinsupp.module.{u_1, u_5, u_3} ι R M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) (Pi.addCommMonoid.{u_1, max u_5 u_6} ι (fun (i : ι) => LinearMap.{u_3, u_3, u_5, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (M i) N (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) (fun (i : ι) => LinearMap.addCommMonoid.{u_3, u_3, u_5, u_6} R R (M i) N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)))) (LinearMap.addCommMonoid.{u_3, u_3, max u_1 u_5, u_6} R R (DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (DFinsupp.addCommMonoid.{u_1, u_5} ι M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (DFinsupp.module.{u_1, u_5, u_3} ι R M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10))) (Pi.module.{u_1, max u_5 u_6, u_4} ι (fun (i : ι) => LinearMap.{u_3, u_3, u_5, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (M i) N (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) S inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42 (fun (i : ι) => LinearMap.addCommMonoid.{u_3, u_3, u_5, u_6} R R (M i) N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10))) (fun (i : ι) => LinearMap.module.{u_3, u_3, u_4, u_5, u_6} R R S (M i) N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.45 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.49)) (LinearMap.module.{u_3, u_3, u_4, max u_1 u_5, u_6} R R S (DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (DFinsupp.addCommMonoid.{u_1, u_5} ι M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (DFinsupp.module.{u_1, u_5, u_3} ι R M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.45 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.49)) (DFinsupp.lsum.{u_1, u_3, u_4, u_5, u_6} ι R S M N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.39 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.45 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.49) (LinearEquiv.mk.{u_4, u_4, max (max u_1 u_5) u_6, max (max u_1 u_5) u_6} S S inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42 (RingHom.id.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42)) (RingHom.id.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42)) (RingHomInvPair.ids.{u_4} S inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42) (RingHomInvPair.ids.{u_4} S inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42) (forall (i : ι), LinearMap.{u_3, u_3, u_5, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (M i) N (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) (LinearMap.{u_3, u_3, max u_5 u_1, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) N (DFinsupp.addCommMonoid.{u_1, u_5} ι M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (DFinsupp.module.{u_1, u_5, u_3} ι R M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) (Pi.addCommMonoid.{u_1, max u_5 u_6} ι (fun (i : ι) => LinearMap.{u_3, u_3, u_5, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (M i) N (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) (fun (i : ι) => LinearMap.addCommMonoid.{u_3, u_3, u_5, u_6} R R (M i) N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)))) (LinearMap.addCommMonoid.{u_3, u_3, max u_1 u_5, u_6} R R (DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (DFinsupp.addCommMonoid.{u_1, u_5} ι M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (DFinsupp.module.{u_1, u_5, u_3} ι R M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10))) (Pi.module.{u_1, max u_5 u_6, u_4} ι (fun (i : ι) => LinearMap.{u_3, u_3, u_5, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (M i) N (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) S inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42 (fun (i : ι) => LinearMap.addCommMonoid.{u_3, u_3, u_5, u_6} R R (M i) N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10))) (fun (i : ι) => LinearMap.module.{u_3, u_3, u_4, u_5, u_6} R R S (M i) N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.45 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.49)) (LinearMap.module.{u_3, u_3, u_4, max u_1 u_5, u_6} R R S (DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (DFinsupp.addCommMonoid.{u_1, u_5} ι M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (DFinsupp.module.{u_1, u_5, u_3} ι R M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.45 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.49) (LinearMap.mk.{u_4, u_4, max (max u_1 u_5) u_6, max (max u_1 u_5) u_6} S S inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42 (RingHom.id.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42)) (forall (i : ι), LinearMap.{u_3, u_3, u_5, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (M i) N (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) (LinearMap.{u_3, u_3, max u_5 u_1, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) N (DFinsupp.addCommMonoid.{u_1, u_5} ι M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (DFinsupp.module.{u_1, u_5, u_3} ι R M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) (Pi.addCommMonoid.{u_1, max u_5 u_6} ι (fun (i : ι) => LinearMap.{u_3, u_3, u_5, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (M i) N (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) (fun (i : ι) => LinearMap.addCommMonoid.{u_3, u_3, u_5, u_6} R R (M i) N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)))) (LinearMap.addCommMonoid.{u_3, u_3, max u_1 u_5, u_6} R R (DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (DFinsupp.addCommMonoid.{u_1, u_5} ι M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (DFinsupp.module.{u_1, u_5, u_3} ι R M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10))) (Pi.module.{u_1, max u_5 u_6, u_4} ι (fun (i : ι) => LinearMap.{u_3, u_3, u_5, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (M i) N (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) S inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42 (fun (i : ι) => LinearMap.addCommMonoid.{u_3, u_3, u_5, u_6} R R (M i) N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10))) (fun (i : ι) => LinearMap.module.{u_3, u_3, u_4, u_5, u_6} R R S (M i) N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.45 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.49)) (LinearMap.module.{u_3, u_3, u_4, max u_1 u_5, u_6} R R S (DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (DFinsupp.addCommMonoid.{u_1, u_5} ι M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (DFinsupp.module.{u_1, u_5, u_3} ι R M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.45 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.49) (AddHom.mk.{max (max u_1 u_5) u_6, max (max u_1 u_5) u_6} (forall (i : ι), LinearMap.{u_3, u_3, u_5, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (M i) N (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) (LinearMap.{u_3, u_3, max u_5 u_1, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) N (DFinsupp.addCommMonoid.{u_1, u_5} ι M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (DFinsupp.module.{u_1, u_5, u_3} ι R M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) (AddCommMagma.toAdd.{max (max u_1 u_5) u_6} (forall (i : ι), LinearMap.{u_3, u_3, u_5, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (M i) N (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) (AddCommSemigroup.toAddCommMagma.{max (max u_1 u_5) u_6} (forall (i : ι), LinearMap.{u_3, u_3, u_5, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (M i) N (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) (AddCommMonoid.toAddCommSemigroup.{max (max u_1 u_5) u_6} (forall (i : ι), LinearMap.{u_3, u_3, u_5, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (M i) N (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) (Pi.addCommMonoid.{u_1, max u_5 u_6} ι (fun (i : ι) => LinearMap.{u_3, u_3, u_5, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (M i) N (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) (fun (i : ι) => LinearMap.addCommMonoid.{u_3, u_3, u_5, u_6} R R (M i) N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10))))))) (AddCommMagma.toAdd.{max (max u_1 u_5) u_6} (LinearMap.{u_3, u_3, max u_5 u_1, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) N (DFinsupp.addCommMonoid.{u_1, u_5} ι M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (DFinsupp.module.{u_1, u_5, u_3} ι R M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) (AddCommSemigroup.toAddCommMagma.{max (max u_1 u_5) u_6} (LinearMap.{u_3, u_3, max u_5 u_1, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) N (DFinsupp.addCommMonoid.{u_1, u_5} ι M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (DFinsupp.module.{u_1, u_5, u_3} ι R M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) (AddCommMonoid.toAddCommSemigroup.{max (max u_1 u_5) u_6} (LinearMap.{u_3, u_3, max u_5 u_1, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) N (DFinsupp.addCommMonoid.{u_1, u_5} ι M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (DFinsupp.module.{u_1, u_5, u_3} ι R M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) (LinearMap.addCommMonoid.{u_3, u_3, max u_1 u_5, u_6} R R (DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (DFinsupp.addCommMonoid.{u_1, u_5} ι M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (DFinsupp.module.{u_1, u_5, u_3} ι R M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)))))) (fun (F : forall (i : ι), LinearMap.{u_3, u_3, u_5, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (M i) N (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) => LinearMap.mk.{u_3, u_3, max u_1 u_5, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) N (DFinsupp.addCommMonoid.{u_1, u_5} ι M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (DFinsupp.module.{u_1, u_5, u_3} ι R M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 (AddHom.mk.{max u_1 u_5, u_6} (DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) N (AddCommMagma.toAdd.{max u_1 u_5} (DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) (AddCommSemigroup.toAddCommMagma.{max u_1 u_5} (DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) (AddCommMonoid.toAddCommSemigroup.{max u_1 u_5} (DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) (DFinsupp.addCommMonoid.{u_1, u_5} ι M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13)))) (AddCommMagma.toAdd.{u_6} N (AddCommSemigroup.toAddCommMagma.{u_6} N (AddCommMonoid.toAddCommSemigroup.{u_6} N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32))) (DFunLike.coe.{max (max (succ u_1) (succ u_5)) (succ u_6), max (succ u_1) (succ u_5), succ u_6} (AddMonoidHom.{max u_5 u_1, u_6} (DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) N (AddZeroClass.toAddZero.{max u_1 u_5} (DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) (DFinsupp.addZeroClass.{u_1, u_5} ι M (fun (i : ι) => AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i))))) (AddZeroClass.toAddZero.{u_6} N (AddMonoid.toAddZeroClass.{u_6} N (AddCommMonoid.toAddMonoid.{u_6} N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32)))) (DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) => N) (AddMonoidHom.instFunLike.{max u_1 u_5, u_6} (DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) N (AddZeroClass.toAddZero.{max u_1 u_5} (DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) (DFinsupp.addZeroClass.{u_1, u_5} ι M (fun (i : ι) => AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i))))) (AddZeroClass.toAddZero.{u_6} N (AddMonoid.toAddZeroClass.{u_6} N (AddCommMonoid.toAddMonoid.{u_6} N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32)))) (DFinsupp.sumAddHom.{u_1, u_5, u_6} ι N M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.39 (fun (i : ι) => AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i))) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (fun (i : ι) => LinearMap.toAddMonoidHom.{u_3, u_3, u_5, u_6} R R (M i) N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (F i)))) (DFinsupp.lsum._proof_6.{u_1, u_5, u_6, u_3} ι R M N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.39 F)) (fun (c : R) (f : DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) => DFinsupp.lsum._proof_1.{u_6, u_1, u_5, u_3} ι R M N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.39 F c f)) (fun (F : forall (i : ι), LinearMap.{u_3, u_3, u_5, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (M i) N (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) (G : forall (i : ι), LinearMap.{u_3, u_3, u_5, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (M i) N (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) => DFinsupp.lsum._proof_2.{u_1, u_5, u_6, u_3} ι R M N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.39 F G)) (fun (c : S) (F : forall (i : ι), LinearMap.{u_3, u_3, u_5, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (M i) N (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) => DFinsupp.lsum._proof_3.{u_1, u_5, u_6, u_3, u_4} ι R S M N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.39 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.42 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.45 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.49 c F)) (fun (F : LinearMap.{u_3, u_3, max u_5 u_1, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) N (DFinsupp.addCommMonoid.{u_1, u_5} ι M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (DFinsupp.module.{u_1, u_5, u_3} ι R M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) (i : ι) => LinearMap.comp.{u_3, u_3, u_3, u_5, max u_1 u_5, u_6} R R R (M i) (DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) (DFinsupp.addCommMonoid.{u_1, u_5} ι M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) (DFinsupp.module.{u_1, u_5, u_3} ι R M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (DFinsupp.lsum._proof_7.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10) F (DFinsupp.lsingle.{u_1, u_3, u_5} ι R M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.39 i)) (fun (F : forall (i : ι), LinearMap.{u_3, u_3, u_5, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (M i) N (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 i) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) => DFinsupp.lsum._proof_4.{u_1, u_5, u_6, u_3} ι R M N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.39 F) (fun (F : LinearMap.{u_3, u_3, max u_5 u_1, u_6} R R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10)) (DFinsupp.{u_1, u_5} ι (fun (i : ι) => M i) (fun (i : ι) => AddZero.toZero.{u_5} (M i) (AddZeroClass.toAddZero.{u_5} (M i) (AddMonoid.toAddZeroClass.{u_5} (M i) (AddCommMonoid.toAddMonoid.{u_5} (M i) (inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 i)))))) N (DFinsupp.addCommMonoid.{u_1, u_5} ι M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 (DFinsupp.module.{u_1, u_5, u_3} ι R M inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22) inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35) => DFinsupp.lsum._proof_5.{u_1, u_5, u_6, u_3} ι R M N inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.10 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.13 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.22 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.32 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.35 inst._@.Mathlib.LinearAlgebra.DFinsupp.706667673._hygCtx._hyg.39 F))","typeFull":"∀ {ι : Type u_1} {R : Type u_3} (S : Type u_4) {M : ι → Type u_5} {N : Type u_6} [inst : Semiring R]\n  [inst_1 : (i : ι) → AddCommMonoid (M i)] [inst_2 : (i : ι) → Module R (M i)] [inst_3 : AddCommMonoid N]\n  [inst_4 : Module R N] [inst_5 : DecidableEq ι] [inst_6 : Semiring S] [inst_7 : Module S N]\n  [inst_8 : SMulCommClass R S N],\n  DFinsupp.lsum S =\n    { toFun := fun F => { toFun := ⇑(DFinsupp.sumAddHom fun i => (F i).toAddMonoidHom), map_add' := ⋯, map_smul' := ⋯ },\n      map_add' := ⋯, map_smul' := ⋯, invFun := fun F i => F ∘ₗ DFinsupp.lsingle i, left_inv := ⋯, right_inv := ⋯ }","typeReadable":"∀ {ι : Type u_1} {R : Type u_3} (S : Type u_4) {M : ι → Type u_5} {N : Type u_6} [inst : Semiring R]\n  [inst_1 : (i : ι) → AddCommMonoid (M i)] [inst_2 : (i : ι) → Module R (M i)] [inst_3 : AddCommMonoid N]\n  [inst_4 : Module R N] [inst_5 : DecidableEq ι] [inst_6 : Semiring S] [inst_7 : Module S N]\n  [inst_8 : SMulCommClass R S N],\n  DFinsupp.lsum S =\n    { toFun := fun F => { toFun := ⇑(DFinsupp.sumAddHom fun i => (F i).toAddMonoidHom), map_add' := ⋯, map_smul' := ⋯ },\n      map_add' := ⋯, map_smul' := ⋯, invFun := fun F i => F ∘ₗ DFinsupp.lsingle i, left_inv := ⋯, right_inv := ⋯ }","typeReferences":[["Pi","addCommMonoid"],["AddHom","mk"],["SMulZeroClass","toSMul"],["Pi","module"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["AddMonoidHom"],["DFinsupp","addZeroClass"],["LinearEquiv","mk"],["AddMonoidHom","instFunLike"],["DistribSMul","toSMulZeroClass"],["DFinsupp","lsum","_proof_6"],["LinearMap","toAddMonoidHom"],["LinearMap","module"],["DistribMulAction","toDistribSMul"],["DFinsupp","lsum","_proof_2"],["LinearMap"],["AddZeroClass","toAddZero"],["DFinsupp","lsingle"],["DFinsupp","module"],["RingHomInvPair","ids"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["DFinsupp","lsum","_proof_1"],["DFinsupp"],["Module"],["DFinsupp","addCommMonoid"],["DecidableEq"],["LinearMap","mk"],["LinearMap","comp"],["AddCommMonoid","toAddMonoid"],["LinearEquiv"],["DFinsupp","lsum","_proof_5"],["DFunLike","coe"],["DFinsupp","lsum","_proof_3"],["MonoidWithZero","toMonoid"],["DFinsupp","sumAddHom"],["DFinsupp","lsum","_proof_4"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["DFinsupp","lsum","_proof_7"],["Eq"],["Semiring","toMonoidWithZero"],["LinearMap","addCommMonoid"],["SMulCommClass"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["DFinsupp","lsum"],["Semiring"]],"valueReferences":[["DFinsupp"],["DFinsupp","addCommMonoid"],["LinearMap","module"],["Pi","addCommMonoid"],["AddCommMonoid","toAddMonoid"],["LinearEquiv"],["LinearMap"],["AddZeroClass","toAddZero"],["Pi","module"],["DFinsupp","module"],["LinearMap","addCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Eq","refl"],["DFinsupp","lsum"],["RingHomInvPair","ids"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["DirectSum","congr_linearEquiv_toLinearMap"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.DirectSum.Module.1339316558._hygCtx._hyg.3 : Semiring.{u_1} R] {ι : Type.{u_2}} {N : ι -> Type.{u_3}} [inst._@.Mathlib.Algebra.DirectSum.Module.1339316558._hygCtx._hyg.10 : forall (i : ι), AddCommMonoid.{u_3} (N i)] [inst._@.Mathlib.Algebra.DirectSum.Module.1339316558._hygCtx._hyg.18 : forall (i : ι), Module.{u_1, u_3} R (N i) inst._@.Mathlib.Algebra.DirectSum.Module.1339316558._hygCtx._hyg.3 (inst._@.Mathlib.Algebra.DirectSum.Module.1339316558._hygCtx._hyg.10 i)] {P : ι -> Type.{u_4}} [inst._@.Mathlib.Algebra.DirectSum.Module.1339316558._hygCtx._hyg.30 : forall (i : ι), AddCommMonoid.{u_4} (P i)] [inst._@.Mathlib.Algebra.DirectSum.Module.1339316558._hygCtx._hyg.39 : forall (i : ι), Module.{u_1, u_4} R (P i) inst._@.Mathlib.Algebra.DirectSum.Module.1339316558._hygCtx._hyg.3 (inst._@.Mathlib.Algebra.DirectSum.Module.1339316558._hygCtx._hyg.30 i)] (u : forall (i : ι), LinearEquiv.{u_1, u_1, u_3, u_4} R R 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↑(DirectSum.congrLinearEquiv u) = DirectSum.lmap fun i => ↑(u i)","typeReadable":"∀ {R : Type u_1} [inst : Semiring R] {ι : Type u_2} {N : ι → Type u_3} [inst_1 : (i : ι) → AddCommMonoid (N i)]\n  [inst_2 : (i : ι) → Module R (N i)] {P : ι → Type u_4} [inst_3 : (i : ι) → AddCommMonoid (P i)]\n  [inst_4 : (i : ι) → Module R (P i)] (u : (i : ι) → N i ≃ₗ[R] P i),\n  ↑(DirectSum.congrLinearEquiv u) = DirectSum.lmap fun i => ↑(u i)","typeReferences":[["DirectSum","lmap"],["Module"],["LinearEquiv","toLinearMap"],["LinearEquiv"],["LinearMap"],["DirectSum"],["DirectSum","congrLinearEquiv"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["instAddCommMonoidDirectSum"],["RingHom","id"],["RingHomInvPair","ids"],["DirectSum","instModule"],["Eq"],["Semiring"]],"valueReferences":[["DirectSum","congrLinearEquiv_toLinearMap"]]},{"isProp":false,"kind":"definition","name":["DirectSum","congr_addEquiv"],"typeFallback":"forall {ι : Type.{u_2}} {N : ι -> Type.{u_3}} 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.DirectSum.Module.1193137464._hygCtx._hyg.3 : Semiring.{u_1} R] {ι : Type.{u_4}} {M : ι -> Type.{u_3}} [inst._@.Mathlib.Algebra.DirectSum.Module.1193137464._hygCtx._hyg.10 : forall (i : ι), AddCommMonoid.{u_3} (M i)] [inst._@.Mathlib.Algebra.DirectSum.Module.1193137464._hygCtx._hyg.19 : forall (i : ι), Module.{u_1, u_3} R (M i) inst._@.Mathlib.Algebra.DirectSum.Module.1193137464._hygCtx._hyg.3 (inst._@.Mathlib.Algebra.DirectSum.Module.1193137464._hygCtx._hyg.10 i)] {S : Type.{u_2}} [inst._@.Mathlib.Algebra.DirectSum.Module.1193137464._hygCtx._hyg.30 : Semiring.{u_2} S] [inst._@.Mathlib.Algebra.DirectSum.Module.1193137464._hygCtx._hyg.33 : forall (i : ι), Module.{u_2, u_3} S (M i) inst._@.Mathlib.Algebra.DirectSum.Module.1193137464._hygCtx._hyg.30 (inst._@.Mathlib.Algebra.DirectSum.Module.1193137464._hygCtx._hyg.10 i)] [inst._@.Mathlib.Algebra.DirectSum.Module.1193137464._hygCtx._hyg.43 : forall (i : ι), SMulCommClass.{u_1, u_2, u_3} R S (M i) 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