diff --git "a/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.GradedModule.sym.json" "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.GradedModule.sym.json"
new file mode 100644--- /dev/null
+++ "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.GradedModule.sym.json"
@@ -0,0 +1 @@
+[{"isProp":true,"kind":"definition","name":["SetLike","gmulAction","match_1"],"typeFallback":"forall {ιM : Type.{u_1}} {M : Type.{u_2}} {σ : Type.{u_3}} (𝓜 : ιM -> σ) [inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.43 : SetLike.{u_3, u_2} σ M] (motive : (GradedMonoid.{u_1, u_2} ιM (fun (i : ιM) => Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_3} M σ (SetLike.instMembership.{u_3, u_2} σ M inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.43) (𝓜 i) x))) -> Prop) (x._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx.83.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.91 : GradedMonoid.{u_1, u_2} ιM (fun (i : ιM) => Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_3} M σ (SetLike.instMembership.{u_3, u_2} σ M inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.43) (𝓜 i) x))), (forall (_i : ιM) (_m : Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_3} M σ (SetLike.instMembership.{u_3, u_2} σ M inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.43) (𝓜 _i) x)), motive (Sigma.mk.{u_1, u_2} ιM (fun (i : ιM) => Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_3} M σ (SetLike.instMembership.{u_3, u_2} σ M inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.43) (𝓜 i) x)) _i _m)) -> (motive x._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx.83.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.91)","typeFull":"∀ {ιM : Type u_1} {M : Type u_2} {σ : Type u_3} (𝓜 : ιM → σ) [inst : SetLike σ M]\n  (motive : (GradedMonoid fun i => ↥(𝓜 i)) → Prop) (x : GradedMonoid fun i => ↥(𝓜 i)),\n  (∀ (_i : ιM) (_m : ↥(𝓜 _i)), motive ⟨_i, _m⟩) → motive x","typeReadable":"∀ {ιM : Type u_1} {M : Type u_2} {σ : Type u_3} (𝓜 : ιM → σ) [inst : SetLike σ M]\n  (motive : (GradedMonoid fun i => ↥(𝓜 i)) → Prop) (x : GradedMonoid fun i => ↥(𝓜 i)),\n  (∀ (_i : ιM) (_m : ↥(𝓜 _i)), motive ⟨_i, _m⟩) → motive x","typeReferences":[["GradedMonoid"],["SetLike","instMembership"],["Subtype"],["Sigma","mk"],["Membership","mem"],["SetLike"]],"valueReferences":[["Sigma","casesOn"],["SetLike","instMembership"],["Subtype"],["Membership","mem"]]},{"isProp":true,"kind":"theorem","name":["GradedModule","linearEquiv","_proof_1"],"typeFallback":"forall {ιA : Type.{u_5}} {ιM : Type.{u_1}} {A : Type.{u_4}} {M : Type.{u_2}} {σ : Type.{u_3}} {σ' : Type.{u_6}} [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.9 : AddMonoid.{u_5} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.12 : AddAction.{u_5, u_1} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.9] [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.19 : Semiring.{u_4} A] (𝓐 : ιA -> σ') [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.29 : SetLike.{u_6, u_4} σ' A] (𝓜 : ιM -> σ) [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.39 : Module.{u_4, u_2} A M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36] [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43 : SetLike.{u_3, u_2} σ M] [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.47 : AddSubmonoidClass.{u_6, u_4} σ' A (AddMonoid.toAddZeroClass.{u_4} A (AddMonoidWithOne.toAddMonoid.{u_4} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_4} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_4} A (Semiring.toNonAssocSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.19))))) inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.51 : AddSubmonoidClass.{u_3, u_2} σ M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36)) inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43] [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.55 : SetLike.GradedMonoid.{u_5, u_4, u_6} ιA A σ' inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.29 (MonoidWithZero.toMonoid.{u_4} A (Semiring.toMonoidWithZero.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.19)) inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.9 𝓐] [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.58 : SetLike.GradedSMul.{u_5, u_1, u_6, u_4, u_3, u_2} ιA ιM σ' A σ M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43 (SMulZeroClass.toSMul.{u_4, u_2} A M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36)))) (DistribSMul.toSMulZeroClass.{u_4, u_2} A M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36)) (DistribMulAction.toDistribSMul.{u_4, u_2} A M (MonoidWithZero.toMonoid.{u_4} A (Semiring.toMonoidWithZero.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.19)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36) (Module.toDistribMulAction.{u_4, u_2} A M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.39)))) (AddSemigroupAction.toVAdd.{u_5, u_1} ιA ιM (AddMonoid.toAddSemigroup.{u_5} ιA inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_5, u_1} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.12)) 𝓐 𝓜] [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.62 : DecidableEq.{succ u_5} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.65 : DecidableEq.{succ u_1} ιM] [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.68 : GradedRing.{u_5, u_4, u_6} ιA A σ' inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.62 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.47 𝓐] [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.71 : DirectSum.Decomposition.{u_1, u_2, u_3} ιM M σ inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.65 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.51 𝓜] (x : A) (y : M), Eq.{succ (max u_1 u_2)} (DirectSum.{u_1, u_2} ιM (fun (i : ιM) => Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_3} M σ (SetLike.instMembership.{u_3, u_2} σ M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43) (𝓜 i) x)) (fun (i : ιM) => AddSubmonoidClass.toAddCommMonoid.{u_3, u_2} M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36 σ inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.51 (𝓜 i))) (AddHom.toFun.{u_2, max u_1 u_2} M (DirectSum.{u_1, u_2} ιM (fun (i : ιM) => Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_3} M σ 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inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.51 (𝓜 i))) (AddCommSemigroup.toAddCommMagma.{max u_1 u_2} (DirectSum.{u_1, u_2} ιM (fun (i : ιM) => Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_3} M σ (SetLike.instMembership.{u_3, u_2} σ M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43) (𝓜 i) x)) (fun (i : ιM) => AddSubmonoidClass.toAddCommMonoid.{u_3, u_2} M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36 σ inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.51 (𝓜 i))) (AddCommMonoid.toAddCommSemigroup.{max u_1 u_2} (DirectSum.{u_1, u_2} ιM (fun (i : ιM) => Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_3} M σ (SetLike.instMembership.{u_3, u_2} σ M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43) (𝓜 i) x)) (fun (i 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inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.51 (𝓜 i))) (AddCommSemigroup.toAddCommMagma.{max u_1 u_2} (DirectSum.{u_1, u_2} ιM (fun (i : ιM) => Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_3} M σ (SetLike.instMembership.{u_3, u_2} σ M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43) (𝓜 i) x)) (fun (i : ιM) => AddSubmonoidClass.toAddCommMonoid.{u_3, u_2} M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36 σ inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.51 (𝓜 i))) (AddCommMonoid.toAddCommSemigroup.{max u_1 u_2} (DirectSum.{u_1, u_2} ιM (fun (i : ιM) => Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_3} M σ (SetLike.instMembership.{u_3, u_2} σ M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43) (𝓜 i) x)) (fun (i : ιM) => AddSubmonoidClass.toAddCommMonoid.{u_3, u_2} M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36 σ inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.51 (𝓜 i))) (instAddCommMonoidDirectSum.{u_1, u_2} ιM (fun (i : ιM) => Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_3} M σ (SetLike.instMembership.{u_3, u_2} σ M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43) (𝓜 i) x)) (fun (i : ιM) => AddSubmonoidClass.toAddCommMonoid.{u_3, u_2} M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36 σ inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.51 (𝓜 i)))))) (DirectSum.decomposeAddEquiv.{u_1, u_2, u_3} ιM M σ inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.65 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.51 𝓜 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.71)) (HSMul.hSMul.{u_4, u_2, u_2} A M M (instHSMul.{u_4, u_2} A M (SMulZeroClass.toSMul.{u_4, u_2} A M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36)))) (DistribSMul.toSMulZeroClass.{u_4, u_2} A M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36)) (DistribMulAction.toDistribSMul.{u_4, u_2} A M (MonoidWithZero.toMonoid.{u_4} A (Semiring.toMonoidWithZero.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.19)) (AddCommMonoid.toAddMonoid.{u_2} M 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: Type u_4} {A : Type u_2} {M : Type u_5} {σ : Type u_6} {σ' : Type u_3} [inst : AddMonoid ιA]\n  [inst_1 : AddAction ιA ιM] [inst_2 : Semiring A] (𝓐 : ιA → σ') [inst_3 : SetLike σ' A] (𝓜 : ιM → σ)\n  [inst_4 : AddMonoid M] [inst_5 : DistribMulAction A M] [inst_6 : SetLike σ M] [inst_7 : SetLike.GradedMonoid 𝓐]\n  [inst_8 : SetLike.GradedSMul 𝓐 𝓜] (x x_1 : GradedMonoid fun i => ↥(𝓐 i)) (x_2 : GradedMonoid fun i => ↥(𝓜 i)),\n  (x * x_1) • x_2 = x • x_1 • x_2","typeReadable":"∀ {ιA : Type u_1} {ιM : Type u_4} {A : Type u_2} {M : Type u_5} {σ : Type u_6} {σ' : Type u_3} [inst : AddMonoid ιA]\n  [inst_1 : AddAction ιA ιM] [inst_2 : Semiring A] (𝓐 : ιA → σ') [inst_3 : SetLike σ' A] (𝓜 : ιM → σ)\n  [inst_4 : AddMonoid M] [inst_5 : DistribMulAction A M] [inst_6 : SetLike σ M] [inst_7 : SetLike.GradedMonoid 𝓐]\n  [inst_8 : SetLike.GradedSMul 𝓐 𝓜] (x x_1 : GradedMonoid fun i => ↥(𝓐 i)) (x_2 : GradedMonoid fun i => ↥(𝓜 i)),\n  (x * x_1) • x_2 = x • x_1 • 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{ιA : Type.{u_3}} {ιM : Type.{u_4}} {A : Type.{u_5}} {M : Type.{u_1}} {σ : Type.{u_2}} {σ' : Type.{u_6}} [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 : AddMonoid.{u_3} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12 : AddAction.{u_3, u_4} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19 : Semiring.{u_5} A] (𝓐 : ιA -> σ') [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29 : SetLike.{u_6, u_5} σ' A] (𝓜 : ιM -> σ) [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.39 : Module.{u_5, u_1} A M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43 : SetLike.{u_2, u_1} σ M] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.47 : AddSubmonoidClass.{u_6, u_5} σ' A (AddMonoid.toAddZeroClass.{u_5} A (AddMonoidWithOne.toAddMonoid.{u_5} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_5} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_5} A (Semiring.toNonAssocSemiring.{u_5} A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19))))) inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.51 : AddSubmonoidClass.{u_2, u_1} σ M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36)) inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.58 : SetLike.GradedSMul.{u_3, u_4, u_6, u_5, u_2, u_1} ιA ιM σ' A σ M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43 (SMulZeroClass.toSMul.{u_5, u_1} A M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36)))) (DistribSMul.toSMulZeroClass.{u_5, u_1} A M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36)) (DistribMulAction.toDistribSMul.{u_5, u_1} A M (MonoidWithZero.toMonoid.{u_5} A (Semiring.toMonoidWithZero.{u_5} A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36) (Module.toDistribMulAction.{u_5, u_1} A M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.39)))) (AddSemigroupAction.toVAdd.{u_3, u_4} ιA ιM (AddMonoid.toAddSemigroup.{u_3} ιA inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_3, u_4} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12)) 𝓐 𝓜] {i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.128 : ιA} {j._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.129 : ιM} (_b : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M σ (SetLike.instMembership.{u_2, u_1} σ M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43) (𝓜 j._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.129) x)), Eq.{succ u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M σ 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inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_3, u_4} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12))) i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.128 j._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.129)) x) (HSMul.hSMul.{u_5, u_1, u_1} A M M (instHSMul.{u_5, u_1} A M (SMulZeroClass.toSMul.{u_5, u_1} A M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36)))) (DistribSMul.toSMulZeroClass.{u_5, u_1} A M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36)) (DistribMulAction.toDistribSMul.{u_5, u_1} A M (MonoidWithZero.toMonoid.{u_5} A (Semiring.toMonoidWithZero.{u_5} A 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inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12))) i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.128 j._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.129)) x)) (AddMonoid.toAddZeroClass.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M σ (SetLike.instMembership.{u_2, u_1} σ M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43) (𝓜 (HVAdd.hVAdd.{u_3, u_4, u_4} ιA ιM ιM (instHVAdd.{u_3, u_4} ιA ιM (AddSemigroupAction.toVAdd.{u_3, u_4} ιA ιM (AddMonoid.toAddSemigroup.{u_3} ιA inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_3, u_4} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12))) i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.128 j._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.129)) x)) 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inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.51 (𝓜 (HVAdd.hVAdd.{u_3, u_4, u_4} ιA ιM ιM (instHVAdd.{u_3, u_4} ιA ιM (AddSemigroupAction.toVAdd.{u_3, u_4} ιA ιM (AddMonoid.toAddSemigroup.{u_3} ιA inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_3, u_4} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12))) i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.128 j._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.129)))))))))","typeFull":"∀ {ιA : Type u_3} {ιM : Type u_4} {A : Type u_5} {M : Type u_1} {σ : Type u_2} {σ' : Type u_6} [inst : AddMonoid ιA]\n  [inst_1 : AddAction ιA ιM] [inst_2 : Semiring A] (𝓐 : ιA → σ') [inst_3 : SetLike σ' A] (𝓜 : ιM → σ)\n  [inst_4 : AddCommMonoid M] [inst_5 : Module A M] [inst_6 : SetLike σ M] [inst_7 : AddSubmonoidClass σ' A]\n  [inst_8 : AddSubmonoidClass σ M] [inst_9 : SetLike.GradedSMul 𝓐 𝓜] {i : ιA} {j : ιM} (_b : ↥(𝓜 j)), ⟨↑0 • ↑_b, ⋯⟩ = 0","typeReadable":"∀ {ιA : Type u_3} {ιM : Type u_4} {A : Type u_5} {M : Type u_1} {σ : Type u_2} {σ' : Type u_6} [inst : AddMonoid ιA]\n  [inst_1 : AddAction ιA ιM] [inst_2 : Semiring A] (𝓐 : ιA → σ') [inst_3 : SetLike σ' A] (𝓜 : ιM → σ)\n  [inst_4 : AddCommMonoid M] [inst_5 : Module A M] [inst_6 : SetLike σ M] [inst_7 : AddSubmonoidClass σ' A]\n  [inst_8 : AddSubmonoidClass σ M] [inst_9 : SetLike.GradedSMul 𝓐 𝓜] {i : ιA} {j : ιM} (_b : ↥(𝓜 j)), ⟨↑0 • ↑_b, ⋯⟩ = 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{ιA : Type.{u_2}} {ιB : Type.{u_3}} (A : ιA -> Type.{u_4}) (M : ιB -> Type.{u_1}) [inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 : AddMonoid.{u_2} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 : VAdd.{u_2, u_3} ιA ιB] [inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.17 : forall (i : ιA), AddCommMonoid.{u_4} (A i)] [inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.28 : forall (i : ιB), AddCommMonoid.{u_1} (M i)] [inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.37 : GradedMonoid.GMonoid.{u_2, u_4} ιA A inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.40 : DirectSum.Gmodule.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 (fun (i : ιA) => AddCommMonoid.toAddMonoid.{u_4} (A i) (inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.17 i)) (fun (i : ιB) => AddCommMonoid.toAddMonoid.{u_1} (M i) (inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.28 i)) inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.37] {i : ιA} {j : ιB} (b : M j) (c : M j), Eq.{succ u_1} (M (HVAdd.hVAdd.{u_2, u_3, u_3} ιA ιB ιB (instHVAdd.{u_2, u_3} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13) i j)) (GradedMonoid.GSMul.smul.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 (GradedMonoid.GMulAction.toGSMul.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.37 (DirectSum.GdistribMulAction.toGMulAction.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.37 (fun (i : ιB) => AddCommMonoid.toAddMonoid.{u_1} (M i) (inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.28 i)) (DirectSum.Gmodule.toGdistribMulAction.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 (fun (i : ιA) => AddCommMonoid.toAddMonoid.{u_4} (A i) (inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.17 i)) (fun (i : ιB) => AddCommMonoid.toAddMonoid.{u_1} (M i) (inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.28 i)) inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.40))) i j (OfNat.ofNat.{u_4} (A i) 0 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inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 (GradedMonoid.GMulAction.toGSMul.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.37 (DirectSum.GdistribMulAction.toGMulAction.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.37 (fun (i : ιB) => AddCommMonoid.toAddMonoid.{u_1} (M i) (inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.28 i)) (DirectSum.Gmodule.toGdistribMulAction.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 (fun (i : ιA) => 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→ Type u_1) [inst : AddMonoid ιA] [inst_1 : VAdd ιA ιB]\n  [inst_2 : (i : ιA) → AddCommMonoid (A i)] [inst_3 : (i : ιB) → AddCommMonoid (M i)] [inst_4 : GradedMonoid.GMonoid A]\n  [inst_5 : DirectSum.Gmodule A M] {i : ιA} {j : ιB} (b c : M j),\n  GradedMonoid.GSMul.smul 0 (b + c) = GradedMonoid.GSMul.smul 0 b + GradedMonoid.GSMul.smul 0 c","typeReferences":[["VAdd"],["instHVAdd"],["instHAdd"],["AddMonoid"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["DirectSum","Gmodule","toGdistribMulAction"],["OfNat","ofNat"],["HVAdd","hVAdd"],["HAdd","hAdd"],["AddCommMonoid"],["AddMonoid","toAddSemigroup"],["GradedMonoid","GMulAction","toGSMul"],["Zero","toOfNat0"],["DirectSum","Gmodule"],["DirectSum","GdistribMulAction","toGMulAction"],["Eq"],["AddZero","toZero"],["GradedMonoid","GMonoid"],["AddSemigroup","toAdd"],["AddMonoid","toAddZeroClass"],["GradedMonoid","GSMul","smul"]],"valueReferences":[["DirectSum","GdistribMulAction","smul_add"],["Zero","toOfNat0"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toZero"],["DirectSum","Gmodule","toGdistribMulAction"],["OfNat","ofNat"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["DirectSum","gsmulHom","_proof_5"],"typeFallback":"forall {ιA : Type.{u_2}} {ιB : Type.{u_3}} (A : ιA -> Type.{u_4}) (M : ιB -> Type.{u_1}) [inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 : AddMonoid.{u_2} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 : VAdd.{u_2, u_3} ιA ιB] [inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.17 : forall (i : ιA), AddCommMonoid.{u_4} (A i)] [inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.28 : forall (i : ιB), AddCommMonoid.{u_1} (M i)] [inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.37 : GradedMonoid.GMonoid.{u_2, u_4} ιA A inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.40 : DirectSum.Gmodule.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 (fun (i : ιA) => AddCommMonoid.toAddMonoid.{u_4} (A i) 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(𝓜 : ιM → σ)\n  [inst_4 : AddMonoid M] [inst_5 : DistribMulAction A M] [inst_6 : SetLike σ M] [inst_7 : AddSubmonoidClass σ M]\n  [inst_8 : SetLike.GradedMonoid 𝓐] [inst_9 : SetLike.GradedSMul 𝓐 𝓜] {i : ιA} {j : ιM} (_a : ↥(𝓐 i)),\n  GradedMonoid.GSMul.smul _a 0 = 0","typeReferences":[["SetLike","gmulAction"],["Subtype"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddAction"],["AddAction","toAddSemigroupAction"],["MonoidWithZero","toMonoid"],["GradedMonoid","GMulAction","toGSMul"],["Zero","toOfNat0"],["SetLike"],["Eq"],["AddSubmonoidClass"],["DistribMulAction"],["GradedMonoid","GSMul","smul"],["DistribSMul","toSMulZeroClass"],["AddSubmonoidClass","toAddMonoid"],["SetLike","instMembership"],["instHVAdd"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddMonoid"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["HVAdd","hVAdd"],["AddSemigroupAction","toVAdd"],["AddMonoid","toAddSemigroup"],["AddZero","toZero"],["SetLike","GradedSMul"],["SetLike","gMonoid"],["SetLike","GradedMonoid"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["smul_zero"],["SetLike","gmulAction"],["Subtype"],["Membership","mem"],["Subtype","val"],["AddAction","toAddSemigroupAction"],["MonoidWithZero","toMonoid"],["GradedMonoid","GMulAction","toGSMul"],["Zero","toOfNat0"],["GradedMonoid","GSMul","smul"],["DistribSMul","toSMulZeroClass"],["AddSubmonoidClass","toAddMonoid"],["SetLike","instMembership"],["instHVAdd"],["DistribMulAction","toDistribSMul"],["Subtype","ext"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["HVAdd","hVAdd"],["OfNat","ofNat"],["AddSemigroupAction","toVAdd"],["AddMonoid","toAddSemigroup"],["AddZero","toZero"],["SetLike","gMonoid"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["DirectSum","Gmodule","smulAddMonoidHom","eq_1"],"typeFallback":"forall 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u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.13) _i _j)) (DirectSum.{u_2, u_4} ιB (fun (i : ιB) => M i) inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.28) (AddMonoid.toAddZeroClass.{u_4} (M _j) (AddCommMonoid.toAddMonoid.{u_4} (M _j) (inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.28 _j))) (inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.28 (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.13) _i _j)) (instAddCommMonoidDirectSum.{u_2, u_4} ιB (fun (i : ιB) => M i) inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.28)) (DirectSum.of.{u_2, u_4} ιB M inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.40 (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.13) _i _j))) (DirectSum.gsmulHom.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.46 _i _j))))))","typeFull":"∀ {ιA : Type u_1} {ιB : Type u_2} (A : ιA → Type u_3) (M : ιB → Type u_4) [inst : AddMonoid ιA] [inst_1 : VAdd ιA ιB]\n  [inst_2 : (i : ιA) → AddCommMonoid (A i)] [inst_3 : (i : ιB) → AddCommMonoid (M i)] [inst_4 : DecidableEq ιA]\n  [inst_5 : DecidableEq ιB] [inst_6 : GradedMonoid.GMonoid A] [inst_7 : DirectSum.Gmodule A M],\n  DirectSum.Gmodule.smulAddMonoidHom A M =\n    DirectSum.toAddMonoid fun _i =>\n      (DirectSum.toAddMonoid fun _j =>\n          ((AddMonoidHom.compHom (DirectSum.of M (_i +ᵥ _j))).comp (DirectSum.gsmulHom A M)).flip).flip","typeReadable":"∀ {ιA : Type u_1} {ιB : Type u_2} (A : ιA → Type u_3) (M : ιB → Type u_4) [inst : AddMonoid ιA] [inst_1 : VAdd ιA ιB]\n  [inst_2 : (i : ιA) → AddCommMonoid (A i)] [inst_3 : (i : ιB) → AddCommMonoid (M i)] [inst_4 : DecidableEq ιA]\n  [inst_5 : DecidableEq ιB] [inst_6 : GradedMonoid.GMonoid A] [inst_7 : DirectSum.Gmodule A M],\n  DirectSum.Gmodule.smulAddMonoidHom A M =\n    DirectSum.toAddMonoid fun _i =>\n      (DirectSum.toAddMonoid fun _j =>\n          ((AddMonoidHom.compHom (DirectSum.of M (_i +ᵥ _j))).comp (DirectSum.gsmulHom A 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y)","typeReferences":[["VAdd"],["DecidableEq"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["DirectSum"],["AddCommMonoid"],["instAddCommMonoidDirectSum"],["AddMonoidHom"],["GradedMonoid","GMulAction","toGSMul"],["instHSMul"],["AddMonoidHom","instFunLike"],["Eq"],["DirectSum","Gmodule","instSMulOfDecidableEq"],["GradedMonoid","GMonoid"],["GradedMonoid","GSMul","smul"],["instHVAdd"],["DirectSum","of"],["AddMonoid"],["AddZeroClass","toAddZero"],["DirectSum","Gmodule","toGdistribMulAction"],["HVAdd","hVAdd"],["HSMul","hSMul"],["DirectSum","Gmodule"],["DirectSum","GdistribMulAction","toGMulAction"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Eq","trans"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["congrArg"],["DirectSum"],["instAddCommMonoidDirectSum"],["AddMonoidHom"],["GradedMonoid","GMulAction","toGSMul"],["AddMonoidHom","instFunLike"],["instHSMul"],["congrFun'"],["Eq"],["DirectSum","Gmodule","instSMulOfDecidableEq"],["GradedMonoid","GSMul","smul"],["DirectSum","Gmodule","smulAddMonoidHom_apply_of_of"],["instHVAdd"],["True"],["DirectSum","of"],["AddZeroClass","toAddZero"],["DirectSum","Gmodule","toGdistribMulAction"],["HVAdd","hVAdd"],["eq_self"],["of_eq_true"],["HSMul","hSMul"],["DirectSum","GdistribMulAction","toGMulAction"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"definition","name":["DirectSum","Gmodule","mk","_flat_ctor"],"typeFallback":"forall 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inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43)","typeFull":"{ιA : Type u_1} →\n  {ιB : Type u_2} →\n    {A : ιA → Type u_3} →\n      {M : ιB → Type u_4} →\n        [inst : AddMonoid ιA] →\n          [inst_1 : VAdd ιA ιB] →\n            [inst_2 : (i : ιA) → AddMonoid (A i)] →\n              [inst_3 : (i : ιB) → AddMonoid (M i)] →\n                [inst_4 : GradedMonoid.GMonoid A] →\n                  (smul : {i : ιA} → {j : ιB} → A i → M j → M (i +ᵥ j)) →\n                    (∀ (b : GradedMonoid M), 1 • b = b) →\n                      (∀ (a a' : GradedMonoid A) (b : GradedMonoid M), (a * a') • b = a • a' • b) →\n                        (∀ {i : ιA} {j : ιB} (a : A i) (b c : M j), 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inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 toGdistribMulAction add_smul zero_smul) (DirectSum.Gmodule.mk.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 toGdistribMulAction' add_smul' zero_smul')) -> ((HEq.{max (max (max (succ u_1) (succ u_2)) (succ u_3)) (succ u_4)} (DirectSum.GdistribMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34) toGdistribMulAction (DirectSum.GdistribMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34) toGdistribMulAction') -> P) -> P","typeFull":"{ιA : Type u_1} →\n  {ιB : Type u_2} →\n    {A : ιA → Type u_3} →\n      {M : ιB → Type u_4} →\n        {inst : AddMonoid ιA} →\n          {inst_1 : VAdd ιA ιB} →\n            {inst_2 : (i : ιA) → AddMonoid (A i)} →\n              {inst_3 : (i : ιB) → AddMonoid (M i)} →\n                {inst_4 : GradedMonoid.GMonoid A} →\n                  {P : Sort u} →\n                    {toGdistribMulAction : DirectSum.GdistribMulAction A M} →\n                      {add_smul :\n                          ∀ {i : ιA} {j : ιB} (a a' : A i) (b : M j),\n                            GradedMonoid.GSMul.smul (a + a') b =\n                              GradedMonoid.GSMul.smul a b + GradedMonoid.GSMul.smul a' b} →\n                        {zero_smul : ∀ {i : ιA} {j : ιB} (b : M j), GradedMonoid.GSMul.smul 0 b = 0} →\n                          {toGdistribMulAction' : DirectSum.GdistribMulAction A M} →\n                            {add_smul' :\n                                ∀ {i : ιA} {j : ιB} (a a' : A i) (b : M j),\n                                  GradedMonoid.GSMul.smul (a + a') b =\n                                    GradedMonoid.GSMul.smul a b + GradedMonoid.GSMul.smul a' b} →\n                              {zero_smul' : ∀ {i : ιA} {j : ιB} (b : M j), GradedMonoid.GSMul.smul 0 b = 0} →\n                                { toGdistribMulAction := toGdistribMulAction, add_smul := add_smul,\n                                      zero_smul := zero_smul } =\n                                    { toGdistribMulAction := toGdistribMulAction', add_smul := add_smul',\n                                      zero_smul := zero_smul' } →\n                                  (toGdistribMulAction ≍ toGdistribMulAction' → P) → P","typeReadable":"{ιA : Type u_1} →\n  {ιB : Type u_2} →\n    {A : ιA → Type u_3} →\n      {M : ιB → Type u_4} →\n        {inst : AddMonoid ιA} →\n          {inst_1 : VAdd ιA ιB} →\n            {inst_2 : (i : ιA) → AddMonoid (A i)} →\n              {inst_3 : (i : ιB) → AddMonoid (M i)} →\n                {inst_4 : GradedMonoid.GMonoid A} →\n                  {P : Sort u} →\n                    {toGdistribMulAction : DirectSum.GdistribMulAction A M} →\n                      {add_smul :\n                          ∀ {i : ιA} {j : ιB} (a a' : A i) (b : M j),\n                            GradedMonoid.GSMul.smul (a + a') b =\n                              GradedMonoid.GSMul.smul a b + GradedMonoid.GSMul.smul a' b} →\n                        {zero_smul : ∀ {i : ιA} {j : ιB} (b : M j), GradedMonoid.GSMul.smul 0 b = 0} →\n                          {toGdistribMulAction' : DirectSum.GdistribMulAction A M} →\n                            {add_smul' :\n                                ∀ {i : ιA} {j : ιB} (a a' : A i) (b : M j),\n                                  GradedMonoid.GSMul.smul (a + a') b =\n                                    GradedMonoid.GSMul.smul a b + GradedMonoid.GSMul.smul a' b} →\n                              {zero_smul' : ∀ {i : ιA} {j : ιB} (b : M j), GradedMonoid.GSMul.smul 0 b = 0} →\n                                { toGdistribMulAction := toGdistribMulAction, add_smul := add_smul,\n                                      zero_smul := zero_smul } =\n                                    { toGdistribMulAction := toGdistribMulAction', add_smul := add_smul',\n                                      zero_smul := zero_smul' } →\n                                  (toGdistribMulAction ≍ toGdistribMulAction' → P) → P","typeReferences":[["VAdd"],["DirectSum","GdistribMulAction"],["instHVAdd"],["instHAdd"],["AddMonoid"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["HVAdd","hVAdd"],["DirectSum","Gmodule","mk"],["HAdd","hAdd"],["AddMonoid","toAddSemigroup"],["HEq"],["GradedMonoid","GMulAction","toGSMul"],["DirectSum","Gmodule"],["Zero","toOfNat0"],["DirectSum","GdistribMulAction","toGMulAction"],["Eq"],["AddZero","toZero"],["GradedMonoid","GMonoid"],["AddMonoid","toAddZeroClass"],["GradedMonoid","GSMul","smul"],["AddSemigroup","toAdd"]],"valueReferences":[["DirectSum","Gmodule","mk"],["VAdd"],["HEq","refl"],["Eq","refl"],["DirectSum","Gmodule","noConfusion"],["id"],["DirectSum","Gmodule"],["AddMonoid"],["heq_of_eq"],["GradedMonoid","GMonoid"]]},{"isProp":false,"kind":"constructor","name":["DirectSum","Gmodule","mk"],"typeFallback":"forall {ιA : Type.{u_1}} {ιB : Type.{u_2}} {A : ιA -> Type.{u_3}} {M : ιB -> Type.{u_4}} [inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 : AddMonoid.{u_1} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 : VAdd.{u_1, u_2} ιA ιB] [inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.25 : forall (i : ιA), AddMonoid.{u_3} (A i)] [inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 : forall (i : ιB), AddMonoid.{u_4} (M i)] [inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 : GradedMonoid.GMonoid.{u_1, u_3} ιA A inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18] [toGdistribMulAction : DirectSum.GdistribMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34], (forall {i : ιA} {j : ιB} (a : A i) (a' : A i) (b : M j), Eq.{succ u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} 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inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 toGdistribMulAction)) i j a b) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 (GradedMonoid.GMulAction.toGSMul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 (DirectSum.GdistribMulAction.toGMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 toGdistribMulAction)) i j a' b))) -> (forall {i : ιA} {j : ιB} (b 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                       GradedMonoid.GSMul.smul a b + GradedMonoid.GSMul.smul a' b) →\n                      (∀ {i : ιA} {j : ιB} (b : M j), GradedMonoid.GSMul.smul 0 b = 0) → DirectSum.Gmodule A M","typeReferences":[["VAdd"],["DirectSum","GdistribMulAction"],["instHVAdd"],["instHAdd"],["AddMonoid"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["HVAdd","hVAdd"],["HAdd","hAdd"],["AddMonoid","toAddSemigroup"],["GradedMonoid","GMulAction","toGSMul"],["DirectSum","Gmodule"],["Zero","toOfNat0"],["DirectSum","GdistribMulAction","toGMulAction"],["Eq"],["AddZero","toZero"],["GradedMonoid","GMonoid"],["AddMonoid","toAddZeroClass"],["GradedMonoid","GSMul","smul"],["AddSemigroup","toAdd"]],"valueReferences":null},{"isProp":false,"kind":"definition","name":["SetLike","gdistribMulAction"],"typeFallback":"forall {ιA : Type.{u_1}} {ιM : Type.{u_2}} {A : Type.{u_4}} {M : Type.{u_5}} {σ : Type.{u_6}} {σ' : Type.{u_7}} [inst._@.Mathlib.Algebra.Module.GradedModule.1644032540._hygCtx._hyg.9 : AddMonoid.{u_1} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.1644032540._hygCtx._hyg.12 : AddAction.{u_1, u_2} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.1644032540._hygCtx._hyg.9] [inst._@.Mathlib.Algebra.Module.GradedModule.1644032540._hygCtx._hyg.19 : Semiring.{u_4} A] (𝓐 : ιA -> σ') [inst._@.Mathlib.Algebra.Module.GradedModule.1644032540._hygCtx._hyg.29 : SetLike.{u_7, u_4} σ' A] (𝓜 : ιM -> σ) [inst._@.Mathlib.Algebra.Module.GradedModule.1644032540._hygCtx._hyg.36 : AddMonoid.{u_5} M] [inst._@.Mathlib.Algebra.Module.GradedModule.1644032540._hygCtx._hyg.39 : DistribMulAction.{u_4, u_5} A M (MonoidWithZero.toMonoid.{u_4} A (Semiring.toMonoidWithZero.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.1644032540._hygCtx._hyg.19)) inst._@.Mathlib.Algebra.Module.GradedModule.1644032540._hygCtx._hyg.36] [inst._@.Mathlib.Algebra.Module.GradedModule.1644032540._hygCtx._hyg.43 : SetLike.{u_6, u_5} σ M] [inst._@.Mathlib.Algebra.Module.GradedModule.1644032540._hygCtx._hyg.47 : AddSubmonoidClass.{u_6, u_5} σ M (AddMonoid.toAddZeroClass.{u_5} M inst._@.Mathlib.Algebra.Module.GradedModule.1644032540._hygCtx._hyg.36) inst._@.Mathlib.Algebra.Module.GradedModule.1644032540._hygCtx._hyg.43] [inst._@.Mathlib.Algebra.Module.GradedModule.1644032540._hygCtx._hyg.51 : SetLike.GradedMonoid.{u_1, u_4, u_7} ιA A σ' inst._@.Mathlib.Algebra.Module.GradedModule.1644032540._hygCtx._hyg.29 (MonoidWithZero.toMonoid.{u_4} A (Semiring.toMonoidWithZero.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.1644032540._hygCtx._hyg.19)) inst._@.Mathlib.Algebra.Module.GradedModule.1644032540._hygCtx._hyg.9 𝓐] [inst._@.Mathlib.Algebra.Module.GradedModule.1644032540._hygCtx._hyg.54 : SetLike.GradedSMul.{u_1, u_2, u_7, u_4, u_6, u_5} ιA ιM σ' A σ M inst._@.Mathlib.Algebra.Module.GradedModule.1644032540._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.GradedModule.1644032540._hygCtx._hyg.43 (SMulZeroClass.toSMul.{u_4, u_5} A M (AddZero.toZero.{u_5} M (AddZeroClass.toAddZero.{u_5} M 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M] →\n                            [inst_6 : SetLike σ M] →\n                              [inst_7 : AddSubmonoidClass σ M] →\n                                [inst_8 : SetLike.GradedMonoid 𝓐] →\n                                  [SetLike.GradedSMul 𝓐 𝓜] →\n                                    DirectSum.GdistribMulAction (fun i => ↥(𝓐 i)) fun i => ↥(𝓜 i)","typeReadable":"{ιA : Type u_1} →\n  {ιM : Type u_2} →\n    {A : Type u_4} →\n      {M : Type u_5} →\n        {σ : Type u_6} →\n          {σ' : Type u_7} →\n            [inst : AddMonoid ιA] →\n              [inst_1 : AddAction ιA ιM] →\n                [inst_2 : Semiring A] →\n                  (𝓐 : ιA → σ') →\n                    [inst_3 : SetLike σ' A] →\n                      (𝓜 : ιM → σ) →\n                        [inst_4 : AddMonoid M] →\n                          [inst_5 : DistribMulAction A M] →\n                            [inst_6 : SetLike σ M] →\n                              [inst_7 : AddSubmonoidClass σ M] →\n                                [inst_8 : SetLike.GradedMonoid 𝓐] →\n                                  [SetLike.GradedSMul 𝓐 𝓜] →\n                                    DirectSum.GdistribMulAction (fun i => ↥(𝓐 i)) fun i => ↥(𝓜 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{ιA : Type.{u_3}} {ιM : Type.{u_4}} {A : Type.{u_5}} {M : Type.{u_1}} {σ : Type.{u_2}} {σ' : Type.{u_6}} [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 : AddMonoid.{u_3} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12 : AddAction.{u_3, u_4} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19 : Semiring.{u_5} A] (𝓐 : ιA -> σ') [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29 : SetLike.{u_6, u_5} σ' A] (𝓜 : ιM -> σ) [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.39 : Module.{u_5, u_1} A M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43 : SetLike.{u_2, u_1} σ M] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.47 : AddSubmonoidClass.{u_6, u_5} σ' A (AddMonoid.toAddZeroClass.{u_5} A (AddMonoidWithOne.toAddMonoid.{u_5} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_5} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_5} A (Semiring.toNonAssocSemiring.{u_5} A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19))))) inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.58 : SetLike.GradedSMul.{u_3, u_4, u_6, u_5, u_2, u_1} ιA ιM σ' A σ M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43 (SMulZeroClass.toSMul.{u_5, u_1} A M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36)))) (DistribSMul.toSMulZeroClass.{u_5, u_1} A M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36)) (DistribMulAction.toDistribSMul.{u_5, u_1} A M (MonoidWithZero.toMonoid.{u_5} A (Semiring.toMonoidWithZero.{u_5} A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36) (Module.toDistribMulAction.{u_5, u_1} A M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.39)))) (AddSemigroupAction.toVAdd.{u_3, u_4} ιA ιM (AddMonoid.toAddSemigroup.{u_3} ιA inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_3, u_4} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12)) 𝓐 𝓜] {i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.128 : ιA} {j._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.129 : ιM} (_b : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M σ (SetLike.instMembership.{u_2, u_1} σ M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43) (𝓜 j._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.129) x)), Membership.mem.{u_1, u_2} M σ (SetLike.instMembership.{u_2, u_1} σ M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43) (𝓜 (HVAdd.hVAdd.{u_3, u_4, u_4} ιA ιM ιM (instHVAdd.{u_3, u_4} ιA ιM (AddSemigroupAction.toVAdd.{u_3, u_4} ιA ιM (AddMonoid.toAddSemigroup.{u_3} ιA inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_3, u_4} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12))) i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.128 j._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.129)) (HSMul.hSMul.{u_5, u_1, u_1} A M M (instHSMul.{u_5, u_1} A M (SMulZeroClass.toSMul.{u_5, u_1} A M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36)))) (DistribSMul.toSMulZeroClass.{u_5, u_1} A M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36)) (DistribMulAction.toDistribSMul.{u_5, u_1} A M (MonoidWithZero.toMonoid.{u_5} A (Semiring.toMonoidWithZero.{u_5} A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36) (Module.toDistribMulAction.{u_5, u_1} A M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.39))))) (Subtype.val.{succ u_5} A (fun (x : A) => Membership.mem.{u_5, u_6} A σ' (SetLike.instMembership.{u_6, u_5} σ' A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29) (𝓐 i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.128) x) (OfNat.ofNat.{u_5} (Subtype.{succ u_5} A (fun (x : A) => Membership.mem.{u_5, u_6} A σ' (SetLike.instMembership.{u_6, u_5} σ' A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29) (𝓐 i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.128) x)) 0 (Zero.toOfNat0.{u_5} (Subtype.{succ u_5} A (fun (x : A) => Membership.mem.{u_5, u_6} A σ' (SetLike.instMembership.{u_6, u_5} σ' A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29) (𝓐 i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.128) x)) (AddZero.toZero.{u_5} (Subtype.{succ u_5} A (fun (x : A) => Membership.mem.{u_5, u_6} A σ' (SetLike.instMembership.{u_6, u_5} σ' A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29) (𝓐 i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.128) x)) (AddZeroClass.toAddZero.{u_5} (Subtype.{succ u_5} A (fun (x : A) => Membership.mem.{u_5, u_6} A σ' (SetLike.instMembership.{u_6, u_5} σ' A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29) (𝓐 i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.128) x)) (AddMonoid.toAddZeroClass.{u_5} (Subtype.{succ u_5} A (fun (x : A) => Membership.mem.{u_5, u_6} A σ' (SetLike.instMembership.{u_6, u_5} σ' A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29) (𝓐 i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.128) x)) (AddCommMonoid.toAddMonoid.{u_5} (Subtype.{succ u_5} A (fun (x : A) => 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: Type u_4} {A : Type u_5} {M : Type u_1} {σ : Type u_2} {σ' : Type u_6} [inst : AddMonoid ιA]\n  [inst_1 : AddAction ιA ιM] [inst_2 : Semiring A] (𝓐 : ιA → σ') [inst_3 : SetLike σ' A] (𝓜 : ιM → σ)\n  [inst_4 : AddCommMonoid M] [inst_5 : Module A M] [inst_6 : SetLike σ M] [inst_7 : AddSubmonoidClass σ' A]\n  [SetLike.GradedSMul 𝓐 𝓜] {i : ιA} {j : ιM} (_b : ↥(𝓜 j)), ↑0 • ↑_b ∈ 𝓜 (i +ᵥ j)","typeReadable":"∀ {ιA : Type u_3} {ιM : Type u_4} {A : Type u_5} {M : Type u_1} {σ : Type u_2} {σ' : Type u_6} [inst : AddMonoid ιA]\n  [inst_1 : AddAction ιA ιM] [inst_2 : Semiring A] (𝓐 : ιA → σ') [inst_3 : SetLike σ' A] (𝓜 : ιM → σ)\n  [inst_4 : AddCommMonoid M] [inst_5 : Module A M] [inst_6 : SetLike σ M] [inst_7 : AddSubmonoidClass σ' A]\n  [SetLike.GradedSMul 𝓐 𝓜] {i : ιA} {j : ιM} (_b : ↥(𝓜 j)), ↑0 • ↑_b ∈ 𝓜 (i +ᵥ 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{inst_3 : (i : ιB) → AddMonoid (M i)} [self : DirectSum.GdistribMulAction A M]\n  {i : ιA} {j : ιB} (a : A i), GradedMonoid.GSMul.smul a 0 = 0","typeReadable":"∀ {ιA : Type u_1} {ιB : Type u_2} {A : ιA → Type u_3} {M : ιB → Type u_4} {inst : AddMonoid ιA} {inst_1 : VAdd ιA ιB}\n  {inst_2 : GradedMonoid.GMonoid A} {inst_3 : (i : ιB) → AddMonoid (M i)} [self : DirectSum.GdistribMulAction A M]\n  {i : ιA} {j : ιB} (a : A i), GradedMonoid.GSMul.smul a 0 = 0","typeReferences":[["VAdd"],["DirectSum","GdistribMulAction"],["instHVAdd"],["AddMonoid"],["AddZeroClass","toAddZero"],["HVAdd","hVAdd"],["OfNat","ofNat"],["GradedMonoid","GMulAction","toGSMul"],["DirectSum","GdistribMulAction","toGMulAction"],["Zero","toOfNat0"],["Eq"],["AddZero","toZero"],["GradedMonoid","GMonoid"],["AddMonoid","toAddZeroClass"],["GradedMonoid","GSMul","smul"]],"valueReferences":[]},{"isProp":true,"kind":"theorem","name":["GradedModule","linearEquiv","_proof_3"],"typeFallback":"forall {ιM : Type.{u_2}} {M : 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: A j),\n  GradedMonoid.GMul.mul (x + x_1) c = GradedMonoid.GMul.mul x c + GradedMonoid.GMul.mul x_1 c","typeReadable":"∀ {ιA : Type u_2} (A : ιA → Type u_1) [inst : AddMonoid ιA] [inst_1 : (i : ιA) → AddCommMonoid (A i)]\n  [h : DirectSum.GSemiring A] {i j : ιA} (x x_1 : A i) (c : A j),\n  GradedMonoid.GMul.mul (x + x_1) c = GradedMonoid.GMul.mul x c + GradedMonoid.GMul.mul x_1 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inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 toGMulAction) i j a b) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 (GradedMonoid.GMulAction.toGSMul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 toGMulAction) i j a c))) -> (forall {i : ιA} {j : ιB} (a : A i), Eq.{succ u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 (GradedMonoid.GMulAction.toGSMul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 toGMulAction) i j a (OfNat.ofNat.{u_4} (M j) 0 (Zero.toOfNat0.{u_4} (M j) (AddZero.toZero.{u_4} (M j) (AddZeroClass.toAddZero.{u_4} (M j) (AddMonoid.toAddZeroClass.{u_4} (M j) (inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 j))))))) (OfNat.ofNat.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) 0 (Zero.toOfNat0.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (AddZero.toZero.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (AddZeroClass.toAddZero.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA 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ιB] →\n            [inst_2 : GradedMonoid.GMonoid A] →\n              [inst_3 : (i : ιB) → AddMonoid (M i)] →\n                [toGMulAction : GradedMonoid.GMulAction A M] →\n                  (∀ {i : ιA} {j : ιB} (a : A i) (b c : M j),\n                      GradedMonoid.GSMul.smul a (b + c) = GradedMonoid.GSMul.smul a b + GradedMonoid.GSMul.smul a c) →\n                    (∀ {i : ιA} {j : ιB} (a : A i), GradedMonoid.GSMul.smul a 0 = 0) → DirectSum.GdistribMulAction A M","typeReadable":"{ιA : Type u_1} →\n  {ιB : Type u_2} →\n    {A : ιA → Type u_3} →\n      {M : ιB → Type u_4} →\n        [inst : AddMonoid ιA] →\n          [inst_1 : VAdd ιA ιB] →\n            [inst_2 : GradedMonoid.GMonoid A] →\n              [inst_3 : (i : ιB) → AddMonoid (M i)] →\n                [toGMulAction : GradedMonoid.GMulAction A M] →\n                  (∀ {i : ιA} {j : ιB} (a : A i) (b c : M j),\n                      GradedMonoid.GSMul.smul a (b + c) = GradedMonoid.GSMul.smul a b + GradedMonoid.GSMul.smul a c) →\n                    (∀ {i : ιA} {j : ιB} (a : A i), GradedMonoid.GSMul.smul a 0 = 0) → DirectSum.GdistribMulAction A M","typeReferences":[["DirectSum","GdistribMulAction"],["VAdd"],["instHVAdd"],["instHAdd"],["AddMonoid"],["AddZeroClass","toAddZero"],["GradedMonoid","GMulAction"],["OfNat","ofNat"],["HVAdd","hVAdd"],["HAdd","hAdd"],["AddMonoid","toAddSemigroup"],["GradedMonoid","GMulAction","toGSMul"],["Zero","toOfNat0"],["Eq"],["AddZero","toZero"],["GradedMonoid","GMonoid"],["AddMonoid","toAddZeroClass"],["GradedMonoid","GSMul","smul"],["AddSemigroup","toAdd"]],"valueReferences":null},{"isProp":false,"kind":"definition","name":["DirectSum","Gmodule","recOn"],"typeFallback":"forall {ιA : Type.{u_1}} {ιB : Type.{u_2}} {A : ιA -> Type.{u_3}} {M : ιB -> Type.{u_4}} [inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 : AddMonoid.{u_1} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 : VAdd.{u_1, u_2} ιA ιB] [inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.25 : forall (i : ιA), AddMonoid.{u_3} (A i)] [inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 : forall (i : ιB), AddMonoid.{u_4} (M i)] [inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 : GradedMonoid.GMonoid.{u_1, u_3} ιA A inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18] {motive : (DirectSum.Gmodule.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43) -> Sort.{u}} (t : DirectSum.Gmodule.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43), (forall [toGdistribMulAction : DirectSum.GdistribMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34] (add_smul : forall {i : ιA} {j : ιB} (a : A i) (a' : A i) (b : M j), Eq.{succ u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M 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(HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (instHAdd.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (AddSemigroup.toAdd.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (AddMonoid.toAddSemigroup.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j))))) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 (GradedMonoid.GMulAction.toGSMul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 (DirectSum.GdistribMulAction.toGMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 toGdistribMulAction)) i j a b) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 (GradedMonoid.GMulAction.toGSMul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 (DirectSum.GdistribMulAction.toGMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 toGdistribMulAction)) i j a' b))) (zero_smul : forall {i : ιA} {j : ιB} (b : M j), Eq.{succ u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 (GradedMonoid.GMulAction.toGSMul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 (DirectSum.GdistribMulAction.toGMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 toGdistribMulAction)) i j (OfNat.ofNat.{u_3} (A i) 0 (Zero.toOfNat0.{u_3} (A i) (AddZero.toZero.{u_3} (A i) (AddZeroClass.toAddZero.{u_3} (A i) (AddMonoid.toAddZeroClass.{u_3} (A i) (inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.25 i)))))) b) (OfNat.ofNat.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) 0 (Zero.toOfNat0.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (AddZero.toZero.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (AddZeroClass.toAddZero.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (AddMonoid.toAddZeroClass.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)))))))), motive (DirectSum.Gmodule.mk.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 toGdistribMulAction add_smul zero_smul)) -> (motive t)","typeFull":"{ιA : Type u_1} →\n  {ιB : Type u_2} →\n    {A : ιA → Type u_3} →\n      {M : ιB → Type u_4} →\n        [inst : AddMonoid ιA] →\n          [inst_1 : VAdd ιA ιB] →\n            [inst_2 : (i : ιA) → AddMonoid (A i)] →\n              [inst_3 : (i : ιB) → AddMonoid (M i)] →\n                [inst_4 : GradedMonoid.GMonoid A] →\n                  {motive : DirectSum.Gmodule A M → Sort u} →\n                    (t : DirectSum.Gmodule A M) →\n                      ([toGdistribMulAction : DirectSum.GdistribMulAction A M] →\n                          (add_smul :\n                              ∀ {i : ιA} {j : ιB} (a a' : A i) (b : M j),\n                                GradedMonoid.GSMul.smul (a + a') b =\n                                  GradedMonoid.GSMul.smul a b + GradedMonoid.GSMul.smul a' b) →\n                            (zero_smul : ∀ {i : ιA} {j : ιB} (b : M j), GradedMonoid.GSMul.smul 0 b = 0) →\n                              motive\n                                { toGdistribMulAction := toGdistribMulAction, add_smul := add_smul,\n                                  zero_smul := zero_smul }) →\n                        motive t","typeReadable":"{ιA : Type u_1} →\n  {ιB : Type u_2} →\n    {A : ιA → Type u_3} →\n      {M : ιB → Type u_4} →\n        [inst : AddMonoid ιA] →\n          [inst_1 : VAdd ιA ιB] →\n            [inst_2 : (i : ιA) → AddMonoid (A i)] →\n              [inst_3 : (i : ιB) → AddMonoid (M i)] →\n                [inst_4 : GradedMonoid.GMonoid A] →\n                  {motive : DirectSum.Gmodule A M → Sort u} →\n                    (t : DirectSum.Gmodule A M) →\n                      ([toGdistribMulAction : DirectSum.GdistribMulAction A M] →\n                          (add_smul :\n                              ∀ {i : ιA} {j : ιB} (a a' : A i) (b : M j),\n                                GradedMonoid.GSMul.smul (a + a') b =\n                                  GradedMonoid.GSMul.smul a b + GradedMonoid.GSMul.smul a' b) →\n                            (zero_smul : ∀ {i : ιA} {j : ιB} (b : M j), GradedMonoid.GSMul.smul 0 b = 0) →\n                              motive\n                                { toGdistribMulAction := toGdistribMulAction, add_smul := add_smul,\n                                  zero_smul := zero_smul }) →\n                        motive t","typeReferences":[["VAdd"],["DirectSum","GdistribMulAction"],["instHVAdd"],["instHAdd"],["AddMonoid"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["HVAdd","hVAdd"],["DirectSum","Gmodule","mk"],["HAdd","hAdd"],["AddMonoid","toAddSemigroup"],["GradedMonoid","GMulAction","toGSMul"],["Zero","toOfNat0"],["DirectSum","Gmodule"],["DirectSum","GdistribMulAction","toGMulAction"],["Eq"],["AddZero","toZero"],["GradedMonoid","GMonoid"],["AddMonoid","toAddZeroClass"],["AddSemigroup","toAdd"],["GradedMonoid","GSMul","smul"]],"valueReferences":[["DirectSum","Gmodule","rec"]]},{"isProp":false,"kind":"definition","name":["GradedModule","linearEquiv"],"typeFallback":"forall {ιA : Type.{u_1}} {ιM : Type.{u_2}} {A : Type.{u_4}} {M : Type.{u_5}} {σ : Type.{u_6}} {σ' : Type.{u_7}} [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.9 : AddMonoid.{u_1} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.12 : AddAction.{u_1, u_2} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.9] [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.19 : Semiring.{u_4} A] (𝓐 : ιA -> σ') [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.29 : SetLike.{u_7, u_4} σ' A] (𝓜 : ιM -> σ) [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36 : AddCommMonoid.{u_5} M] [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.39 : Module.{u_4, u_5} A M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36] [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43 : SetLike.{u_6, u_5} σ M] [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.47 : AddSubmonoidClass.{u_7, u_4} σ' A (AddMonoid.toAddZeroClass.{u_4} A (AddMonoidWithOne.toAddMonoid.{u_4} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_4} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_4} A (Semiring.toNonAssocSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.19))))) inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.51 : AddSubmonoidClass.{u_6, u_5} σ M (AddMonoid.toAddZeroClass.{u_5} M (AddCommMonoid.toAddMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36)) inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43] [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.55 : SetLike.GradedMonoid.{u_1, u_4, u_7} ιA A σ' inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.29 (MonoidWithZero.toMonoid.{u_4} A (Semiring.toMonoidWithZero.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.19)) inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.9 𝓐] [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.58 : SetLike.GradedSMul.{u_1, u_2, u_7, u_4, u_6, u_5} ιA ιM σ' A σ M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43 (SMulZeroClass.toSMul.{u_4, u_5} A M (AddZero.toZero.{u_5} M (AddZeroClass.toAddZero.{u_5} M (AddMonoid.toAddZeroClass.{u_5} M (AddCommMonoid.toAddMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36)))) (DistribSMul.toSMulZeroClass.{u_4, u_5} A M (AddMonoid.toAddZeroClass.{u_5} M (AddCommMonoid.toAddMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36)) (DistribMulAction.toDistribSMul.{u_4, u_5} A M (MonoidWithZero.toMonoid.{u_4} A (Semiring.toMonoidWithZero.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.19)) (AddCommMonoid.toAddMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36) (Module.toDistribMulAction.{u_4, u_5} A M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.39)))) (AddSemigroupAction.toVAdd.{u_1, u_2} ιA ιM (AddMonoid.toAddSemigroup.{u_1} ιA inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_1, u_2} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.12)) 𝓐 𝓜] [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.62 : DecidableEq.{succ u_1} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.65 : DecidableEq.{succ u_2} ιM] [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.68 : GradedRing.{u_1, u_4, u_7} ιA A σ' inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.62 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.47 𝓐] [inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.71 : DirectSum.Decomposition.{u_2, u_5, u_6} ιM M σ inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.65 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.51 𝓜], LinearEquiv.{u_4, u_4, u_5, max u_5 u_2} A A inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.19 (RingHom.id.{u_4} A (Semiring.toNonAssocSemiring.{u_4} A 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(instAddCommMonoidDirectSum.{u_2, u_5} ιM (fun (i : ιM) => Subtype.{succ u_5} M (fun (x : M) => Membership.mem.{u_5, u_6} M σ (SetLike.instMembership.{u_6, u_5} σ M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43) (𝓜 i) x)) (fun (i : ιM) => AddSubmonoidClass.toAddCommMonoid.{u_6, u_5} M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36 σ inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.51 (𝓜 i))) inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.39 (inferInstance.{max (max (succ u_2) (succ u_4)) (succ u_5)} ([mdata noImplicitLambda:1 Module.{u_4, max u_5 u_2} A (DirectSum.{u_2, u_5} ιM (fun (i : ιM) => Subtype.{succ u_5} M (fun (x : M) => Membership.mem.{u_5, u_6} M σ (SetLike.instMembership.{u_6, u_5} σ M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43) (𝓜 i) x)) (fun (i : ιM) => AddSubmonoidClass.toAddCommMonoid.{u_6, u_5} M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36 σ inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.51 (𝓜 i))) inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.19 (instAddCommMonoidDirectSum.{u_2, u_5} ιM (fun (i : ιM) => Subtype.{succ u_5} M (fun (x : M) => Membership.mem.{u_5, u_6} M σ (SetLike.instMembership.{u_6, u_5} σ M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43) (𝓜 i) x)) (fun (i : ιM) => AddSubmonoidClass.toAddCommMonoid.{u_6, u_5} M inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36 σ inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.51 (𝓜 i)))]) (GradedModule.isModule.{u_1, u_2, u_4, u_5, u_6, u_7} ιA ιM A M σ σ' inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.19 𝓐 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.29 𝓜 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.47 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.51 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.55 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.58 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.62 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.65 inst._@.Mathlib.Algebra.Module.GradedModule.2994524416._hygCtx._hyg.68))","typeFull":"{ιA : Type u_1} →\n  {ιM : Type u_2} →\n    {A : Type u_4} →\n      {M : Type u_5} →\n        {σ : Type u_6} →\n          {σ' : Type u_7} →\n            [inst : AddMonoid ιA] →\n              [inst_1 : AddAction ιA ιM] →\n                [inst_2 : Semiring A] →\n                  (𝓐 : ιA → σ') ���\n                    [inst_3 : SetLike σ' A] →\n                      (𝓜 : ιM → σ) →\n                        [inst_4 : AddCommMonoid M] →\n                          [inst_5 : Module A M] →\n                            [inst_6 : SetLike σ M] →\n                              [inst_7 : AddSubmonoidClass σ' A] →\n                                [inst_8 : AddSubmonoidClass σ M] →\n                                  [inst_9 : SetLike.GradedMonoid 𝓐] →\n                                    [inst_10 : SetLike.GradedSMul 𝓐 𝓜] →\n                                      [inst_11 : DecidableEq ιA] →\n                                        [inst_12 : DecidableEq ιM] →\n                                          [inst_13 : GradedRing 𝓐] →\n                                            [DirectSum.Decomposition 𝓜] → M ≃ₗ[A] DirectSum ιM fun i => ↥(𝓜 i)","typeReadable":"{ιA : Type u_1} →\n  {ιM : Type u_2} →\n    {A : Type u_4} →\n      {M : Type u_5} →\n        {σ : Type u_6} →\n          {σ' : Type u_7} →\n            [inst : AddMonoid ιA] →\n              [inst_1 : AddAction ιA ιM] →\n                [inst_2 : Semiring A] →\n                  (𝓐 : ιA → σ') →\n                    [inst_3 : SetLike σ' A] →\n                      (𝓜 : ιM → σ) →\n                        [inst_4 : AddCommMonoid M] →\n                          [inst_5 : Module A M] →\n                            [inst_6 : SetLike σ M] →\n                              [inst_7 : AddSubmonoidClass σ' A] →\n                                [inst_8 : AddSubmonoidClass σ M] →\n                                  [inst_9 : SetLike.GradedMonoid 𝓐] →\n                                    [inst_10 : SetLike.GradedSMul 𝓐 𝓜] →\n                                      [inst_11 : DecidableEq ιA] →\n                                        [inst_12 : DecidableEq ιM] →\n                                          [inst_13 : GradedRing 𝓐] →\n                                            [DirectSum.Decomposition 𝓜] → M ≃ₗ[A] DirectSum ιM fun i => ↥(𝓜 i)","typeReferences":[["Subtype"],["Module"],["DecidableEq"],["Membership","mem"],["DirectSum","Decomposition"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["LinearEquiv"],["AddMonoidWithOne","toAddMonoid"],["GradedRing"],["AddAction"],["AddAction","toAddSemigroupAction"],["DirectSum"],["AddCommMonoid"],["GradedModule","isModule"],["Semiring","toNonAssocSemiring"],["instAddCommMonoidDirectSum"],["RingHom","id"],["MonoidWithZero","toMonoid"],["SetLike"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["NonAssocSemiring","toAddCommMonoidWithOne"],["AddSubmonoidClass"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddMonoid"],["AddZeroClass","toAddZero"],["AddSubmonoidClass","toAddCommMonoid"],["Module","toDistribMulAction"],["AddSemigroupAction","toVAdd"],["AddMonoid","toAddSemigroup"],["RingHomInvPair","ids"],["inferInstance"],["SetLike","GradedSMul"],["AddZero","toZero"],["SetLike","GradedMonoid"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Subtype"],["Module"],["LinearMap","mk"],["Membership","mem"],["AddEquiv","toAddHom"],["GradedModule","linearEquiv","_proof_3"],["DirectSum"],["GradedModule","isModule"],["Semiring","toNonAssocSemiring"],["instAddCommMonoidDirectSum"],["RingHom","id"],["LinearEquiv","mk"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Equiv","toFun"],["AddEquiv","symm"],["SetLike","instMembership"],["GradedModule","linearEquiv","_proof_2"],["DirectSum","decomposeAddEquiv"],["AddEquiv","toEquiv"],["AddSubmonoidClass","toAddCommMonoid"],["AddCommMonoid","toAddCommSemigroup"],["GradedModule","linearEquiv","_proof_1"],["RingHomInvPair","ids"],["inferInstance"]]},{"isProp":true,"kind":"theorem","name":["DirectSum","Gmodule","module","_proof_1"],"typeFallback":"forall {ιA : Type.{u_3}} {ιB : Type.{u_1}} (A : ιA -> Type.{u_4}) (M : ιB -> Type.{u_2}) [inst._@.Mathlib.Algebra.Module.GradedModule.1565340096._hygCtx._hyg.10 : AddMonoid.{u_3} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.1565340096._hygCtx._hyg.13 : VAdd.{u_3, u_1} ιA ιB] [inst._@.Mathlib.Algebra.Module.GradedModule.1565340096._hygCtx._hyg.17 : forall (i : ιA), AddCommMonoid.{u_4} (A i)] [inst._@.Mathlib.Algebra.Module.GradedModule.1565340096._hygCtx._hyg.28 : forall (i : ιB), AddCommMonoid.{u_2} (M i)] [inst._@.Mathlib.Algebra.Module.GradedModule.1565340096._hygCtx._hyg.37 : DecidableEq.{succ u_3} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.1565340096._hygCtx._hyg.40 : DecidableEq.{succ u_1} ιB] [inst._@.Mathlib.Algebra.Module.GradedModule.1565340096._hygCtx._hyg.43 : DirectSum.GSemiring.{u_3, u_4} ιA A inst._@.Mathlib.Algebra.Module.GradedModule.1565340096._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.1565340096._hygCtx._hyg.17] 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AddCommMonoid (A i)] [inst_3 : (i : ιB) → AddCommMonoid (M i)] [inst_4 : DecidableEq ιA]\n  [inst_5 : DecidableEq ιB] [inst_6 : DirectSum.GSemiring A] [inst_7 : DirectSum.Gmodule A M]\n  (r s : DirectSum ιA fun i => A i) (x : DirectSum ιB fun i => M i), (r + s) • x = r • x + s • x","typeReadable":"∀ {ιA : Type u_3} {ιB : Type u_1} (A : ιA → Type u_4) (M : ιB → Type u_2) [inst : AddMonoid ιA] [inst_1 : VAdd ιA ιB]\n  [inst_2 : (i : ιA) → AddCommMonoid (A i)] [inst_3 : (i : ιB) → AddCommMonoid (M i)] [inst_4 : DecidableEq ιA]\n  [inst_5 : DecidableEq ιB] [inst_6 : DirectSum.GSemiring A] [inst_7 : DirectSum.Gmodule A M]\n  (r s : DirectSum ιA fun i => A i) (x : DirectSum ιB fun i => M i), (r + s) • x = r • x + s • 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inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25} {smul_add' : forall {i : ιA} {j : ιB} (a : A i) (b : M j) (c : M j), Eq.{succ u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 (GradedMonoid.GMulAction.toGSMul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 toGMulAction') i j a (HAdd.hAdd.{u_4, u_4, u_4} (M j) (M j) (M j) (instHAdd.{u_4} (M j) (AddSemigroup.toAdd.{u_4} (M j) (AddMonoid.toAddSemigroup.{u_4} (M j) (inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 j)))) b c)) (HAdd.hAdd.{u_4, u_4, u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (instHAdd.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (AddSemigroup.toAdd.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (AddMonoid.toAddSemigroup.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j))))) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 (GradedMonoid.GMulAction.toGSMul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 toGMulAction') i j a b) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 (GradedMonoid.GMulAction.toGSMul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 toGMulAction') i j a c))} {smul_zero' : forall {i : ιA} {j : ιB} (a : A i), Eq.{succ u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 (GradedMonoid.GMulAction.toGSMul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 toGMulAction') i j a (OfNat.ofNat.{u_4} (M j) 0 (Zero.toOfNat0.{u_4} (M j) (AddZero.toZero.{u_4} (M j) (AddZeroClass.toAddZero.{u_4} (M j) (AddMonoid.toAddZeroClass.{u_4} (M j) (inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 j))))))) (OfNat.ofNat.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) 0 (Zero.toOfNat0.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (AddZero.toZero.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (AddZeroClass.toAddZero.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (AddMonoid.toAddZeroClass.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)))))))}, (Eq.{max (max (max (succ u_1) (succ u_2)) (succ u_3)) (succ u_4)} (DirectSum.GdistribMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28) (DirectSum.GdistribMulAction.mk.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 toGMulAction smul_add smul_zero) (DirectSum.GdistribMulAction.mk.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 toGMulAction' smul_add' smul_zero')) -> ((HEq.{max (max (max (succ u_1) (succ u_2)) (succ u_3)) (succ u_4)} (GradedMonoid.GMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25) toGMulAction (GradedMonoid.GMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25) toGMulAction') -> P) -> P","typeFull":"{ιA : Type u_1} →\n  {ιB : Type u_2} →\n    {A : ιA → Type u_3} →\n      {M : ιB → Type u_4} →\n        {inst : AddMonoid ιA} →\n          {inst_1 : VAdd ιA ιB} →\n            {inst_2 : GradedMonoid.GMonoid A} →\n              {inst_3 : (i : ιB) → AddMonoid (M i)} →\n                {P : Sort u} →\n                  {toGMulAction : GradedMonoid.GMulAction A M} →\n                    {smul_add :\n                        ∀ {i : ιA} {j : ιB} (a : A i) (b c : M j),\n                          GradedMonoid.GSMul.smul a (b + c) =\n                            GradedMonoid.GSMul.smul a b + GradedMonoid.GSMul.smul a c} →\n                      {smul_zero : ∀ {i : ιA} {j : ιB} (a : A i), GradedMonoid.GSMul.smul a 0 = 0} →\n                        {toGMulAction' : GradedMonoid.GMulAction A M} →\n                          {smul_add' :\n                              ∀ {i : ιA} {j : ιB} (a : A i) (b c : M j),\n                                GradedMonoid.GSMul.smul a (b + c) =\n                                  GradedMonoid.GSMul.smul a b + GradedMonoid.GSMul.smul a c} →\n                            {smul_zero' : ∀ {i : ιA} {j : ιB} (a : A i), GradedMonoid.GSMul.smul a 0 = 0} →\n                              { toGMulAction := toGMulAction, smul_add := smul_add, smul_zero := smul_zero } =\n                                  { toGMulAction := toGMulAction', smul_add := smul_add', smul_zero := smul_zero' } →\n                                (toGMulAction ≍ toGMulAction' → P) → P","typeReadable":"{ιA : Type u_1} →\n  {ιB : Type u_2} →\n    {A : ιA → Type u_3} →\n      {M : ιB → Type u_4} →\n        {inst : AddMonoid ιA} →\n          {inst_1 : VAdd ιA ιB} →\n            {inst_2 : GradedMonoid.GMonoid A} →\n              {inst_3 : (i : ιB) → AddMonoid (M i)} →\n                {P : Sort u} →\n                  {toGMulAction : GradedMonoid.GMulAction A M} →\n                    {smul_add :\n                        ∀ {i : ιA} {j : ιB} (a : A i) (b c : M j),\n                          GradedMonoid.GSMul.smul a (b + c) =\n                            GradedMonoid.GSMul.smul a b + GradedMonoid.GSMul.smul a c} →\n                      {smul_zero : ∀ {i : ιA} {j : ιB} (a : A i), GradedMonoid.GSMul.smul a 0 = 0} →\n                        {toGMulAction' : GradedMonoid.GMulAction A M} →\n                          {smul_add' :\n                              ∀ {i : ιA} {j : ιB} (a : A i) (b c : M j),\n                                GradedMonoid.GSMul.smul a (b + c) =\n                                  GradedMonoid.GSMul.smul a b + GradedMonoid.GSMul.smul a c} →\n                            {smul_zero' : ∀ {i : ιA} {j : ιB} (a : A i), GradedMonoid.GSMul.smul a 0 = 0} →\n                              { toGMulAction := toGMulAction, smul_add := smul_add, smul_zero := smul_zero } =\n                                  { toGMulAction := toGMulAction', smul_add := smul_add', smul_zero := smul_zero' } →\n                                (toGMulAction ≍ toGMulAction' → P) → P","typeReferences":[["DirectSum","GdistribMulAction"],["VAdd"],["instHVAdd"],["instHAdd"],["AddMonoid"],["DirectSum","GdistribMulAction","mk"],["AddZeroClass","toAddZero"],["GradedMonoid","GMulAction"],["OfNat","ofNat"],["HVAdd","hVAdd"],["HAdd","hAdd"],["AddMonoid","toAddSemigroup"],["HEq"],["GradedMonoid","GMulAction","toGSMul"],["Zero","toOfNat0"],["Eq"],["AddZero","toZero"],["GradedMonoid","GMonoid"],["AddMonoid","toAddZeroClass"],["GradedMonoid","GSMul","smul"],["AddSemigroup","toAdd"]],"valueReferences":[["DirectSum","GdistribMulAction","noConfusion"],["DirectSum","GdistribMulAction"],["VAdd"],["HEq","refl"],["Eq","refl"],["id"],["AddMonoid"],["heq_of_eq"],["DirectSum","GdistribMulAction","mk"],["GradedMonoid","GMonoid"]]},{"isProp":true,"kind":"theorem","name":["GradedModule","isModule","_proof_6"],"typeFallback":"forall {ιA : Type.{u_5}} {ιM : Type.{u_1}} {A : Type.{u_4}} {M : Type.{u_2}} {σ : Type.{u_3}} {σ' : Type.{u_6}} [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9 : AddMonoid.{u_5} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.12 : AddAction.{u_5, u_1} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19 : Semiring.{u_4} A] (𝓐 : ιA -> σ') [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29 : SetLike.{u_6, u_4} σ' A] (𝓜 : ιM -> σ) [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.39 : Module.{u_4, u_2} A M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.43 : SetLike.{u_3, u_2} σ M] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.47 : AddSubmonoidClass.{u_6, u_4} σ' A (AddMonoid.toAddZeroClass.{u_4} A (AddMonoidWithOne.toAddMonoid.{u_4} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_4} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_4} A (Semiring.toNonAssocSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19))))) inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.51 : AddSubmonoidClass.{u_3, u_2} σ M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36)) inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.43] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.55 : SetLike.GradedMonoid.{u_5, u_4, u_6} ιA A σ' inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29 (MonoidWithZero.toMonoid.{u_4} A (Semiring.toMonoidWithZero.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19)) inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9 𝓐] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.58 : SetLike.GradedSMul.{u_5, u_1, u_6, u_4, u_3, u_2} ιA ιM σ' A σ M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.43 (SMulZeroClass.toSMul.{u_4, u_2} A M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36)))) (DistribSMul.toSMulZeroClass.{u_4, u_2} A M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36)) (DistribMulAction.toDistribSMul.{u_4, u_2} A M (MonoidWithZero.toMonoid.{u_4} A (Semiring.toMonoidWithZero.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36) (Module.toDistribMulAction.{u_4, u_2} A M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.39)))) (AddSemigroupAction.toVAdd.{u_5, u_1} ιA ιM (AddMonoid.toAddSemigroup.{u_5} ιA inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_5, u_1} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.12)) 𝓐 𝓜] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.62 : DecidableEq.{succ u_5} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.65 : DecidableEq.{succ u_1} ιM] 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[inst_5 : DecidableEq ιB] [inst_6 : GradedMonoid.GMonoid A] [inst_7 : DirectSum.Gmodule A M]\n  (x : DirectSum ιA fun i => A i) (y : DirectSum ιB fun i => M i),\n  x • y = ((DirectSum.Gmodule.smulAddMonoidHom A M) x) y","typeReadable":"∀ {ιA : Type u_1} {ιB : Type u_2} (A : ιA → Type u_3) (M : ιB → Type u_4) [inst : AddMonoid ιA] [inst_1 : VAdd ιA ιB]\n  [inst_2 : (i : ιA) → AddCommMonoid (A i)] [inst_3 : (i : ιB) → AddCommMonoid (M i)] [inst_4 : DecidableEq ιA]\n  [inst_5 : DecidableEq ιB] [inst_6 : GradedMonoid.GMonoid A] [inst_7 : DirectSum.Gmodule A M]\n  (x : DirectSum ιA fun i => A i) (y : DirectSum ιB fun i => M i),\n  x • y = ((DirectSum.Gmodule.smulAddMonoidHom A M) x) 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u_1} →\n  {ιB : Type u_2} →\n    {A : ιA → Type u_3} →\n      {M : ιB → Type u_4} →\n        {inst : AddMonoid ιA} →\n          {inst_1 : VAdd ιA ιB} →\n            {inst_2 : GradedMonoid.GMonoid A} →\n              {inst_3 : (i : ιB) → AddMonoid (M i)} → DirectSum.GdistribMulAction A M → ℕ","typeReferences":[["Nat"],["DirectSum","GdistribMulAction"],["VAdd"],["AddMonoid"],["GradedMonoid","GMonoid"]],"valueReferences":[]},{"isProp":false,"kind":"definition","name":["DirectSum","GdistribMulAction","noConfusion"],"typeFallback":"forall {P : Sort.{u}} {ιA : Type.{u_1}} {ιB : Type.{u_2}} {A : ιA -> Type.{u_3}} {M : ιB -> Type.{u_4}} {inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 : AddMonoid.{u_1} ιA} {inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 : VAdd.{u_1, u_2} ιA ιB} {inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 : GradedMonoid.GMonoid.{u_1, u_3} ιA A inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18} {inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 : forall (i : ιB), AddMonoid.{u_4} (M i)} {t : DirectSum.GdistribMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28} {ιA' : Type.{u_1}} {ιB' : Type.{u_2}} {A' : ιA' -> Type.{u_3}} {M' : ιB' -> Type.{u_4}} {inst'._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 : AddMonoid.{u_1} ιA'} {inst'._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 : VAdd.{u_1, u_2} ιA' ιB'} {inst'._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 : GradedMonoid.GMonoid.{u_1, u_3} ιA' A' inst'._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18} {inst'._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 : forall (i : ιB'), AddMonoid.{u_4} (M' i)} {t' : DirectSum.GdistribMulAction.{u_1, u_2, u_3, u_4} ιA' ιB' A' M' inst'._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst'._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst'._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 inst'._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28}, (Eq.{succ (succ u_1)} Type.{u_1} ιA ιA') -> (Eq.{succ (succ u_2)} Type.{u_2} ιB ιB') -> (HEq.{max (succ u_1) (succ (succ u_3))} (ιA -> Type.{u_3}) A (ιA' -> Type.{u_3}) A') -> (HEq.{max (succ u_2) (succ (succ u_4))} (ιB -> Type.{u_4}) M (ιB' -> Type.{u_4}) M') -> (HEq.{succ u_1} (AddMonoid.{u_1} ιA) inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 (AddMonoid.{u_1} ιA') 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inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 t ιA' ιB' A' M' inst'._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst'._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst'._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 inst'._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 t')","typeFull":"{P : Sort u} →\n  {ιA : Type u_1} →\n    {ιB : Type u_2} →\n      {A : ιA → Type u_3} →\n        {M : ιB → Type u_4} →\n          {inst : AddMonoid ιA} →\n            {inst_1 : VAdd ιA ιB} →\n              {inst_2 : GradedMonoid.GMonoid A} →\n                {inst_3 : (i : ιB) → AddMonoid (M i)} →\n                  {t : DirectSum.GdistribMulAction A M} →\n                    {ιA' : Type u_1} →\n                      {ιB' : Type u_2} →\n                        {A' : ιA' → Type u_3} →\n                          {M' : ιB' → Type u_4} →\n                            {inst' : AddMonoid ιA'} →\n                              {inst'_1 : VAdd ιA' ιB'} →\n                                {inst'_2 : GradedMonoid.GMonoid A'} →\n                                  {inst'_3 : (i : ιB') → AddMonoid (M' i)} →\n                                    {t' : DirectSum.GdistribMulAction A' M'} →\n                                      ιA = ιA' →\n                                        ιB = ιB' →\n                                          A ≍ A' →\n                                            M ≍ M' →\n                                              inst ≍ inst' →\n                                                inst_1 ≍ inst'_1 →\n                                                  inst_2 ≍ inst'_2 →\n                                                    inst_3 ≍ inst'_3 →\n                                                      t ≍ t' → DirectSum.GdistribMulAction.noConfusionType P t t'","typeReadable":"{P : Sort u} →\n  {ιA : Type u_1} →\n    {ιB : Type u_2} →\n      {A : ιA → Type u_3} →\n        {M : ιB → Type u_4} →\n          {inst : AddMonoid ιA} →\n            {inst_1 : VAdd ιA ιB} →\n              {inst_2 : GradedMonoid.GMonoid A} →\n                {inst_3 : (i : ιB) → AddMonoid (M i)} →\n                  {t : DirectSum.GdistribMulAction A M} →\n                    {ιA' : Type u_1} →\n                      {ιB' : Type u_2} →\n                        {A' : ιA' → Type u_3} →\n                          {M' : ιB' → Type u_4} →\n                            {inst' : AddMonoid ιA'} →\n                              {inst'_1 : VAdd ιA' ιB'} →\n                                {inst'_2 : GradedMonoid.GMonoid A'} →\n                                  {inst'_3 : (i : ιB') → AddMonoid (M' i)} →\n                                    {t' : DirectSum.GdistribMulAction A' M'} →\n                                      ιA = ιA' →\n                                        ιB = ιB' →\n                                          A ≍ A' →\n                                            M ≍ M' →\n                                              inst ≍ inst' →\n                                                inst_1 ≍ inst'_1 →\n                                                  inst_2 ≍ inst'_2 →\n                                                    inst_3 ≍ inst'_3 →\n                                                      t ≍ t' → DirectSum.GdistribMulAction.noConfusionType P t t'","typeReferences":[["DirectSum","GdistribMulAction"],["VAdd"],["HEq"],["DirectSum","GdistribMulAction","noConfusionType"],["AddMonoid"],["Eq"],["GradedMonoid","GMonoid"]],"valueReferences":[["DirectSum","GdistribMulAction"],["VAdd"],["DirectSum","GdistribMulAction","casesOn"],["HEq","refl"],["eq_of_heq"],["HEq"],["DirectSum","GdistribMulAction","noConfusionType"],["AddMonoid"],["GradedMonoid","GMulAction"],["Eq","ndrec"],["GradedMonoid","GMonoid"]]},{"isProp":true,"kind":"theorem","name":["DirectSum","gsmulHom","_proof_7"],"typeFallback":"forall {ιA : Type.{u_2}} {ιB : Type.{u_3}} (A : ιA -> Type.{u_4}) (M : ιB -> Type.{u_1}) [inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 : AddMonoid.{u_2} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 : VAdd.{u_2, u_3} ιA ιB] [inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.17 : forall (i : ιA), AddCommMonoid.{u_4} (A i)] [inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.28 : forall (i : ιB), AddCommMonoid.{u_1} (M i)] [inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.37 : GradedMonoid.GMonoid.{u_2, u_4} ιA A inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.40 : DirectSum.Gmodule.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 (fun (i : ιA) => AddCommMonoid.toAddMonoid.{u_4} (A i) (inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.17 i)) (fun (i : ιB) => AddCommMonoid.toAddMonoid.{u_1} (M i) (inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.28 i)) inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.37] {i : ιA} {j : ιB} (_a₁ : A i) (_a₂ : A i) (b : M j) (c : M j), Eq.{succ u_1} (M (HVAdd.hVAdd.{u_2, u_3, u_3} ιA ιB ιB (instHVAdd.{u_2, u_3} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13) i j)) (GradedMonoid.GSMul.smul.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 (GradedMonoid.GMulAction.toGSMul.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.37 (DirectSum.GdistribMulAction.toGMulAction.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.37 (fun (i : ιB) => AddCommMonoid.toAddMonoid.{u_1} (M i) (inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.28 i)) (DirectSum.Gmodule.toGdistribMulAction.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 (fun (i : ιA) => AddCommMonoid.toAddMonoid.{u_4} (A i) (inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.17 i)) (fun (i : ιB) => AddCommMonoid.toAddMonoid.{u_1} (M i) (inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.28 i)) inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.40))) i j (HAdd.hAdd.{u_4, u_4, u_4} (A i) (A i) (A i) (instHAdd.{u_4} (A i) (AddZero.toAdd.{u_4} (A i) (AddZeroClass.toAddZero.{u_4} (A i) (AddMonoid.toAddZeroClass.{u_4} (A i) (AddCommMonoid.toAddMonoid.{u_4} (A i) (inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.17 i)))))) _a₁ _a₂) (HAdd.hAdd.{u_1, u_1, u_1} (M j) (M j) (M j) (instHAdd.{u_1} (M j) (AddSemigroup.toAdd.{u_1} (M j) (AddMonoid.toAddSemigroup.{u_1} (M j) (AddCommMonoid.toAddMonoid.{u_1} (M j) (inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.28 j))))) b c)) (HAdd.hAdd.{u_1, u_1, u_1} (M (HVAdd.hVAdd.{u_2, u_3, u_3} ιA ιB ιB (instHVAdd.{u_2, u_3} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13) i j)) (M (HVAdd.hVAdd.{u_2, u_3, u_3} ιA ιB ιB (instHVAdd.{u_2, u_3} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13) i j)) (M (HVAdd.hVAdd.{u_2, u_3, u_3} ιA ιB ιB (instHVAdd.{u_2, u_3} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13) i j)) (instHAdd.{u_1} (M (HVAdd.hVAdd.{u_2, u_3, u_3} ιA ιB ιB (instHVAdd.{u_2, u_3} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13) i j)) (AddSemigroup.toAdd.{u_1} (M (HVAdd.hVAdd.{u_2, u_3, u_3} ιA ιB ιB (instHVAdd.{u_2, u_3} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13) i j)) (AddMonoid.toAddSemigroup.{u_1} (M (HVAdd.hVAdd.{u_2, u_3, u_3} ιA ιB ιB (instHVAdd.{u_2, u_3} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13) i j)) (AddCommMonoid.toAddMonoid.{u_1} (M (HVAdd.hVAdd.{u_2, u_3, u_3} ιA ιB ιB (instHVAdd.{u_2, u_3} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13) i j)) (inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.28 (HVAdd.hVAdd.{u_2, u_3, u_3} ιA ιB ιB (instHVAdd.{u_2, u_3} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13) i j)))))) (GradedMonoid.GSMul.smul.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 (GradedMonoid.GMulAction.toGSMul.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.37 (DirectSum.GdistribMulAction.toGMulAction.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.37 (fun (i : ιB) => AddCommMonoid.toAddMonoid.{u_1} (M i) (inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.28 i)) (DirectSum.Gmodule.toGdistribMulAction.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 (fun (i : ιA) => AddCommMonoid.toAddMonoid.{u_4} (A i) (inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.17 i)) (fun (i : ιB) => AddCommMonoid.toAddMonoid.{u_1} (M i) (inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.28 i)) inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.40))) i j (HAdd.hAdd.{u_4, u_4, u_4} (A i) (A i) (A i) (instHAdd.{u_4} (A i) (AddZero.toAdd.{u_4} (A i) (AddZeroClass.toAddZero.{u_4} (A i) (AddMonoid.toAddZeroClass.{u_4} (A i) (AddCommMonoid.toAddMonoid.{u_4} (A i) (inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.17 i)))))) _a₁ _a₂) b) (GradedMonoid.GSMul.smul.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 (GradedMonoid.GMulAction.toGSMul.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.37 (DirectSum.GdistribMulAction.toGMulAction.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.37 (fun (i : ιB) => AddCommMonoid.toAddMonoid.{u_1} (M i) (inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.28 i)) (DirectSum.Gmodule.toGdistribMulAction.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 (fun (i : ιA) => AddCommMonoid.toAddMonoid.{u_4} (A i) (inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.17 i)) (fun (i : ιB) => AddCommMonoid.toAddMonoid.{u_1} (M i) (inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.28 i)) inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.40))) i j (HAdd.hAdd.{u_4, u_4, u_4} (A i) (A i) (A i) (instHAdd.{u_4} (A i) (AddZero.toAdd.{u_4} (A i) (AddZeroClass.toAddZero.{u_4} (A i) (AddMonoid.toAddZeroClass.{u_4} (A i) (AddCommMonoid.toAddMonoid.{u_4} (A i) (inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.17 i)))))) _a₁ _a₂) c))","typeFull":"∀ {ιA : Type u_2} {ιB : Type u_3} (A : ιA → Type u_4) (M : ιB → Type u_1) [inst : AddMonoid ιA] [inst_1 : VAdd ιA ιB]\n  [inst_2 : (i : ιA) → AddCommMonoid (A i)] [inst_3 : (i : ιB) → AddCommMonoid (M i)] [inst_4 : GradedMonoid.GMonoid A]\n  [inst_5 : DirectSum.Gmodule A M] {i : ιA} {j : ιB} (_a₁ _a₂ : A i) (b c : M j),\n  GradedMonoid.GSMul.smul (_a₁ + _a₂) (b + c) =\n    GradedMonoid.GSMul.smul (_a₁ + _a₂) b + GradedMonoid.GSMul.smul (_a₁ + _a₂) c","typeReadable":"∀ {ιA : Type u_2} {ιB : Type u_3} (A : ιA → Type u_4) (M : ιB → Type u_1) [inst : AddMonoid ιA] [inst_1 : VAdd ιA ιB]\n  [inst_2 : (i : ιA) → AddCommMonoid (A i)] [inst_3 : (i : ιB) → AddCommMonoid (M i)] [inst_4 : GradedMonoid.GMonoid A]\n  [inst_5 : DirectSum.Gmodule A M] {i : ιA} {j : ιB} (_a₁ _a₂ : A i) (b c : M j),\n  GradedMonoid.GSMul.smul (_a₁ + _a₂) (b + c) =\n    GradedMonoid.GSMul.smul (_a₁ + _a₂) b + GradedMonoid.GSMul.smul (_a₁ + _a₂) c","typeReferences":[["VAdd"],["instHVAdd"],["instHAdd"],["AddMonoid"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["DirectSum","Gmodule","toGdistribMulAction"],["HVAdd","hVAdd"],["HAdd","hAdd"],["AddCommMonoid"],["AddMonoid","toAddSemigroup"],["GradedMonoid","GMulAction","toGSMul"],["DirectSum","Gmodule"],["DirectSum","GdistribMulAction","toGMulAction"],["Eq"],["GradedMonoid","GMonoid"],["AddSemigroup","toAdd"],["AddMonoid","toAddZeroClass"],["GradedMonoid","GSMul","smul"]],"valueReferences":[["HAdd","hAdd"],["DirectSum","GdistribMulAction","smul_add"],["instHAdd"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["DirectSum","Gmodule","toGdistribMulAction"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"definition","name":["DirectSum","Gmodule","noConfusionType"],"typeFallback":"Sort.{u} -> (forall {ιA : Type.{u_1}} {ιB : Type.{u_2}} {A : ιA -> Type.{u_3}} {M : ιB -> Type.{u_4}} [inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 : AddMonoid.{u_1} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 : VAdd.{u_1, u_2} ιA ιB] [inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.25 : forall (i : ιA), AddMonoid.{u_3} (A i)] [inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 : forall (i : ιB), AddMonoid.{u_4} (M i)] [inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 : GradedMonoid.GMonoid.{u_1, u_3} ιA A inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18], (DirectSum.Gmodule.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43) -> (forall {ιA' : Type.{u_1}} {ιB' : Type.{u_2}} {A' : ιA' -> Type.{u_3}} {M' : ιB' -> Type.{u_4}} [inst'._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 : AddMonoid.{u_1} ιA'] [inst'._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 : VAdd.{u_1, u_2} ιA' ιB'] [inst'._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.25 : forall (i : ιA'), AddMonoid.{u_3} (A' i)] [inst'._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 : forall (i : ιB'), AddMonoid.{u_4} (M' i)] [inst'._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 : GradedMonoid.GMonoid.{u_1, u_3} ιA' A' inst'._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18], (DirectSum.Gmodule.{u_1, u_2, u_3, u_4} ιA' ιB' A' M' inst'._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst'._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst'._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.25 inst'._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 inst'._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43) -> Sort.{u}))","typeFull":"Sort u →\n  {ιA : Type u_1} →\n    {ιB : Type u_2} →\n      {A : ιA → Type u_3} →\n        {M : ιB → Type u_4} →\n          [inst : AddMonoid ιA] →\n            [inst_1 : VAdd ιA ιB] →\n              [inst_2 : (i : ιA) → AddMonoid (A i)] →\n                [inst_3 : (i : ιB) → AddMonoid (M i)] →\n                  [inst_4 : GradedMonoid.GMonoid A] →\n                    DirectSum.Gmodule A M →\n                      {ιA' : Type u_1} →\n                        {ιB' : Type u_2} →\n                          {A' : ιA' → Type u_3} →\n                            {M' : ιB' → Type u_4} →\n                              [inst' : AddMonoid ιA'] →\n                                [inst'_1 : VAdd ιA' ιB'] →\n                                  [inst'_2 : (i : ιA') → AddMonoid (A' i)] →\n                                    [inst'_3 : (i : ιB') → AddMonoid (M' i)] →\n                                      [inst'_4 : GradedMonoid.GMonoid A'] → DirectSum.Gmodule A' M' → Sort u","typeReadable":"Sort u →\n  {ιA : Type u_1} →\n    {ιB : Type u_2} →\n      {A : ιA → Type u_3} →\n        {M : ιB → Type u_4} →\n          [inst : AddMonoid ιA] →\n            [inst_1 : VAdd ιA ιB] →\n              [inst_2 : (i : ιA) → AddMonoid (A i)] →\n                [inst_3 : (i : ιB) → AddMonoid (M i)] →\n                  [inst_4 : GradedMonoid.GMonoid A] →\n                    DirectSum.Gmodule A M →\n                      {ιA' : Type u_1} →\n                        {ιB' : Type u_2} →\n                          {A' : ιA' → Type u_3} →\n                            {M' : ιB' → Type u_4} →\n                              [inst' : AddMonoid ιA'] →\n                                [inst'_1 : VAdd ιA' ιB'] →\n                                  [inst'_2 : (i : ιA') → AddMonoid (A' i)] →\n                                    [inst'_3 : (i : ιB') → AddMonoid (M' i)] →\n                                      [inst'_4 : GradedMonoid.GMonoid A'] → DirectSum.Gmodule A' M' → Sort u","typeReferences":[["VAdd"],["DirectSum","Gmodule"],["AddMonoid"],["GradedMonoid","GMonoid"]],"valueReferences":[["DirectSum","GdistribMulAction"],["DirectSum","Gmodule","casesOn"],["HEq"]]},{"isProp":true,"kind":"theorem","name":["GradedModule","isModule","_proof_5"],"typeFallback":"forall {ιA : Type.{u_5}} {ιM : Type.{u_1}} {A : Type.{u_4}} {M : Type.{u_2}} {σ : Type.{u_3}} {σ' : Type.{u_6}} [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9 : AddMonoid.{u_5} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.12 : AddAction.{u_5, u_1} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19 : Semiring.{u_4} A] (𝓐 : ιA -> σ') [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29 : SetLike.{u_6, u_4} σ' A] (𝓜 : ιM -> σ) [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.39 : Module.{u_4, u_2} A M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.43 : SetLike.{u_3, u_2} σ M] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.47 : AddSubmonoidClass.{u_6, u_4} σ' A (AddMonoid.toAddZeroClass.{u_4} A (AddMonoidWithOne.toAddMonoid.{u_4} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_4} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_4} A (Semiring.toNonAssocSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19))))) inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.51 : AddSubmonoidClass.{u_3, u_2} σ M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36)) inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.43] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.55 : SetLike.GradedMonoid.{u_5, u_4, u_6} ιA A σ' inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29 (MonoidWithZero.toMonoid.{u_4} A (Semiring.toMonoidWithZero.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19)) inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9 𝓐] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.58 : SetLike.GradedSMul.{u_5, u_1, u_6, u_4, u_3, u_2} ιA ιM σ' A σ M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.43 (SMulZeroClass.toSMul.{u_4, u_2} A M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36)))) (DistribSMul.toSMulZeroClass.{u_4, u_2} A M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36)) (DistribMulAction.toDistribSMul.{u_4, u_2} A M (MonoidWithZero.toMonoid.{u_4} A (Semiring.toMonoidWithZero.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36) (Module.toDistribMulAction.{u_4, u_2} A M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.39)))) (AddSemigroupAction.toVAdd.{u_5, u_1} ιA ιM (AddMonoid.toAddSemigroup.{u_5} ιA inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_5, u_1} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.12)) 𝓐 𝓜] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.62 : DecidableEq.{succ u_5} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.65 : DecidableEq.{succ u_1} ιM] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.68 : GradedRing.{u_5, u_4, u_6} ιA A σ' inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.62 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.47 𝓐] (r : A) (s : A) (x : DirectSum.{u_1, u_2} ιM 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inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12))) i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.114 j._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.115)) x)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_4} M σ (SetLike.instMembership.{u_4, u_3} σ M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43) (𝓜 (HVAdd.hVAdd.{u_5, u_6, u_6} ιA ιM ιM (instHVAdd.{u_5, u_6} ιA ιM (AddSemigroupAction.toVAdd.{u_5, u_6} ιA ιM (AddMonoid.toAddSemigroup.{u_5} ιA inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_5, u_6} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12))) i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.114 j._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.115)) x)) (instHAdd.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_4} M σ (SetLike.instMembership.{u_4, u_3} σ M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43) (𝓜 (HVAdd.hVAdd.{u_5, u_6, u_6} ιA ιM ιM (instHVAdd.{u_5, u_6} ιA ιM (AddSemigroupAction.toVAdd.{u_5, u_6} ιA ιM (AddMonoid.toAddSemigroup.{u_5} ιA inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_5, u_6} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12))) i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.114 j._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.115)) x)) (AddSemigroup.toAdd.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_4} M σ (SetLike.instMembership.{u_4, u_3} σ M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43) (𝓜 (HVAdd.hVAdd.{u_5, u_6, u_6} ιA ιM ιM (instHVAdd.{u_5, u_6} ιA ιM (AddSemigroupAction.toVAdd.{u_5, u_6} ιA ιM (AddMonoid.toAddSemigroup.{u_5} ιA inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_5, u_6} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12))) i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.114 j._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.115)) x)) (AddMonoid.toAddSemigroup.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_4} M σ (SetLike.instMembership.{u_4, u_3} σ M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43) (𝓜 (HVAdd.hVAdd.{u_5, u_6, u_6} ιA ιM ιM (instHVAdd.{u_5, u_6} ιA ιM (AddSemigroupAction.toVAdd.{u_5, u_6} ιA ιM (AddMonoid.toAddSemigroup.{u_5} ιA inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_5, u_6} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12))) i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.114 j._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.115)) x)) (AddCommMonoid.toAddMonoid.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_4} M σ (SetLike.instMembership.{u_4, u_3} σ M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43) (𝓜 (HVAdd.hVAdd.{u_5, u_6, u_6} ιA ιM ιM (instHVAdd.{u_5, u_6} ιA ιM (AddSemigroupAction.toVAdd.{u_5, u_6} ιA ιM (AddMonoid.toAddSemigroup.{u_5} ιA inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_5, u_6} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12))) i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.114 j._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.115)) x)) (AddSubmonoidClass.toAddCommMonoid.{u_4, u_3} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36 σ inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.51 (𝓜 (HVAdd.hVAdd.{u_5, u_6, u_6} ιA ιM ιM (instHVAdd.{u_5, u_6} ιA ιM (AddSemigroupAction.toVAdd.{u_5, u_6} ιA ιM (AddMonoid.toAddSemigroup.{u_5} ιA inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_5, u_6} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12))) i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.114 j._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.115))))))) (Subtype.mk.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_4} M σ (SetLike.instMembership.{u_4, u_3} σ M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43) (𝓜 (HVAdd.hVAdd.{u_5, u_6, u_6} ιA ιM ιM (instHVAdd.{u_5, u_6} ιA ιM (AddSemigroupAction.toVAdd.{u_5, u_6} ιA ιM (AddMonoid.toAddSemigroup.{u_5} ιA inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_5, u_6} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12))) i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.114 j._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.115)) x) (HSMul.hSMul.{u_1, u_3, u_3} A M M (instHSMul.{u_1, u_3} A M (SMulZeroClass.toSMul.{u_1, u_3} A M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M 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inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29) (𝓐 i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.114) x) _a) (Subtype.val.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_4} M σ (SetLike.instMembership.{u_4, u_3} σ M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43) (𝓜 j._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.115) x) _b)) (SetLike.gmodule._proof_1.{u_3, u_4, u_5, u_6, u_1, u_2} ιA ιM A M σ σ' inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19 𝓐 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29 𝓜 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.58 i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.114 j._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.115 _a _b)) (Subtype.mk.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_4} M σ (SetLike.instMembership.{u_4, u_3} σ M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43) (𝓜 (HVAdd.hVAdd.{u_5, u_6, u_6} ιA ιM ιM (instHVAdd.{u_5, u_6} ιA ιM (AddSemigroupAction.toVAdd.{u_5, u_6} ιA ιM (AddMonoid.toAddSemigroup.{u_5} ιA inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_5, u_6} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12))) i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.114 j._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.115)) x) (HSMul.hSMul.{u_1, u_3, u_3} A M M (instHSMul.{u_1, u_3} A M (SMulZeroClass.toSMul.{u_1, u_3} A M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} A M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36)) (DistribMulAction.toDistribSMul.{u_1, u_3} A M (MonoidWithZero.toMonoid.{u_1} A (Semiring.toMonoidWithZero.{u_1} A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36) (Module.toDistribMulAction.{u_1, u_3} A M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.39))))) (Subtype.val.{succ u_1} A (fun (x : A) => Membership.mem.{u_1, u_2} A σ' (SetLike.instMembership.{u_2, u_1} σ' A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29) (𝓐 i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.114) x) _a') (Subtype.val.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_4} M σ (SetLike.instMembership.{u_4, u_3} σ M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43) (𝓜 j._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.115) x) _b)) (SetLike.gmodule._proof_1.{u_3, u_4, u_5, u_6, u_1, u_2} ιA ιM A M σ σ' inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19 𝓐 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29 𝓜 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.58 i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.114 j._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.115 _a' _b)))","typeFull":"∀ {ιA : Type u_5} {ιM : Type u_6} {A : Type u_1} {M : Type u_3} {σ : Type u_4} {σ' : Type u_2} [inst : AddMonoid ιA]\n  [inst_1 : AddAction ιA ιM] [inst_2 : Semiring A] (𝓐 : ιA → σ') [inst_3 : SetLike σ' A] (𝓜 : ιM → σ)\n  [inst_4 : AddCommMonoid M] [inst_5 : Module A M] [inst_6 : SetLike σ M] [inst_7 : AddSubmonoidClass σ' A]\n  [inst_8 : AddSubmonoidClass σ M] [inst_9 : SetLike.GradedSMul 𝓐 𝓜] {i : ιA} {j : ιM} (_a _a' : ↥(𝓐 i)) (_b : ↥(𝓜 j)),\n  ⟨↑(_a + _a') • ↑_b, ⋯⟩ = ⟨↑_a • ↑_b, ⋯⟩ + ⟨↑_a' • ↑_b, ⋯⟩","typeReadable":"∀ {ιA : Type u_5} {ιM : Type u_6} {A : Type u_1} {M : Type u_3} {σ : Type u_4} {σ' : Type u_2} [inst : 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{ιA : Type.{u_5}} {ιM : Type.{u_6}} {A : Type.{u_1}} {M : Type.{u_3}} {σ : Type.{u_4}} {σ' : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 : AddMonoid.{u_5} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12 : AddAction.{u_5, u_6} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19 : Semiring.{u_1} A] (𝓐 : ιA -> σ') [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29 : SetLike.{u_2, u_1} σ' A] (𝓜 : ιM -> σ) [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.39 : Module.{u_1, u_3} A M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43 : SetLike.{u_4, u_3} σ M] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.51 : AddSubmonoidClass.{u_4, u_3} σ M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36)) inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.55 : SetLike.GradedMonoid.{u_5, u_1, u_2} ιA A σ' inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29 (MonoidWithZero.toMonoid.{u_1} A (Semiring.toMonoidWithZero.{u_1} A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19)) inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 𝓐] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.58 : SetLike.GradedSMul.{u_5, u_6, u_2, u_1, u_4, u_3} ιA ιM σ' A σ M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29 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A] (𝓜 : ιM → σ)\n  [inst_4 : AddCommMonoid M] [inst_5 : Module A M] [inst_6 : SetLike σ M] [inst_7 : AddSubmonoidClass σ M]\n  [inst_8 : SetLike.GradedMonoid 𝓐] [inst_9 : SetLike.GradedSMul 𝓐 𝓜] {i : ιA} {j : ιM} (a : ↥(𝓐 i)) (b c : ↥(𝓜 j)),\n  GradedMonoid.GSMul.smul a (b + c) = GradedMonoid.GSMul.smul a b + GradedMonoid.GSMul.smul a 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{ιA : Type.{u_1}} {ιM : Type.{u_4}} {A : Type.{u_2}} {M : Type.{u_5}} {σ : Type.{u_6}} {σ' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 : AddMonoid.{u_1} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12 : AddAction.{u_1, u_4} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19 : Semiring.{u_2} A] (𝓐 : ιA -> σ') [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29 : SetLike.{u_3, u_2} σ' A] (𝓜 : ιM -> σ) [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36 : AddCommMonoid.{u_5} M] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.39 : Module.{u_2, u_5} A M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43 : SetLike.{u_6, u_5} σ M] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.51 : AddSubmonoidClass.{u_6, u_5} σ M (AddMonoid.toAddZeroClass.{u_5} M (AddCommMonoid.toAddMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36)) inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.55 : SetLike.GradedMonoid.{u_1, u_2, u_3} ιA A σ' inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29 (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19)) inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 𝓐] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.58 : SetLike.GradedSMul.{u_1, u_4, u_3, u_2, u_6, u_5} ιA ιM σ' A σ M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29 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inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29 (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19)) inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 𝓐 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.55) (fun (i : ιM) => AddSubmonoidClass.toAddMonoid.{u_5, u_6} M (AddCommMonoid.toAddMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36) σ inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.51 (𝓜 i)) (SetLike.gdistribMulAction.{u_1, u_4, u_2, u_5, u_6, u_3} ιA ιM A M σ σ' inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19 𝓐 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inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12)) (SetLike.gMonoid.{u_1, u_2, u_3} ιA A σ' inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29 (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19)) inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 𝓐 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.55) (fun (i : ιM) => AddSubmonoidClass.toAddMonoid.{u_5, u_6} M (AddCommMonoid.toAddMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36) σ inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.51 (𝓜 i)) (SetLike.gdistribMulAction.{u_1, u_4, u_2, u_5, u_6, u_3} ιA ιM A M σ σ' inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19 𝓐 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29 𝓜 (AddCommMonoid.toAddMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36) (Module.toDistribMulAction.{u_2, u_5} A M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.39) inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.51 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.55 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.58))))) a' b))","typeFull":"∀ {ιA : Type u_1} {ιM : Type u_4} {A : Type u_2} {M : Type u_5} {σ : Type u_6} {σ' : Type u_3} [inst : AddMonoid ιA]\n  [inst_1 : AddAction ιA ιM] [inst_2 : Semiring A] (𝓐 : ιA → σ') [inst_3 : SetLike σ' A] (𝓜 : ιM → σ)\n  [inst_4 : AddCommMonoid M] [inst_5 : Module A M] [inst_6 : SetLike σ M] [inst_7 : AddSubmonoidClass σ M]\n  [inst_8 : SetLike.GradedMonoid 𝓐] [inst_9 : SetLike.GradedSMul 𝓐 𝓜] (a a' : GradedMonoid fun i => ↥(𝓐 i))\n  (b : GradedMonoid fun i => ↥(𝓜 i)), (a * a') • b = a • a' • b","typeReadable":"∀ {ιA : Type u_1} {ιM : Type u_4} {A : Type u_2} {M : Type u_5} {σ : Type u_6} {σ' : Type u_3} [inst : AddMonoid ιA]\n  [inst_1 : AddAction ιA ιM] [inst_2 : Semiring A] (𝓐 : ιA → σ') [inst_3 : SetLike σ' A] (𝓜 : ιM → σ)\n  [inst_4 : AddCommMonoid M] [inst_5 : Module A M] [inst_6 : SetLike σ M] [inst_7 : AddSubmonoidClass σ M]\n  [inst_8 : SetLike.GradedMonoid 𝓐] [inst_9 : SetLike.GradedSMul 𝓐 𝓜] (a a' : GradedMonoid fun i => ↥(𝓐 i))\n  (b : GradedMonoid fun i => ↥(𝓜 i)), (a * a') • b = a • a' • 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{ιA : Type.{u_5}} {ιM : Type.{u_6}} {A : Type.{u_1}} {M : Type.{u_3}} {σ : Type.{u_4}} {σ' : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 : AddMonoid.{u_5} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12 : AddAction.{u_5, u_6} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19 : Semiring.{u_1} A] (𝓐 : ιA -> σ') [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29 : SetLike.{u_2, u_1} σ' A] (𝓜 : ιM -> σ) [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.39 : Module.{u_1, u_3} A M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43 : SetLike.{u_4, u_3} σ M] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.51 : AddSubmonoidClass.{u_4, u_3} σ M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36)) inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.55 : SetLike.GradedMonoid.{u_5, u_1, u_2} ιA A σ' inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29 (MonoidWithZero.toMonoid.{u_1} A (Semiring.toMonoidWithZero.{u_1} A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19)) inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 𝓐] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.58 : SetLike.GradedSMul.{u_5, u_6, u_2, u_1, u_4, u_3} ιA ιM σ' A σ M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29 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inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_5, u_6} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12))) i j))))))))","typeFull":"∀ {ιA : Type u_5} {ιM : Type u_6} {A : Type u_1} {M : Type u_3} {σ : Type u_4} {σ' : Type u_2} [inst : AddMonoid ιA]\n  [inst_1 : AddAction ιA ιM] [inst_2 : Semiring A] (𝓐 : ιA → σ') [inst_3 : SetLike σ' A] (𝓜 : ιM → σ)\n  [inst_4 : AddCommMonoid M] [inst_5 : Module A M] [inst_6 : SetLike σ M] [inst_7 : AddSubmonoidClass σ M]\n  [inst_8 : SetLike.GradedMonoid 𝓐] [inst_9 : SetLike.GradedSMul 𝓐 𝓜] {i : ιA} {j : ιM} (a : ↥(𝓐 i)),\n  GradedMonoid.GSMul.smul a 0 = 0","typeReadable":"∀ {ιA : Type u_5} {ιM : Type u_6} {A : Type u_1} {M : Type u_3} {σ : Type u_4} {σ' : Type u_2} [inst : AddMonoid ιA]\n  [inst_1 : AddAction ιA ιM] [inst_2 : Semiring A] (𝓐 : ιA → σ') [inst_3 : SetLike σ' A] (𝓜 : ιM → σ)\n  [inst_4 : AddCommMonoid M] [inst_5 : Module A M] [inst_6 : SetLike σ M] [inst_7 : AddSubmonoidClass σ M]\n  [inst_8 : SetLike.GradedMonoid 𝓐] [inst_9 : SetLike.GradedSMul 𝓐 𝓜] {i : ιA} {j : ιM} (a : ↥(𝓐 i)),\n  GradedMonoid.GSMul.smul a 0 = 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{ιA : Type.{u_1}} {ιM : Type.{u_2}} {A : Type.{u_4}} {M : Type.{u_5}} {σ : Type.{u_6}} {σ' : Type.{u_7}} [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9 : AddMonoid.{u_1} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.12 : AddAction.{u_1, u_2} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19 : Semiring.{u_4} A] (𝓐 : ιA -> σ') [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29 : SetLike.{u_7, u_4} σ' A] (𝓜 : ιM -> σ) [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36 : AddCommMonoid.{u_5} M] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.39 : Module.{u_4, u_5} A M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.43 : SetLike.{u_6, u_5} σ M] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.47 : AddSubmonoidClass.{u_7, u_4} σ' A (AddMonoid.toAddZeroClass.{u_4} A (AddMonoidWithOne.toAddMonoid.{u_4} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_4} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_4} A (Semiring.toNonAssocSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19))))) inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.51 : AddSubmonoidClass.{u_6, u_5} σ M (AddMonoid.toAddZeroClass.{u_5} M (AddCommMonoid.toAddMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36)) inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.43] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.55 : SetLike.GradedMonoid.{u_1, u_4, u_7} ιA A σ' inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29 (MonoidWithZero.toMonoid.{u_4} A (Semiring.toMonoidWithZero.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19)) inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9 𝓐] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.58 : SetLike.GradedSMul.{u_1, u_2, u_7, u_4, u_6, u_5} ιA ιM σ' A σ M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.43 (SMulZeroClass.toSMul.{u_4, u_5} A M (AddZero.toZero.{u_5} M (AddZeroClass.toAddZero.{u_5} M (AddMonoid.toAddZeroClass.{u_5} M (AddCommMonoid.toAddMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36)))) (DistribSMul.toSMulZeroClass.{u_4, u_5} A M (AddMonoid.toAddZeroClass.{u_5} M (AddCommMonoid.toAddMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36)) (DistribMulAction.toDistribSMul.{u_4, u_5} A M (MonoidWithZero.toMonoid.{u_4} A (Semiring.toMonoidWithZero.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19)) (AddCommMonoid.toAddMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36) (Module.toDistribMulAction.{u_4, u_5} A M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.39)))) (AddSemigroupAction.toVAdd.{u_1, u_2} ιA ιM (AddMonoid.toAddSemigroup.{u_1} ιA inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_1, u_2} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.12)) 𝓐 𝓜] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.62 : DecidableEq.{succ u_1} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.65 : DecidableEq.{succ u_2} ιM] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.68 : GradedRing.{u_1, u_4, u_7} ιA A σ' inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.62 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.47 𝓐], Module.{u_4, max u_5 u_2} A (DirectSum.{u_2, u_5} ιM (fun (i : ιM) => Subtype.{succ u_5} M (fun (x : M) => Membership.mem.{u_5, u_6} M σ (SetLike.instMembership.{u_6, u_5} σ M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.43) (𝓜 i) x)) (fun (i : ιM) => AddSubmonoidClass.toAddCommMonoid.{u_6, u_5} M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36 σ inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.51 (𝓜 i))) inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19 (instAddCommMonoidDirectSum.{u_2, u_5} ιM (fun (i : ιM) => Subtype.{succ u_5} M (fun (x : M) => Membership.mem.{u_5, u_6} M σ (SetLike.instMembership.{u_6, u_5} σ M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.43) (𝓜 i) x)) (fun (i : ιM) => AddSubmonoidClass.toAddCommMonoid.{u_6, u_5} M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36 σ inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.51 (𝓜 i)))","typeFull":"{ιA : Type u_1} →\n  {ιM : Type u_2} →\n    {A : Type u_4} →\n      {M : Type u_5} →\n        {σ : Type u_6} →\n          {σ' : Type u_7} →\n            [inst : AddMonoid ιA] →\n              [inst_1 : AddAction ιA ιM] →\n                [inst_2 : Semiring A] →\n                  (𝓐 : ιA → σ') →\n                    [inst_3 : SetLike σ' A] →\n                      (𝓜 : ιM → σ) →\n          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                 [inst_4 : AddCommMonoid M] →\n                          [inst_5 : Module A M] →\n                            [inst_6 : SetLike σ M] →\n                              [inst_7 : AddSubmonoidClass σ' A] →\n                                [inst_8 : AddSubmonoidClass σ M] →\n                                  [SetLike.GradedMonoid 𝓐] →\n                                    [SetLike.GradedSMul 𝓐 𝓜] →\n                                      [inst_11 : DecidableEq ιA] →\n                                        [DecidableEq ιM] → [GradedRing 𝓐] → Module A (DirectSum ιM fun i => ↥(𝓜 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(inst._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.13 j._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.45))))) x._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.48 x._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.50)) (HAdd.hAdd.{u_1, u_1, u_1} (A (HAdd.hAdd.{u_2, u_2, u_2} ιA ιA ιA (instHAdd.{u_2} ιA (AddSemigroup.toAdd.{u_2} ιA (AddMonoid.toAddSemigroup.{u_2} ιA inst._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.10))) i._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.44 j._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.45)) (A (HAdd.hAdd.{u_2, u_2, u_2} ιA ιA ιA (instHAdd.{u_2} ιA (AddSemigroup.toAdd.{u_2} ιA (AddMonoid.toAddSemigroup.{u_2} ιA inst._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.10))) i._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.44 j._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.45)) (A (HAdd.hAdd.{u_2, 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(AddMonoid.toAddSemigroup.{u_2} ιA inst._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.10))) i._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.44 j._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.45)))))) (GradedMonoid.GMul.mul.{u_2, u_1} ιA A (AddSemigroup.toAdd.{u_2} ιA (AddMonoid.toAddSemigroup.{u_2} ιA inst._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.10)) (DirectSum.GNonUnitalNonAssocSemiring.toGMul.{u_2, u_1} ιA A (AddSemigroup.toAdd.{u_2} ιA (AddMonoid.toAddSemigroup.{u_2} ιA inst._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.10)) inst._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.13 (DirectSum.GSemiring.toGNonUnitalNonAssocSemiring.{u_2, u_1} ιA A inst._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.13 h)) i._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.44 j._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.45 x._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.46 x._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.48) (GradedMonoid.GMul.mul.{u_2, u_1} ιA A (AddSemigroup.toAdd.{u_2} ιA (AddMonoid.toAddSemigroup.{u_2} ιA inst._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.10)) (DirectSum.GNonUnitalNonAssocSemiring.toGMul.{u_2, u_1} ιA A (AddSemigroup.toAdd.{u_2} ιA (AddMonoid.toAddSemigroup.{u_2} ιA inst._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.10)) inst._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.13 (DirectSum.GSemiring.toGNonUnitalNonAssocSemiring.{u_2, u_1} ιA A inst._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.13 h)) i._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.44 j._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.45 x._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.46 x._@.Mathlib.Algebra.Module.GradedModule.1743805578._hygCtx._hyg.50))","typeFull":"∀ {ιA : Type u_2} (A : ιA → Type u_1) [inst : AddMonoid ιA] [inst_1 : (i : ιA) → AddCommMonoid (A i)]\n  [h : DirectSum.GSemiring A] {i j : ιA} (x : A i) (x_1 x_2 : A j),\n  GradedMonoid.GMul.mul x (x_1 + x_2) = GradedMonoid.GMul.mul x x_1 + GradedMonoid.GMul.mul x x_2","typeReadable":"∀ {ιA : Type u_2} (A : ιA → Type u_1) [inst : AddMonoid ιA] [inst_1 : (i : ιA) → AddCommMonoid (A i)]\n  [h : DirectSum.GSemiring A] {i j : ιA} (x : A i) (x_1 x_2 : A j),\n  GradedMonoid.GMul.mul x (x_1 + x_2) = GradedMonoid.GMul.mul x x_1 + GradedMonoid.GMul.mul x x_2","typeReferences":[["instHAdd"],["DirectSum","GSemiring","toGNonUnitalNonAssocSemiring"],["AddMonoid"],["HAdd","hAdd"],["AddCommMonoid"],["DirectSum","GSemiring"],["DirectSum","GNonUnitalNonAssocSemiring","toGMul"],["AddCommMonoid","toAddCommSemigroup"],["AddMonoid","toAddSemigroup"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Eq"],["GradedMonoid","GMul","mul"],["AddSemigroup","toAdd"]],"valueReferences":[["AddMonoid","toAddSemigroup"],["DirectSum","GNonUnitalNonAssocSemiring","mul_add"],["DirectSum","GSemiring","toGNonUnitalNonAssocSemiring"],["AddSemigroup","toAdd"]]},{"isProp":false,"kind":"definition","name":["DirectSum","GdistribMulAction","recOn"],"typeFallback":"forall {ιA : Type.{u_1}} {ιB : Type.{u_2}} {A : ιA -> Type.{u_3}} {M : ιB -> Type.{u_4}} [inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 : AddMonoid.{u_1} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 : VAdd.{u_1, u_2} ιA ιB] [inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 : GradedMonoid.GMonoid.{u_1, u_3} ιA A inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18] [inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 : forall (i : ιB), AddMonoid.{u_4} (M i)] {motive : (DirectSum.GdistribMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28) -> Sort.{u}} (t : DirectSum.GdistribMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28), (forall [toGMulAction : GradedMonoid.GMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25] (smul_add : forall {i : ιA} {j : ιB} (a : A i) (b : M j) (c : M j), Eq.{succ u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 (GradedMonoid.GMulAction.toGSMul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 toGMulAction) i j a (HAdd.hAdd.{u_4, u_4, u_4} (M j) (M j) (M j) (instHAdd.{u_4} (M j) (AddSemigroup.toAdd.{u_4} (M j) (AddMonoid.toAddSemigroup.{u_4} (M j) (inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 j)))) b c)) (HAdd.hAdd.{u_4, u_4, u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (instHAdd.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (AddSemigroup.toAdd.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (AddMonoid.toAddSemigroup.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j))))) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 (GradedMonoid.GMulAction.toGSMul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 toGMulAction) i j a b) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M 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i j a (OfNat.ofNat.{u_4} (M j) 0 (Zero.toOfNat0.{u_4} (M j) (AddZero.toZero.{u_4} (M j) (AddZeroClass.toAddZero.{u_4} (M j) (AddMonoid.toAddZeroClass.{u_4} (M j) (inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 j))))))) (OfNat.ofNat.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) 0 (Zero.toOfNat0.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (AddZero.toZero.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (AddZeroClass.toAddZero.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (AddMonoid.toAddZeroClass.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)))))))), motive (DirectSum.GdistribMulAction.mk.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 toGMulAction smul_add smul_zero)) -> (motive t)","typeFull":"{ιA : Type u_1} →\n  {ιB : Type u_2} →\n    {A : ιA → Type u_3} →\n      {M : ιB → Type u_4} →\n        [inst : AddMonoid ιA] →\n          [inst_1 : VAdd ιA ιB] →\n            [inst_2 : GradedMonoid.GMonoid A] →\n              [inst_3 : (i : ιB) → AddMonoid (M i)] →\n                {motive : DirectSum.GdistribMulAction A M → Sort u} →\n                  (t : DirectSum.GdistribMulAction A M) →\n                    ([toGMulAction : GradedMonoid.GMulAction A M] →\n                        (smul_add :\n                            ∀ {i : ιA} {j : ιB} (a : A i) (b c : M j),\n                              GradedMonoid.GSMul.smul a (b + c) =\n                                GradedMonoid.GSMul.smul a b + GradedMonoid.GSMul.smul a c) →\n                          (smul_zero : ∀ {i : ιA} {j : ιB} (a : A i), GradedMonoid.GSMul.smul a 0 = 0) →\n                            motive { toGMulAction := toGMulAction, smul_add := smul_add, smul_zero := smul_zero }) →\n                      motive t","typeReadable":"{ιA : Type u_1} →\n  {ιB : Type u_2} →\n    {A : ιA → Type u_3} →\n      {M : ιB → Type u_4} →\n        [inst : AddMonoid ιA] →\n          [inst_1 : VAdd ιA ιB] →\n            [inst_2 : GradedMonoid.GMonoid A] →\n              [inst_3 : (i : ιB) → AddMonoid (M i)] →\n                {motive : DirectSum.GdistribMulAction A M → Sort u} →\n                  (t : DirectSum.GdistribMulAction A M) →\n                    ([toGMulAction : GradedMonoid.GMulAction A M] →\n                        (smul_add :\n                            ∀ {i : ιA} {j : ιB} (a : A i) (b c : M j),\n                              GradedMonoid.GSMul.smul a (b + c) =\n                                GradedMonoid.GSMul.smul a b + GradedMonoid.GSMul.smul a c) →\n                          (smul_zero : ∀ {i : ιA} {j : ιB} (a : A i), GradedMonoid.GSMul.smul a 0 = 0) →\n                            motive { toGMulAction := toGMulAction, smul_add := smul_add, smul_zero := smul_zero }) →\n                      motive t","typeReferences":[["VAdd"],["DirectSum","GdistribMulAction"],["instHVAdd"],["instHAdd"],["AddMonoid"],["DirectSum","GdistribMulAction","mk"],["AddZeroClass","toAddZero"],["GradedMonoid","GMulAction"],["OfNat","ofNat"],["HVAdd","hVAdd"],["HAdd","hAdd"],["AddMonoid","toAddSemigroup"],["GradedMonoid","GMulAction","toGSMul"],["Zero","toOfNat0"],["Eq"],["AddZero","toZero"],["GradedMonoid","GMonoid"],["AddMonoid","toAddZeroClass"],["GradedMonoid","GSMul","smul"],["AddSemigroup","toAdd"]],"valueReferences":[["DirectSum","GdistribMulAction","rec"]]},{"isProp":true,"kind":"theorem","name":["GradedModule","isModule","_proof_1"],"typeFallback":"forall {ιA : Type.{u_5}} {ιM : Type.{u_1}} {A : Type.{u_4}} {M : Type.{u_2}} {σ : Type.{u_3}} {σ' : Type.{u_6}} [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9 : AddMonoid.{u_5} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.12 : AddAction.{u_5, u_1} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19 : Semiring.{u_4} A] (𝓐 : ιA -> σ') [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29 : SetLike.{u_6, u_4} σ' A] (𝓜 : ιM -> σ) [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.39 : Module.{u_4, u_2} A M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.43 : SetLike.{u_3, u_2} σ M] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.47 : AddSubmonoidClass.{u_6, u_4} σ' A (AddMonoid.toAddZeroClass.{u_4} A (AddMonoidWithOne.toAddMonoid.{u_4} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_4} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_4} A (Semiring.toNonAssocSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19))))) inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.51 : AddSubmonoidClass.{u_3, u_2} σ M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36)) inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.43] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.55 : SetLike.GradedMonoid.{u_5, u_4, u_6} ιA A σ' inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29 (MonoidWithZero.toMonoid.{u_4} A (Semiring.toMonoidWithZero.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19)) inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9 𝓐] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.58 : SetLike.GradedSMul.{u_5, u_1, u_6, 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AddCommMonoid (M i)] [inst_4 : GradedMonoid.GMonoid A]\n  [inst_5 : DirectSum.Gmodule A M] {i : ιA} {j : ιB} (a : A i) (b : M j),\n  ((DirectSum.gsmulHom A M) a) b = GradedMonoid.GSMul.smul a b","typeReferences":[["VAdd"],["instHVAdd"],["DirectSum","gsmulHom"],["AddMonoid"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["DirectSum","Gmodule","toGdistribMulAction"],["DFunLike","coe"],["HVAdd","hVAdd"],["AddCommMonoid"],["AddMonoidHom"],["GradedMonoid","GMulAction","toGSMul"],["AddMonoidHom","instFunLike"],["DirectSum","GdistribMulAction","toGMulAction"],["DirectSum","Gmodule"],["AddMonoidHom","instAddCommMonoid"],["Eq"],["GradedMonoid","GMonoid"],["GradedMonoid","GSMul","smul"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["instHVAdd"],["AddMonoidHom"],["Eq","refl"],["AddMonoidHom","instFunLike"],["DirectSum","gsmulHom"],["AddMonoidHom","instAddCommMonoid"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["HVAdd","hVAdd"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"recursor","name":["DirectSum","GdistribMulAction","rec"],"typeFallback":"forall {ιA : Type.{u_1}} {ιB : Type.{u_2}} {A : ιA -> Type.{u_3}} {M : ιB -> Type.{u_4}} [inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 : AddMonoid.{u_1} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 : VAdd.{u_1, u_2} ιA ιB] [inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 : GradedMonoid.GMonoid.{u_1, u_3} ιA A inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18] [inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 : forall (i : ιB), AddMonoid.{u_4} (M i)] {motive : (DirectSum.GdistribMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28) -> Sort.{u}}, (forall [toGMulAction : GradedMonoid.GMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25] (smul_add : forall {i : ιA} {j : ιB} (a : A i) (b : M j) (c : M j), Eq.{succ u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 (GradedMonoid.GMulAction.toGSMul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 toGMulAction) i j a (HAdd.hAdd.{u_4, u_4, u_4} (M j) (M j) (M j) (instHAdd.{u_4} (M j) (AddSemigroup.toAdd.{u_4} (M j) (AddMonoid.toAddSemigroup.{u_4} (M j) (inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 j)))) b c)) (HAdd.hAdd.{u_4, u_4, u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (instHAdd.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (AddSemigroup.toAdd.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (AddMonoid.toAddSemigroup.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j))))) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 (GradedMonoid.GMulAction.toGSMul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 toGMulAction) i j a b) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 (GradedMonoid.GMulAction.toGSMul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 toGMulAction) i j a c))) (smul_zero : forall {i : ιA} {j : ιB} (a : A i), Eq.{succ u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 (GradedMonoid.GMulAction.toGSMul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 toGMulAction) i j a (OfNat.ofNat.{u_4} (M j) 0 (Zero.toOfNat0.{u_4} (M j) (AddZero.toZero.{u_4} (M j) (AddZeroClass.toAddZero.{u_4} (M j) (AddMonoid.toAddZeroClass.{u_4} (M j) (inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 j))))))) (OfNat.ofNat.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) 0 (Zero.toOfNat0.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (AddZero.toZero.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (AddZeroClass.toAddZero.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (AddMonoid.toAddZeroClass.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)))))))), motive (DirectSum.GdistribMulAction.mk.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 toGMulAction smul_add smul_zero)) -> (forall (t : DirectSum.GdistribMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28), motive t)","typeFull":"{ιA : Type u_1} →\n  {ιB : Type u_2} →\n    {A : ιA → Type u_3} →\n      {M : ιB → Type u_4} →\n        [inst : AddMonoid ιA] →\n          [inst_1 : VAdd ιA ιB] →\n            [inst_2 : GradedMonoid.GMonoid A] →\n              [inst_3 : (i : ιB) → AddMonoid (M i)] →\n                {motive : DirectSum.GdistribMulAction A M → Sort u} →\n                  ([toGMulAction : GradedMonoid.GMulAction A M] →\n                      (smul_add :\n                          ∀ {i : ιA} {j : ιB} (a : A i) (b c : M j),\n                            GradedMonoid.GSMul.smul a (b + c) =\n                              GradedMonoid.GSMul.smul a b + GradedMonoid.GSMul.smul a c) →\n                        (smul_zero : ∀ {i : ιA} {j : ιB} (a : A i), GradedMonoid.GSMul.smul a 0 = 0) →\n                          motive { toGMulAction := toGMulAction, smul_add := smul_add, smul_zero := smul_zero }) →\n                    (t : DirectSum.GdistribMulAction A M) → motive t","typeReadable":"{ιA : Type u_1} →\n  {ιB : Type u_2} →\n    {A : ιA → Type u_3} →\n      {M : ιB → Type u_4} →\n        [inst : AddMonoid ιA] →\n          [inst_1 : VAdd ιA ιB] →\n            [inst_2 : GradedMonoid.GMonoid A] →\n              [inst_3 : (i : ιB) → AddMonoid (M i)] →\n                {motive : DirectSum.GdistribMulAction A M → Sort u} →\n                  ([toGMulAction : GradedMonoid.GMulAction A M] →\n                      (smul_add :\n                          ∀ {i : ιA} {j : ιB} (a : A i) (b c : M j),\n                            GradedMonoid.GSMul.smul a (b + c) =\n                              GradedMonoid.GSMul.smul a b + GradedMonoid.GSMul.smul a c) →\n                        (smul_zero : ∀ {i : ιA} {j : ιB} (a : A i), GradedMonoid.GSMul.smul a 0 = 0) →\n                          motive { toGMulAction := toGMulAction, smul_add := smul_add, smul_zero := smul_zero }) →\n                    (t : DirectSum.GdistribMulAction A M) → motive t","typeReferences":[["VAdd"],["DirectSum","GdistribMulAction"],["instHVAdd"],["instHAdd"],["AddMonoid"],["DirectSum","GdistribMulAction","mk"],["AddZeroClass","toAddZero"],["GradedMonoid","GMulAction"],["OfNat","ofNat"],["HVAdd","hVAdd"],["HAdd","hAdd"],["AddMonoid","toAddSemigroup"],["GradedMonoid","GMulAction","toGSMul"],["Zero","toOfNat0"],["Eq"],["AddZero","toZero"],["GradedMonoid","GMonoid"],["AddMonoid","toAddZeroClass"],["GradedMonoid","GSMul","smul"],["AddSemigroup","toAdd"]],"valueReferences":null},{"isProp":false,"kind":"definition","name":["DirectSum","Gmodule","smulAddMonoidHom"],"typeFallback":"forall {ιA : Type.{u_1}} {ιB : Type.{u_2}} (A : ιA -> Type.{u_3}) (M : ιB -> Type.{u_4}) [inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.10 : AddMonoid.{u_1} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.13 : VAdd.{u_1, u_2} ιA ιB] [inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.17 : forall (i : ιA), AddCommMonoid.{u_3} (A i)] [inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.28 : forall (i : ιB), AddCommMonoid.{u_4} (M i)] [inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.37 : DecidableEq.{succ u_1} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.40 : DecidableEq.{succ u_2} ιB] [inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.43 : GradedMonoid.GMonoid.{u_1, u_3} ιA A inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.46 : DirectSum.Gmodule.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.13 (fun (i : ιA) => AddCommMonoid.toAddMonoid.{u_3} (A i) (inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.17 i)) (fun (i : ιB) => AddCommMonoid.toAddMonoid.{u_4} (M i) (inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.28 i)) inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.43], AddMonoidHom.{max u_3 u_1, max u_4 u_2} (DirectSum.{u_1, u_3} ιA (fun (i : ιA) => A i) inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.17) (AddMonoidHom.{max u_4 u_2, max u_4 u_2} (DirectSum.{u_2, u_4} ιB (fun (i : ιB) => M i) inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.28) (DirectSum.{u_2, u_4} ιB (fun (i : ιB) => M i) inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.28) (AddZeroClass.toAddZero.{max u_2 u_4} (DirectSum.{u_2, u_4} ιB (fun (i : ιB) => M i) inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.28) (AddMonoid.toAddZeroClass.{max u_2 u_4} (DirectSum.{u_2, u_4} ιB (fun (i : ιB) => M i) inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.28) (AddCommMonoid.toAddMonoid.{max u_2 u_4} (DirectSum.{u_2, u_4} ιB (fun (i : ιB) => M i) inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.28) (instAddCommMonoidDirectSum.{u_2, u_4} ιB (fun (i : ιB) => M i) inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.28)))) (AddZeroClass.toAddZero.{max u_2 u_4} (DirectSum.{u_2, u_4} ιB (fun (i : ιB) => M i) inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.28) (AddMonoid.toAddZeroClass.{max u_2 u_4} (DirectSum.{u_2, u_4} ιB (fun (i : ιB) => M i) inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.28) (AddCommMonoid.toAddMonoid.{max u_2 u_4} (DirectSum.{u_2, u_4} ιB (fun (i : ιB) => M i) inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.28) (instAddCommMonoidDirectSum.{u_2, u_4} ιB (fun (i : ιB) => M i) inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.28))))) (AddZeroClass.toAddZero.{max u_1 u_3} (DirectSum.{u_1, u_3} ιA (fun (i : ιA) => A i) inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.17) (AddMonoid.toAddZeroClass.{max u_1 u_3} (DirectSum.{u_1, u_3} ιA (fun (i : ιA) => A i) inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.17) (AddCommMonoid.toAddMonoid.{max u_1 u_3} (DirectSum.{u_1, u_3} ιA (fun (i : ιA) => A i) inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.17) (instAddCommMonoidDirectSum.{u_1, u_3} ιA (fun (i : ιA) => A i) inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.17)))) (AddZeroClass.toAddZero.{max u_2 u_4} (AddMonoidHom.{max u_4 u_2, max u_4 u_2} (DirectSum.{u_2, u_4} ιB (fun (i : ιB) => M i) inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.28) (DirectSum.{u_2, u_4} ιB (fun (i : ιB) => M i) inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.28) (AddZeroClass.toAddZero.{max u_2 u_4} (DirectSum.{u_2, u_4} ιB (fun (i : ιB) => M i) inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.28) (AddMonoid.toAddZeroClass.{max u_2 u_4} (DirectSum.{u_2, u_4} ιB (fun (i : ιB) => M i) 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inst._@.Mathlib.Algebra.Module.GradedModule.82327297._hygCtx._hyg.28)))))","typeFull":"{ιA : Type u_1} →\n  {ιB : Type u_2} →\n    (A : ιA → Type u_3) →\n      (M : ιB → Type u_4) →\n        [inst : AddMonoid ιA] →\n          [inst_1 : VAdd ιA ιB] →\n            [inst_2 : (i : ιA) → AddCommMonoid (A i)] →\n              [inst_3 : (i : ιB) → AddCommMonoid (M i)] →\n                [DecidableEq ιA] →\n                  [DecidableEq ιB] →\n                    [inst_6 : GradedMonoid.GMonoid A] →\n                      [DirectSum.Gmodule A M] →\n                        (DirectSum ιA fun i => A i) →+ (DirectSum ιB fun i => M i) →+ DirectSum ιB fun i => M i","typeReadable":"{ιA : Type u_1} →\n  {ιB : Type u_2} →\n    (A : ιA → Type u_3) →\n      (M : ιB → Type u_4) →\n        [inst : AddMonoid ιA] →\n          [inst_1 : VAdd ιA ιB] →\n            [inst_2 : (i : ιA) → AddCommMonoid (A i)] →\n              [inst_3 : (i : ιB) → AddCommMonoid (M i)] →\n                [DecidableEq ιA] →\n                  [DecidableEq ιB] →\n                    [inst_6 : GradedMonoid.GMonoid A] →\n                      [DirectSum.Gmodule A M] →\n                        (DirectSum ιA fun i => A i) →+ (DirectSum ιB fun i => M i) →+ DirectSum ιB fun i => M i","typeReferences":[["VAdd"],["DecidableEq"],["AddMonoid"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["DirectSum"],["AddCommMonoid"],["instAddCommMonoidDirectSum"],["AddMonoidHom"],["DirectSum","Gmodule"],["AddMonoidHom","instAddCommMonoid"],["GradedMonoid","GMonoid"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["DirectSum","toAddMonoid"],["instHVAdd"],["DirectSum","of"],["DirectSum","gsmulHom"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["HVAdd","hVAdd"],["DirectSum"],["AddMonoidHom","compHom"],["instAddCommMonoidDirectSum"],["AddMonoidHom","comp"],["AddMonoidHom"],["AddMonoidHom","flip"],["AddMonoidHom","instFunLike"],["AddMonoidHom","instAddCommMonoid"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"definition","name":["SetLike","gmodule"],"typeFallback":"forall {ιA : Type.{u_1}} {ιM : Type.{u_2}} {A : Type.{u_4}} {M : Type.{u_5}} {σ : Type.{u_6}} {σ' : Type.{u_7}} [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 : AddMonoid.{u_1} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12 : AddAction.{u_1, u_2} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19 : Semiring.{u_4} A] (𝓐 : ιA -> σ') [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29 : SetLike.{u_7, u_4} σ' A] (𝓜 : ιM -> σ) [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36 : AddCommMonoid.{u_5} M] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.39 : Module.{u_4, u_5} A M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43 : SetLike.{u_6, u_5} σ M] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.47 : AddSubmonoidClass.{u_7, u_4} σ' A (AddMonoid.toAddZeroClass.{u_4} A (AddMonoidWithOne.toAddMonoid.{u_4} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_4} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_4} A (Semiring.toNonAssocSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19))))) inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.51 : AddSubmonoidClass.{u_6, u_5} σ M (AddMonoid.toAddZeroClass.{u_5} M (AddCommMonoid.toAddMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36)) inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.55 : SetLike.GradedMonoid.{u_1, u_4, u_7} ιA A σ' inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29 (MonoidWithZero.toMonoid.{u_4} A (Semiring.toMonoidWithZero.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19)) inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 𝓐] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.58 : SetLike.GradedSMul.{u_1, u_2, u_7, u_4, u_6, u_5} ιA ιM σ' A σ M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43 (SMulZeroClass.toSMul.{u_4, u_5} A M (AddZero.toZero.{u_5} M (AddZeroClass.toAddZero.{u_5} M (AddMonoid.toAddZeroClass.{u_5} M (AddCommMonoid.toAddMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36)))) (DistribSMul.toSMulZeroClass.{u_4, u_5} A M (AddMonoid.toAddZeroClass.{u_5} M (AddCommMonoid.toAddMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36)) (DistribMulAction.toDistribSMul.{u_4, u_5} A M (MonoidWithZero.toMonoid.{u_4} A (Semiring.toMonoidWithZero.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19)) (AddCommMonoid.toAddMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36) (Module.toDistribMulAction.{u_4, u_5} A M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.39)))) (AddSemigroupAction.toVAdd.{u_1, u_2} ιA ιM (AddMonoid.toAddSemigroup.{u_1} ιA inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_1, u_2} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12)) 𝓐 𝓜], DirectSum.Gmodule.{u_1, u_2, u_4, u_5} ιA ιM (fun (i : ιA) => Subtype.{succ u_4} A (fun (x : A) => Membership.mem.{u_4, u_7} A σ' (SetLike.instMembership.{u_7, u_4} σ' A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29) (𝓐 i) x)) (fun (i : ιM) => Subtype.{succ u_5} M (fun (x : M) => Membership.mem.{u_5, u_6} M σ (SetLike.instMembership.{u_6, u_5} σ M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43) (𝓜 i) x)) inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 (AddSemigroupAction.toVAdd.{u_1, u_2} ιA ιM (AddMonoid.toAddSemigroup.{u_1} ιA inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_1, u_2} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12)) (fun (i : ιA) => AddCommMonoid.toAddMonoid.{u_4} (Subtype.{succ u_4} A (fun (x : A) => Membership.mem.{u_4, u_7} A σ' (SetLike.instMembership.{u_7, u_4} σ' A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29) (𝓐 i) x)) (AddSubmonoidClass.toAddCommMonoid.{u_7, u_4} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} A (Semiring.toNonAssocSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19))) σ' inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.47 (𝓐 i))) (fun (i : ιM) => AddCommMonoid.toAddMonoid.{u_5} (Subtype.{succ u_5} M (fun (x : M) => Membership.mem.{u_5, u_6} M σ (SetLike.instMembership.{u_6, u_5} σ M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43) (𝓜 i) x)) (AddSubmonoidClass.toAddCommMonoid.{u_6, u_5} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36 σ inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.51 (𝓜 i))) (SetLike.gMonoid.{u_1, u_4, u_7} ιA A σ' inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29 (MonoidWithZero.toMonoid.{u_4} A (Semiring.toMonoidWithZero.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19)) inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 𝓐 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.55)","typeFull":"{ιA : Type u_1} →\n  {ιM : Type u_2} →\n    {A : Type u_4} →\n      {M : Type u_5} →\n        {σ : Type u_6} →\n          {σ' : Type u_7} →\n            [inst : AddMonoid ιA] →\n              [inst_1 : AddAction ιA ιM] →\n                [inst_2 : Semiring A] →\n                  (𝓐 : ιA → σ') →\n                    [inst_3 : SetLike σ' A] →\n                      (𝓜 : ιM → σ) →\n                        [inst_4 : AddCommMonoid M] →\n                          [inst_5 : Module A M] →\n                            [inst_6 : SetLike σ M] →\n                              [inst_7 : AddSubmonoidClass σ' A] →\n                                [inst_8 : AddSubmonoidClass σ M] →\n                                  [inst_9 : SetLike.GradedMonoid 𝓐] →\n                                    [SetLike.GradedSMul 𝓐 𝓜] → DirectSum.Gmodule (fun i => ↥(𝓐 i)) fun i => ↥(𝓜 i)","typeReadable":"{ιA : Type u_1} →\n  {ιM : Type u_2} →\n    {A : Type u_4} →\n      {M : Type u_5} →\n        {σ : Type u_6} →\n          {σ' : Type u_7} →\n            [inst : AddMonoid ιA] →\n              [inst_1 : AddAction ιA ιM] →\n                [inst_2 : Semiring A] →\n                  (𝓐 : ιA → σ') →\n                    [inst_3 : SetLike σ' A] →\n                      (𝓜 : ιM → σ) →\n                        [inst_4 : AddCommMonoid M] →\n                          [inst_5 : Module A M] →\n                            [inst_6 : SetLike σ M] →\n                              [inst_7 : AddSubmonoidClass σ' A] →\n                                [inst_8 : AddSubmonoidClass σ M] →\n                                  [inst_9 : SetLike.GradedMonoid 𝓐] →\n                                    [SetLike.GradedSMul 𝓐 𝓜] → DirectSum.Gmodule (fun i => ↥(𝓐 i)) fun i => ↥(𝓜 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{ιA : Type.{u_1}} {ιB : Type.{u_2}} (A : ιA -> Type.{u_3}) (M : ιB -> Type.{u_4}) [inst._@.Mathlib.Algebra.Module.GradedModule.4183123856._hygCtx._hyg.10 : AddMonoid.{u_1} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.4183123856._hygCtx._hyg.13 : VAdd.{u_1, u_2} ιA ιB] [inst._@.Mathlib.Algebra.Module.GradedModule.4183123856._hygCtx._hyg.17 : forall (i : ιA), AddCommMonoid.{u_3} (A i)] [inst._@.Mathlib.Algebra.Module.GradedModule.4183123856._hygCtx._hyg.28 : forall (i : ιB), AddCommMonoid.{u_4} (M i)] [inst._@.Mathlib.Algebra.Module.GradedModule.4183123856._hygCtx._hyg.37 : DecidableEq.{succ u_1} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.4183123856._hygCtx._hyg.40 : DecidableEq.{succ u_2} ιB] [inst._@.Mathlib.Algebra.Module.GradedModule.4183123856._hygCtx._hyg.43 : GradedMonoid.GMonoid.{u_1, u_3} ιA A inst._@.Mathlib.Algebra.Module.GradedModule.4183123856._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.GradedModule.4183123856._hygCtx._hyg.46 : DirectSum.Gmodule.{u_1, u_2, 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j._@.Mathlib.Algebra.Module.GradedModule.1644032540._hygCtx._hyg.88 _a _c))","typeFull":"∀ {ιA : Type u_5} {ιM : Type u_6} {A : Type u_1} {M : Type u_3} {σ : Type u_4} {σ' : Type u_2} [inst : AddMonoid ιA]\n  [inst_1 : AddAction ιA ιM] [inst_2 : Semiring A] (𝓐 : ιA → σ') [inst_3 : SetLike σ' A] (𝓜 : ιM → σ)\n  [inst_4 : AddMonoid M] [inst_5 : DistribMulAction A M] [inst_6 : SetLike σ M] [inst_7 : AddSubmonoidClass σ M]\n  [inst_8 : SetLike.GradedMonoid 𝓐] [inst_9 : SetLike.GradedSMul 𝓐 𝓜] {i : ιA} {j : ιM} (_a : ↥(𝓐 i)) (_b _c : ↥(𝓜 j)),\n  GradedMonoid.GSMul.smul _a (_b + _c) = GradedMonoid.GSMul.smul _a _b + GradedMonoid.GSMul.smul _a _c","typeReadable":"∀ {ιA : Type u_5} {ιM : Type u_6} {A : Type u_1} {M : Type u_3} {σ : Type u_4} {σ' : Type u_2} [inst : AddMonoid ιA]\n  [inst_1 : AddAction ιA ιM] [inst_2 : Semiring A] (𝓐 : ιA → σ') [inst_3 : SetLike σ' A] (𝓜 : ιM → σ)\n  [inst_4 : AddMonoid M] [inst_5 : DistribMulAction A M] [inst_6 : SetLike σ M] [inst_7 : AddSubmonoidClass σ M]\n  [inst_8 : SetLike.GradedMonoid 𝓐] [inst_9 : SetLike.GradedSMul 𝓐 𝓜] {i : ιA} {j : ιM} (_a : ↥(𝓐 i)) (_b _c : ↥(𝓜 j)),\n  GradedMonoid.GSMul.smul _a (_b + _c) = GradedMonoid.GSMul.smul _a _b + GradedMonoid.GSMul.smul _a 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{ιA : Type.{u_1}} {ιB : Type.{u_2}} (A : ιA -> Type.{u_3}) (M : ιB -> Type.{u_4}) [inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 : AddMonoid.{u_1} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 : VAdd.{u_1, u_2} ιA ιB] [inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 : GradedMonoid.GMonoid.{u_1, u_3} ιA A inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18] [inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 : forall (i : ιB), AddMonoid.{u_4} (M i)], Sort.{max (max (max (succ u_1) (succ u_2)) (succ u_3)) (succ u_4)}","typeFull":"{ιA : Type u_1} →\n  {ιB : Type u_2} →\n    (A : ιA → Type u_3) →\n      (M : ιB → Type u_4) →\n        [inst : AddMonoid ιA] →\n          [VAdd ιA ιB] →\n            [GradedMonoid.GMonoid A] → [(i : ιB) → AddMonoid (M i)] → Type (max (max (max u_1 u_2) u_3) u_4)","typeReadable":"{ιA : Type u_1} →\n  {ιB : Type u_2} →\n    (A : ιA → Type u_3) →\n      (M : ιB → Type u_4) →\n        [inst : AddMonoid ιA] →\n          [VAdd ιA ιB] →\n            [GradedMonoid.GMonoid A] → [(i : ιB) → AddMonoid (M i)] → Type (max (max (max u_1 u_2) u_3) u_4)","typeReferences":[["VAdd"],["AddMonoid"],["GradedMonoid","GMonoid"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["SetLike","gmodule","_proof_2"],"typeFallback":"forall {ιA : Type.{u_4}} {ιM : Type.{u_1}} {A : Type.{u_5}} {M : Type.{u_2}} {σ : Type.{u_3}} {σ' : Type.{u_6}} [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 : AddMonoid.{u_4} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12 : AddAction.{u_4, u_1} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19 : Semiring.{u_5} A] (𝓐 : ιA -> σ') [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29 : SetLike.{u_6, u_5} σ' A] (𝓜 : ιM -> σ) 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σ' A (AddMonoid.toAddZeroClass.{u_4} A (AddMonoidWithOne.toAddMonoid.{u_4} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_4} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_4} A (Semiring.toNonAssocSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19))))) inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.51 : AddSubmonoidClass.{u_3, u_2} σ M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36)) inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.43] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.55 : SetLike.GradedMonoid.{u_5, u_4, u_6} ιA A σ' inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29 (MonoidWithZero.toMonoid.{u_4} A (Semiring.toMonoidWithZero.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19)) 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→\n            [inst_2 : GradedMonoid.GMonoid A] →\n              [inst_3 : (i : ιB) → AddMonoid (M i)] →\n                (smul : {i : ιA} → {j : ιB} → A i → M j → M (i +ᵥ j)) →\n                  (∀ (b : GradedMonoid M), 1 • b = b) →\n                    (∀ (a a' : GradedMonoid A) (b : GradedMonoid M), (a * a') • b = a • a' • b) →\n                      (∀ {i : ιA} {j : ιB} (a : A i) (b c : M j), smul a (b + c) = smul a b + smul a c) →\n                        (∀ {i : ιA} {j : ιB} (a : A i), smul a 0 = 0) → DirectSum.GdistribMulAction A M","typeReadable":"{ιA : Type u_1} →\n  {ιB : Type u_2} →\n    {A : ιA → Type u_3} →\n      {M : ιB → Type u_4} →\n        [inst : AddMonoid ιA] →\n          [inst_1 : VAdd ιA ιB] →\n            [inst_2 : GradedMonoid.GMonoid A] →\n              [inst_3 : (i : ιB) → AddMonoid (M i)] →\n                (smul : {i : ιA} → {j : ιB} → A i → M j → M (i +ᵥ j)) →\n                  (∀ (b : GradedMonoid M), 1 • b = b) →\n                    (∀ (a a' : GradedMonoid A) (b : GradedMonoid M), (a * a') • b = a • a' • b) →\n                      (∀ {i : ιA} {j : ιB} (a : A i) (b c : M j), smul a (b + c) = smul a b + smul a c) →\n                        (∀ {i : ιA} {j : ιB} (a : A i), smul a 0 = 0) → DirectSum.GdistribMulAction A 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{inst_4 : GradedMonoid.GMonoid A} → DirectSum.Gmodule A M → ℕ","typeReferences":[["Nat"],["VAdd"],["DirectSum","Gmodule"],["AddMonoid"],["GradedMonoid","GMonoid"]],"valueReferences":[]},{"isProp":false,"kind":"definition","name":["SetLike","gmulAction"],"typeFallback":"forall {ιA : Type.{u_1}} {ιM : Type.{u_2}} {A : Type.{u_4}} {M : Type.{u_5}} {σ : Type.{u_6}} {σ' : Type.{u_7}} [inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.9 : AddMonoid.{u_1} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.12 : AddAction.{u_1, u_2} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.9] [inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.19 : Semiring.{u_4} A] (𝓐 : ιA -> σ') [inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.29 : SetLike.{u_7, u_4} σ' A] (𝓜 : ιM -> σ) [inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.36 : AddMonoid.{u_5} M] [inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.39 : DistribMulAction.{u_4, u_5} A M (MonoidWithZero.toMonoid.{u_4} A (Semiring.toMonoidWithZero.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.19)) inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.36] [inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.43 : SetLike.{u_6, u_5} σ M] [inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.47 : SetLike.GradedMonoid.{u_1, u_4, u_7} ιA A σ' inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.29 (MonoidWithZero.toMonoid.{u_4} A (Semiring.toMonoidWithZero.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.19)) inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.9 𝓐] [inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.50 : SetLike.GradedSMul.{u_1, u_2, u_7, u_4, u_6, u_5} ιA ιM σ' A σ M inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.43 (SMulZeroClass.toSMul.{u_4, u_5} A M (AddZero.toZero.{u_5} M (AddZeroClass.toAddZero.{u_5} M (AddMonoid.toAddZeroClass.{u_5} M inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.36))) (DistribSMul.toSMulZeroClass.{u_4, u_5} A M (AddMonoid.toAddZeroClass.{u_5} M inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.36) (DistribMulAction.toDistribSMul.{u_4, u_5} A M (MonoidWithZero.toMonoid.{u_4} A (Semiring.toMonoidWithZero.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.19)) inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.39))) (AddSemigroupAction.toVAdd.{u_1, u_2} ιA ιM (AddMonoid.toAddSemigroup.{u_1} ιA inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_1, u_2} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.12)) 𝓐 𝓜], GradedMonoid.GMulAction.{u_1, u_2, u_4, u_5} ιA ιM (fun (i : ιA) => Subtype.{succ u_4} A (fun (x : A) => Membership.mem.{u_4, u_7} A σ' (SetLike.instMembership.{u_7, u_4} σ' A inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.29) (𝓐 i) x)) (fun (i : ιM) => Subtype.{succ u_5} M (fun (x : M) => Membership.mem.{u_5, u_6} M σ (SetLike.instMembership.{u_6, u_5} σ M inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.43) (𝓜 i) x)) inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.9 (AddSemigroupAction.toVAdd.{u_1, u_2} ιA ιM (AddMonoid.toAddSemigroup.{u_1} ιA inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_1, u_2} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.12)) (SetLike.gMonoid.{u_1, u_4, u_7} ιA A σ' inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.29 (MonoidWithZero.toMonoid.{u_4} A (Semiring.toMonoidWithZero.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.19)) inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.9 𝓐 inst._@.Mathlib.Algebra.Module.GradedModule.2370764869._hygCtx._hyg.47)","typeFull":"{ιA : Type u_1} →\n  {ιM : Type u_2} →\n    {A : Type u_4} →\n      {M : Type u_5} →\n        {σ : Type u_6} →\n          {σ' : Type u_7} →\n            [inst : AddMonoid ιA] →\n              [inst_1 : AddAction ιA ιM] →\n                [inst_2 : Semiring A] →\n                  (𝓐 : ιA → σ') →\n                    [inst_3 : SetLike σ' A] →\n                      (𝓜 : ιM → σ) →\n                        [inst_4 : AddMonoid M] →\n                          [inst_5 : DistribMulAction A M] →\n                            [inst_6 : SetLike σ M] →\n                              [inst_7 : SetLike.GradedMonoid 𝓐] →\n                                [SetLike.GradedSMul 𝓐 𝓜] → GradedMonoid.GMulAction (fun i => ↥(𝓐 i)) fun i => ↥(𝓜 i)","typeReadable":"{ιA : Type u_1} →\n  {ιM : Type u_2} →\n    {A : Type u_4} →\n      {M : Type u_5} →\n        {σ : Type u_6} →\n          {σ' : Type u_7} →\n            [inst : AddMonoid ιA] →\n              [inst_1 : AddAction ιA ιM] →\n                [inst_2 : Semiring A] →\n                  (𝓐 : ιA → σ') →\n                    [inst_3 : SetLike σ' A] →\n                      (𝓜 : ιM → σ) →\n                        [inst_4 : AddMonoid M] →\n                          [inst_5 : DistribMulAction A M] →\n                            [inst_6 : SetLike σ M] →\n                              [inst_7 : SetLike.GradedMonoid 𝓐] →\n                                [SetLike.GradedSMul 𝓐 𝓜] → GradedMonoid.GMulAction (fun i => ↥(𝓐 i)) fun i => ↥(𝓜 i)","typeReferences":[["SetLike","instMembership"],["Subtype"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddMonoid"],["SMulZeroClass","toSMul"],["GradedMonoid","GMulAction"],["AddZeroClass","toAddZero"],["AddAction"],["AddAction","toAddSemigroupAction"],["AddSemigroupAction","toVAdd"],["AddMonoid","toAddSemigroup"],["MonoidWithZero","toMonoid"],["SetLike"],["SetLike","GradedSMul"],["AddZero","toZero"],["SetLike","gMonoid"],["DistribMulAction"],["SetLike","GradedMonoid"],["DistribSMul","toSMulZeroClass"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["SetLike","instMembership"],["Subtype"],["SetLike","gmulAction","_proof_2"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddZeroClass","toAddZero"],["AddAction","toAddSemigroupAction"],["SetLike","gmulAction","_proof_1"],["GradedMonoid","GMulAction","mk"],["AddSemigroupAction","toVAdd"],["AddMonoid","toAddSemigroup"],["MonoidWithZero","toMonoid"],["SetLike","toGSMul"],["AddZero","toZero"],["SetLike","gMonoid"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["GradedModule","isModule","_proof_3"],"typeFallback":"forall {ιA : Type.{u_5}} {ιM : Type.{u_1}} {A : Type.{u_4}} {M : Type.{u_2}} {σ : Type.{u_3}} {σ' : Type.{u_6}} [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9 : AddMonoid.{u_5} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.12 : AddAction.{u_5, u_1} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19 : Semiring.{u_4} A] (𝓐 : ιA -> σ') [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29 : SetLike.{u_6, u_4} σ' A] (𝓜 : ιM -> σ) [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.39 : Module.{u_4, u_2} A M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.43 : SetLike.{u_3, u_2} σ M] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.47 : AddSubmonoidClass.{u_6, u_4} σ' A (AddMonoid.toAddZeroClass.{u_4} A (AddMonoidWithOne.toAddMonoid.{u_4} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_4} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_4} A (Semiring.toNonAssocSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19))))) inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.51 : AddSubmonoidClass.{u_3, u_2} σ M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36)) inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.43] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.55 : SetLike.GradedMonoid.{u_5, u_4, u_6} ιA A σ' inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29 (MonoidWithZero.toMonoid.{u_4} A (Semiring.toMonoidWithZero.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19)) inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9 𝓐] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.58 : SetLike.GradedSMul.{u_5, u_1, u_6, u_4, u_3, u_2} ιA ιM σ' A σ M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.43 (SMulZeroClass.toSMul.{u_4, u_2} A M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36)))) (DistribSMul.toSMulZeroClass.{u_4, u_2} A M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36)) (DistribMulAction.toDistribSMul.{u_4, u_2} A M (MonoidWithZero.toMonoid.{u_4} A (Semiring.toMonoidWithZero.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36) (Module.toDistribMulAction.{u_4, u_2} A M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.39)))) (AddSemigroupAction.toVAdd.{u_5, u_1} ιA ιM (AddMonoid.toAddSemigroup.{u_5} ιA inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_5, u_1} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.12)) 𝓐 𝓜] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.62 : DecidableEq.{succ u_5} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.65 : DecidableEq.{succ u_1} ιM] 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inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43) -> Sort.{u}}, (forall [toGdistribMulAction : DirectSum.GdistribMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34] (add_smul : forall {i : ιA} {j : ιB} (a : A i) (a' : A i) (b : M j), Eq.{succ u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 (GradedMonoid.GMulAction.toGSMul.{u_1, u_2, u_3, u_4} ιA ιB A M 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(HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (instHAdd.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (AddSemigroup.toAdd.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (AddMonoid.toAddSemigroup.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j))))) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 (GradedMonoid.GMulAction.toGSMul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 (DirectSum.GdistribMulAction.toGMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 toGdistribMulAction)) i j a b) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 (GradedMonoid.GMulAction.toGSMul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 (DirectSum.GdistribMulAction.toGMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 toGdistribMulAction)) i j a' b))) (zero_smul : forall {i : ιA} {j : ιB} (b : M j), Eq.{succ u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 (GradedMonoid.GMulAction.toGSMul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 (DirectSum.GdistribMulAction.toGMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 toGdistribMulAction)) i j (OfNat.ofNat.{u_3} (A i) 0 (Zero.toOfNat0.{u_3} (A i) (AddZero.toZero.{u_3} (A i) (AddZeroClass.toAddZero.{u_3} (A i) (AddMonoid.toAddZeroClass.{u_3} (A i) (inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.25 i)))))) b) (OfNat.ofNat.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) 0 (Zero.toOfNat0.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (AddZero.toZero.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (AddZeroClass.toAddZero.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (AddMonoid.toAddZeroClass.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)))))))), motive (DirectSum.Gmodule.mk.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 toGdistribMulAction add_smul zero_smul)) -> (forall (t : DirectSum.Gmodule.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43), motive t)","typeFull":"{ιA : Type u_1} →\n  {ιB : Type u_2} →\n    {A : ιA → Type u_3} →\n      {M : ιB → Type u_4} →\n        [inst : AddMonoid ιA] →\n          [inst_1 : VAdd ιA ιB] →\n            [inst_2 : (i : ιA) → AddMonoid (A i)] →\n              [inst_3 : (i : ιB) → AddMonoid (M i)] →\n                [inst_4 : GradedMonoid.GMonoid A] →\n                  {motive : DirectSum.Gmodule A M → Sort u} →\n                    ([toGdistribMulAction : DirectSum.GdistribMulAction A M] →\n                        (add_smul :\n                            ∀ {i : ιA} {j : ιB} (a a' : A i) (b : M j),\n                              GradedMonoid.GSMul.smul (a + a') b =\n                                GradedMonoid.GSMul.smul a b + GradedMonoid.GSMul.smul a' b) →\n                          (zero_smul : ∀ {i : ιA} {j : ιB} (b : M j), GradedMonoid.GSMul.smul 0 b = 0) →\n                            motive\n                              { toGdistribMulAction := toGdistribMulAction, add_smul := add_smul,\n                                zero_smul := zero_smul }) →\n                      (t : DirectSum.Gmodule A M) → motive t","typeReadable":"{ιA : Type u_1} →\n  {ιB : Type u_2} →\n    {A : ιA → Type u_3} →\n      {M : ιB → Type u_4} →\n        [inst : AddMonoid ιA] →\n          [inst_1 : VAdd ιA ιB] →\n            [inst_2 : (i : ιA) → AddMonoid (A i)] →\n              [inst_3 : (i : ιB) → AddMonoid (M i)] →\n                [inst_4 : GradedMonoid.GMonoid A] →\n                  {motive : DirectSum.Gmodule A M → Sort u} →\n                    ([toGdistribMulAction : DirectSum.GdistribMulAction A M] →\n                        (add_smul :\n                            ∀ {i : ιA} {j : ιB} (a a' : A i) (b : M j),\n                              GradedMonoid.GSMul.smul (a + a') b =\n                                GradedMonoid.GSMul.smul a b + GradedMonoid.GSMul.smul a' b) →\n                          (zero_smul : ∀ {i : ιA} {j : ιB} (b : M j), GradedMonoid.GSMul.smul 0 b = 0) →\n                            motive\n                              { toGdistribMulAction := toGdistribMulAction, add_smul := add_smul,\n                                zero_smul := zero_smul }) →\n                      (t : DirectSum.Gmodule A M) → motive t","typeReferences":[["VAdd"],["DirectSum","GdistribMulAction"],["instHVAdd"],["instHAdd"],["AddMonoid"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["HVAdd","hVAdd"],["DirectSum","Gmodule","mk"],["HAdd","hAdd"],["AddMonoid","toAddSemigroup"],["GradedMonoid","GMulAction","toGSMul"],["Zero","toOfNat0"],["DirectSum","Gmodule"],["DirectSum","GdistribMulAction","toGMulAction"],["Eq"],["AddZero","toZero"],["GradedMonoid","GMonoid"],["AddMonoid","toAddZeroClass"],["AddSemigroup","toAdd"],["GradedMonoid","GSMul","smul"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["SetLike","gmodule","_proof_6"],"typeFallback":"forall {ιA : Type.{u_3}} {ιM : Type.{u_4}} {A : Type.{u_5}} {M : Type.{u_1}} {σ : Type.{u_2}} {σ' : Type.{u_6}} [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 : AddMonoid.{u_3} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12 : AddAction.{u_3, u_4} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19 : Semiring.{u_5} A] (𝓐 : ιA -> σ') [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29 : SetLike.{u_6, u_5} σ' A] (𝓜 : ιM -> σ) [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.39 : Module.{u_5, u_1} A M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43 : SetLike.{u_2, u_1} σ M] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.47 : AddSubmonoidClass.{u_6, u_5} σ' A (AddMonoid.toAddZeroClass.{u_5} A (AddMonoidWithOne.toAddMonoid.{u_5} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_5} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_5} A (Semiring.toNonAssocSemiring.{u_5} A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19))))) inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.58 : SetLike.GradedSMul.{u_3, u_4, u_6, u_5, u_2, u_1} ιA ιM σ' A σ M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43 (SMulZeroClass.toSMul.{u_5, u_1} A M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36)))) (DistribSMul.toSMulZeroClass.{u_5, u_1} A M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36)) (DistribMulAction.toDistribSMul.{u_5, u_1} A M (MonoidWithZero.toMonoid.{u_5} A (Semiring.toMonoidWithZero.{u_5} A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36) (Module.toDistribMulAction.{u_5, u_1} A M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.39)))) (AddSemigroupAction.toVAdd.{u_3, u_4} ιA ιM (AddMonoid.toAddSemigroup.{u_3} ιA inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_3, u_4} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.12)) 𝓐 𝓜] {i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.114 : ιA} {j._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.115 : ιM} (_a : Subtype.{succ u_5} A (fun (x : A) => Membership.mem.{u_5, u_6} A σ' (SetLike.instMembership.{u_6, u_5} σ' A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29) (𝓐 i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.114) x)) (_a' : Subtype.{succ u_5} A (fun (x : A) => Membership.mem.{u_5, u_6} A σ' (SetLike.instMembership.{u_6, u_5} σ' A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29) (𝓐 i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.114) x)) (_b : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M σ (SetLike.instMembership.{u_2, u_1} σ M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.43) (𝓜 j._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.115) x)), Membership.mem.{u_1, u_2} M σ (SetLike.instMembership.{u_2, u_1} σ 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(AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36)) (DistribMulAction.toDistribSMul.{u_5, u_1} A M (MonoidWithZero.toMonoid.{u_5} A (Semiring.toMonoidWithZero.{u_5} A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36) (Module.toDistribMulAction.{u_5, u_1} A M inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.39))))) (Subtype.val.{succ u_5} A (fun (x : A) => Membership.mem.{u_5, u_6} A σ' (SetLike.instMembership.{u_6, u_5} σ' A inst._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.29) (𝓐 i._@.Mathlib.Algebra.Module.GradedModule.389950551._hygCtx._hyg.114) x) (HAdd.hAdd.{u_5, u_5, u_5} (Subtype.{succ u_5} A (fun (x : A) => 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Type.{u_4}) [inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 : AddMonoid.{u_1} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 : VAdd.{u_1, u_2} ιA ιB] [inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.17 : forall (i : ιA), AddCommMonoid.{u_3} (A i)] [inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.28 : forall (i : ιB), AddCommMonoid.{u_4} (M i)] [inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.37 : GradedMonoid.GMonoid.{u_1, u_3} ιA A inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.40 : DirectSum.Gmodule.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 (fun (i : ιA) => AddCommMonoid.toAddMonoid.{u_3} (A i) 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AddMonoid ιA] →\n          [inst_1 : VAdd ιA ιB] →\n            [inst_2 : (i : ιA) → AddCommMonoid (A i)] →\n              [inst_3 : (i : ιB) → AddCommMonoid (M i)] →\n                [inst_4 : GradedMonoid.GMonoid A] →\n                  [DirectSum.Gmodule A M] → {i : ιA} → {j : ιB} → A i →+ M j →+ M (i +ᵥ j)","typeReadable":"{ιA : Type u_1} →\n  {ιB : Type u_2} →\n    (A : ιA → Type u_3) →\n      (M : ιB → Type u_4) →\n        [inst : AddMonoid ιA] →\n          [inst_1 : VAdd ιA ιB] →\n            [inst_2 : (i : ιA) → AddCommMonoid (A i)] →\n              [inst_3 : (i : ιB) → AddCommMonoid (M i)] →\n                [inst_4 : GradedMonoid.GMonoid A] →\n                  [DirectSum.Gmodule A M] → {i : ιA} → {j : ιB} → A i ���+ M j →+ M (i +ᵥ 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{A : ιA -> Type.{u_3}} {M : ιB -> Type.{u_4}} [inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 : AddMonoid.{u_1} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 : VAdd.{u_1, u_2} ιA ιB] [inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.25 : forall (i : ιA), AddMonoid.{u_3} (A i)] [inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 : forall (i : ιB), AddMonoid.{u_4} (M i)] [inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 : GradedMonoid.GMonoid.{u_1, u_3} ιA A inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18] {motive : (DirectSum.Gmodule.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43) -> Sort.{u}} (t : DirectSum.Gmodule.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43), (forall [toGdistribMulAction : DirectSum.GdistribMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34] (add_smul : forall {i : ιA} {j : ιB} (a : A i) (a' : A i) (b : M j), Eq.{succ u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB 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inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 toGdistribMulAction)) i j a b) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 (GradedMonoid.GMulAction.toGSMul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 (DirectSum.GdistribMulAction.toGMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 toGdistribMulAction)) i j a' b))) (zero_smul : forall {i : ιA} {j : ιB} (b : M j), Eq.{succ u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 (GradedMonoid.GMulAction.toGSMul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 (DirectSum.GdistribMulAction.toGMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 toGdistribMulAction)) i j (OfNat.ofNat.{u_3} (A i) 0 (Zero.toOfNat0.{u_3} (A i) (AddZero.toZero.{u_3} (A i) (AddZeroClass.toAddZero.{u_3} (A i) (AddMonoid.toAddZeroClass.{u_3} (A i) (inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.25 i)))))) b) (OfNat.ofNat.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) 0 (Zero.toOfNat0.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (AddZero.toZero.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (AddZeroClass.toAddZero.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (AddMonoid.toAddZeroClass.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)) (inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21) i j)))))))), motive (DirectSum.Gmodule.mk.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.GradedModule.1371148586._hygCtx._hyg.43 toGdistribMulAction add_smul zero_smul)) -> (motive t)","typeFull":"{ιA : Type u_1} →\n  {ιB : Type u_2} →\n    {A : ιA → Type u_3} →\n      {M : ιB → Type u_4} →\n        [inst : AddMonoid ιA] →\n          [inst_1 : VAdd ιA ιB] →\n            [inst_2 : (i : ιA) → AddMonoid (A i)] →\n              [inst_3 : (i : ιB) → AddMonoid (M i)] →\n                [inst_4 : GradedMonoid.GMonoid A] →\n                  {motive : DirectSum.Gmodule A M → Sort u} →\n                    (t : DirectSum.Gmodule A M) →\n                      ([toGdistribMulAction : DirectSum.GdistribMulAction A M] →\n                          (add_smul :\n                              ∀ {i : ιA} {j : ιB} (a a' : A i) (b : M j),\n                                GradedMonoid.GSMul.smul (a + a') b =\n                                  GradedMonoid.GSMul.smul a b + GradedMonoid.GSMul.smul a' b) →\n                            (zero_smul : ∀ {i : ιA} {j : ιB} (b : M j), GradedMonoid.GSMul.smul 0 b = 0) →\n                              motive\n                                { toGdistribMulAction := toGdistribMulAction, add_smul := add_smul,\n                                  zero_smul := zero_smul }) →\n                        motive t","typeReadable":"{ιA : Type u_1} →\n  {ιB : Type u_2} →\n    {A : ιA → Type u_3} →\n      {M : ιB → Type u_4} →\n        [inst : AddMonoid ιA] →\n          [inst_1 : VAdd ιA ιB] →\n            [inst_2 : (i : ιA) → AddMonoid (A i)] →\n              [inst_3 : (i : ιB) → AddMonoid (M i)] →\n                [inst_4 : GradedMonoid.GMonoid A] →\n                  {motive : DirectSum.Gmodule A M → Sort u} →\n                    (t : DirectSum.Gmodule A M) →\n                      ([toGdistribMulAction : DirectSum.GdistribMulAction A M] →\n                          (add_smul :\n                              ∀ {i : ιA} {j : ιB} (a a' : A i) (b : M j),\n                                GradedMonoid.GSMul.smul (a + a') b =\n                                  GradedMonoid.GSMul.smul a b + GradedMonoid.GSMul.smul a' b) →\n                            (zero_smul : ∀ {i : ιA} {j : ιB} (b : M j), GradedMonoid.GSMul.smul 0 b = 0) →\n                              motive\n                                { toGdistribMulAction := toGdistribMulAction, add_smul := add_smul,\n                                  zero_smul := zero_smul }) →\n                        motive t","typeReferences":[["VAdd"],["DirectSum","GdistribMulAction"],["instHVAdd"],["instHAdd"],["AddMonoid"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["HVAdd","hVAdd"],["DirectSum","Gmodule","mk"],["HAdd","hAdd"],["AddMonoid","toAddSemigroup"],["GradedMonoid","GMulAction","toGSMul"],["Zero","toOfNat0"],["DirectSum","Gmodule"],["DirectSum","GdistribMulAction","toGMulAction"],["Eq"],["AddZero","toZero"],["GradedMonoid","GMonoid"],["AddMonoid","toAddZeroClass"],["AddSemigroup","toAdd"],["GradedMonoid","GSMul","smul"]],"valueReferences":[["DirectSum","Gmodule","rec"]]},{"isProp":false,"kind":"definition","name":["DirectSum","GdistribMulAction","casesOn"],"typeFallback":"forall {ιA : Type.{u_1}} {ιB : Type.{u_2}} {A : ιA -> Type.{u_3}} {M : ιB -> Type.{u_4}} [inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 : AddMonoid.{u_1} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 : VAdd.{u_1, u_2} ιA ιB] [inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 : GradedMonoid.GMonoid.{u_1, u_3} ιA A inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18] [inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 : forall (i : ιB), AddMonoid.{u_4} (M i)] {motive : (DirectSum.GdistribMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28) -> Sort.{u}} (t : DirectSum.GdistribMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28), (forall [toGMulAction : GradedMonoid.GMulAction.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25] (smul_add : forall {i : ιA} {j : ιB} (a : A i) (b : M j) (c : M j), Eq.{succ u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 (GradedMonoid.GMulAction.toGSMul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 toGMulAction) i j a (HAdd.hAdd.{u_4, u_4, u_4} (M j) (M j) (M j) (instHAdd.{u_4} (M j) (AddSemigroup.toAdd.{u_4} (M j) (AddMonoid.toAddSemigroup.{u_4} (M j) (inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 j)))) b c)) (HAdd.hAdd.{u_4, u_4, u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (instHAdd.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (AddSemigroup.toAdd.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (AddMonoid.toAddSemigroup.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j))))) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 (GradedMonoid.GMulAction.toGSMul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 toGMulAction) i j a b) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 (GradedMonoid.GMulAction.toGSMul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 toGMulAction) i j a c))) (smul_zero : forall {i : ιA} {j : ιB} (a : A i), Eq.{succ u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (GradedMonoid.GSMul.smul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 (GradedMonoid.GMulAction.toGSMul.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 toGMulAction) i j a (OfNat.ofNat.{u_4} (M j) 0 (Zero.toOfNat0.{u_4} (M j) (AddZero.toZero.{u_4} (M j) (AddZeroClass.toAddZero.{u_4} (M j) (AddMonoid.toAddZeroClass.{u_4} (M j) (inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 j))))))) (OfNat.ofNat.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) 0 (Zero.toOfNat0.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (AddZero.toZero.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (AddZeroClass.toAddZero.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (AddMonoid.toAddZeroClass.{u_4} (M (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)) (inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 (HVAdd.hVAdd.{u_1, u_2, u_2} ιA ιB ιB (instHVAdd.{u_1, u_2} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21) i j)))))))), motive (DirectSum.GdistribMulAction.mk.{u_1, u_2, u_3, u_4} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.GradedModule.1701233059._hygCtx._hyg.28 toGMulAction smul_add smul_zero)) -> (motive t)","typeFull":"{ιA : Type u_1} →\n  {ιB : Type u_2} →\n    {A : ιA → Type u_3} →\n      {M : ιB → Type u_4} →\n        [inst : AddMonoid ιA] →\n          [inst_1 : VAdd ιA ιB] →\n            [inst_2 : GradedMonoid.GMonoid A] →\n              [inst_3 : (i : ιB) → AddMonoid (M i)] →\n                {motive : DirectSum.GdistribMulAction A M → Sort u} →\n                  (t : DirectSum.GdistribMulAction A M) →\n                    ([toGMulAction : GradedMonoid.GMulAction A M] →\n                        (smul_add :\n                            ∀ {i : ιA} {j : ιB} (a : A i) (b c : M j),\n                              GradedMonoid.GSMul.smul a (b + c) =\n                                GradedMonoid.GSMul.smul a b + GradedMonoid.GSMul.smul a c) →\n                          (smul_zero : ∀ {i : ιA} {j : ιB} (a : A i), GradedMonoid.GSMul.smul a 0 = 0) →\n                            motive { toGMulAction := toGMulAction, smul_add := smul_add, smul_zero := smul_zero }) →\n                      motive t","typeReadable":"{ιA : Type u_1} →\n  {ιB : Type u_2} →\n    {A : ιA → Type u_3} →\n      {M : ιB → Type u_4} →\n        [inst : AddMonoid ιA] →\n          [inst_1 : VAdd ιA ιB] →\n            [inst_2 : GradedMonoid.GMonoid A] →\n              [inst_3 : (i : ιB) → AddMonoid (M i)] →\n                {motive : DirectSum.GdistribMulAction A M → Sort u} →\n                  (t : DirectSum.GdistribMulAction A M) →\n                    ([toGMulAction : GradedMonoid.GMulAction A M] →\n                        (smul_add :\n                            ∀ {i : ιA} {j : ιB} (a : A i) (b c : M j),\n                              GradedMonoid.GSMul.smul a (b + c) =\n                                GradedMonoid.GSMul.smul a b + GradedMonoid.GSMul.smul a c) →\n                          (smul_zero : ∀ {i : ιA} {j : ιB} (a : A i), GradedMonoid.GSMul.smul a 0 = 0) →\n                            motive { toGMulAction := toGMulAction, smul_add := smul_add, smul_zero := smul_zero }) →\n                      motive t","typeReferences":[["VAdd"],["DirectSum","GdistribMulAction"],["instHVAdd"],["instHAdd"],["AddMonoid"],["DirectSum","GdistribMulAction","mk"],["AddZeroClass","toAddZero"],["GradedMonoid","GMulAction"],["OfNat","ofNat"],["HVAdd","hVAdd"],["HAdd","hAdd"],["AddMonoid","toAddSemigroup"],["GradedMonoid","GMulAction","toGSMul"],["Zero","toOfNat0"],["Eq"],["AddZero","toZero"],["GradedMonoid","GMonoid"],["AddMonoid","toAddZeroClass"],["GradedMonoid","GSMul","smul"],["AddSemigroup","toAdd"]],"valueReferences":[["DirectSum","GdistribMulAction","rec"]]},{"isProp":true,"kind":"theorem","name":["GradedModule","isModule","_proof_4"],"typeFallback":"forall {ιA : Type.{u_5}} {ιM : Type.{u_1}} {A : Type.{u_4}} {M : Type.{u_2}} {σ : Type.{u_3}} {σ' : Type.{u_6}} [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9 : AddMonoid.{u_5} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.12 : AddAction.{u_5, u_1} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19 : Semiring.{u_4} A] (𝓐 : ιA -> σ') [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29 : SetLike.{u_6, u_4} σ' A] (𝓜 : ιM -> σ) [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.39 : Module.{u_4, u_2} A M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.43 : SetLike.{u_3, u_2} σ M] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.47 : AddSubmonoidClass.{u_6, u_4} σ' A (AddMonoid.toAddZeroClass.{u_4} A (AddMonoidWithOne.toAddMonoid.{u_4} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_4} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_4} A 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inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.39)))) (AddSemigroupAction.toVAdd.{u_5, u_1} ιA ιM (AddMonoid.toAddSemigroup.{u_5} ιA inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_5, u_1} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.12)) 𝓐 𝓜] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.62 : DecidableEq.{succ u_5} ιA] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.65 : DecidableEq.{succ u_1} ιM] [inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.68 : GradedRing.{u_5, u_4, u_6} ιA A σ' inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.62 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.47 𝓐] (a : A) (x : DirectSum.{u_1, u_2} ιM (fun (i : ιM) => Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_3} M σ (SetLike.instMembership.{u_3, u_2} σ M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.43) (𝓜 i) x)) (fun (i : ιM) => AddSubmonoidClass.toAddCommMonoid.{u_3, u_2} M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36 σ inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.51 (𝓜 i))) (y : DirectSum.{u_1, u_2} ιM (fun (i : ιM) => Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_3} M σ (SetLike.instMembership.{u_3, u_2} σ M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.43) (𝓜 i) x)) (fun (i : ιM) => AddSubmonoidClass.toAddCommMonoid.{u_3, u_2} M 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inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.43) (𝓜 i) x)) inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9 (AddSemigroupAction.toVAdd.{u_5, u_1} ιA ιM (AddMonoid.toAddSemigroup.{u_5} ιA inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9) (AddAction.toAddSemigroupAction.{u_5, u_1} ιA ιM inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.12)) (fun (i : ιA) => AddSubmonoidClass.toAddCommMonoid.{u_6, u_4} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} A (Semiring.toNonAssocSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19))) σ' inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.47 (𝓐 i)) (fun (i : ιM) => AddSubmonoidClass.toAddCommMonoid.{u_3, u_2} M inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36 σ inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.51 (𝓜 i)) inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.62 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.65 (SetLike.gsemiring.{u_5, u_6, u_4} ιA σ' A inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.47 𝓐 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.55) (SetLike.gmodule.{u_5, u_1, u_4, u_2, u_3, u_6} ιA ιM A M σ σ' inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19 𝓐 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29 𝓜 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.47 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.51 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.55 inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.58)) inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.19 (RingEquiv.toRingHom.{u_4, max u_5 u_4} A (DirectSum.{u_5, u_4} ιA (fun (i : ιA) => Subtype.{succ u_4} A (fun (x : A) => Membership.mem.{u_4, u_6} A σ' (SetLike.instMembership.{u_6, u_4} σ' A inst._@.Mathlib.Algebra.Module.GradedModule.86298187._hygCtx._hyg.29) (𝓐 i) x)) (fun (i : ιA) => 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j)))))) (fun (b : M j) => GradedMonoid.GSMul.smul.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 (GradedMonoid.GMulAction.toGSMul.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.37 (DirectSum.GdistribMulAction.toGMulAction.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.37 (fun (i : ιB) => AddCommMonoid.toAddMonoid.{u_1} (M i) (inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.28 i)) (DirectSum.Gmodule.toGdistribMulAction.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 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j)) (inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.28 (HVAdd.hVAdd.{u_2, u_3, u_3} ιA ιB ιB (instHVAdd.{u_2, u_3} ιA ιB inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13) i j)))))) (fun (b : M j) => GradedMonoid.GSMul.smul.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 (GradedMonoid.GMulAction.toGSMul.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.37 (DirectSum.GdistribMulAction.toGMulAction.{u_2, u_3, u_4, u_1} ιA ιB A M inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.GradedModule.3141205250._hygCtx._hyg.37 (fun (i : ιB) => AddCommMonoid.toAddMonoid.{u_1} (M i) 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DirectSum.Gmodule A M] {i : ιA} {j : ιB} (_a₁ _a₂ : A i),\n  { toFun := fun b => GradedMonoid.GSMul.smul (_a₁ + _a₂) b, map_zero' := ⋯, map_add' := ⋯ } =\n    { toFun := fun b => GradedMonoid.GSMul.smul _a₁ b, map_zero' := ⋯, map_add' := ⋯ } +\n      { toFun := fun b => GradedMonoid.GSMul.smul _a₂ b, map_zero' := ⋯, map_add' := ⋯ }","typeReadable":"∀ {ιA : Type u_2} {ιB : Type u_3} (A : ιA → Type u_4) (M : ιB → Type u_1) [inst : AddMonoid ιA] [inst_1 : VAdd ιA ιB]\n  [inst_2 : (i : ιA) → AddCommMonoid (A i)] [inst_3 : (i : ιB) → AddCommMonoid (M i)] [inst_4 : GradedMonoid.GMonoid A]\n  [inst_5 : DirectSum.Gmodule A M] {i : ιA} {j : ιB} (_a₁ _a₂ : A i),\n  { toFun := fun b => GradedMonoid.GSMul.smul (_a₁ + _a₂) b, map_zero' := ⋯, map_add' := ⋯ } =\n    { toFun := fun b => GradedMonoid.GSMul.smul _a₁ b, map_zero' := ⋯, map_add' := ⋯ } +\n      { toFun := fun b => GradedMonoid.GSMul.smul _a₂ b, map_zero' := ⋯, map_add' := ⋯ 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