diff --git "a/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.Submodule.Defs.sym.json" "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.Submodule.Defs.sym.json"
new file mode 100644--- /dev/null
+++ "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.Submodule.Defs.sym.json"
@@ -0,0 +1 @@
+[{"isProp":false,"kind":"definition","name":["Submodule","zero"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.24028081._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.24028081._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.24028081._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.24028081._hygCtx._hyg.10} (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.24028081._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.24028081._hygCtx._hyg.10 module_M), Zero.{v} (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.24028081._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.24028081._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.24028081._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.24028081._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.24028081._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.24028081._hygCtx._hyg.10 module_M)) p x))","typeFull":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] → [inst_1 : AddCommMonoid M] → {module_M : Module R M} → (p : Submodule R M) → Zero ↥p","typeReadable":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] → [inst_1 : AddCommMonoid M] → {module_M : Module R M} → (p : Submodule R M) → Zero ↥p","typeReferences":[["AddCommMonoid"],["SetLike","instMembership"],["Subtype"],["Submodule","setLike"],["Module"],["Membership","mem"],["Zero"],["Submodule"],["Semiring"]],"valueReferences":[["Subtype"],["SetLike","instMembership"],["Submodule","zero","_proof_1"],["Membership","mem"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Submodule"],["Zero","mk"],["Submodule","setLike"],["Zero","toOfNat0"],["Subtype","mk"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","nonempty"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.148012321._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.148012321._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.148012321._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.148012321._hygCtx._hyg.10} (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.148012321._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.148012321._hygCtx._hyg.10 module_M), Set.Nonempty.{v} M (SetLike.coe.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.148012321._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.148012321._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.148012321._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.148012321._hygCtx._hyg.10 module_M) p)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M),\n  (↑p).Nonempty","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M),\n  (↑p).Nonempty","typeReferences":[["Set","Nonempty"],["AddCommMonoid"],["SetLike","coe"],["Submodule","setLike"],["Module"],["Submodule"],["Semiring"]],"valueReferences":[["Set"],["Membership","mem"],["Exists","intro"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Submodule","zero_mem"],["Set","instMembership"],["Submodule"],["SetLike","coe"],["Submodule","setLike"],["Zero","toOfNat0"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","module","_proof_1"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.7 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.10 : AddCommMonoid.{u_2} M] {module_M : Module.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.10}, IsScalarTower.{u_1, u_1, u_2} R R M (SemigroupAction.toSMul.{u_1, u_1} R R (Monoid.toSemigroup.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.7)) (Monoid.toMulAction.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.7))))) (SemigroupAction.toSMul.{u_1, u_2} R M (Monoid.toSemigroup.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.7)) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.10) (Module.toDistribMulAction.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.10 module_M)))) (SemigroupAction.toSMul.{u_1, u_2} R M (Monoid.toSemigroup.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.7)) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.10) (Module.toDistribMulAction.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.10 module_M))))","typeFull":"∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M},\n  IsScalarTower R R M","typeReadable":"∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M},\n  IsScalarTower R R M","typeReferences":[["Module"],["IsScalarTower"],["SemigroupAction","toSMul"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"],["Monoid","toMulAction"],["AddCommMonoid"],["Module","toDistribMulAction"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Monoid","toSemigroup"],["MulAction","toSemigroupAction"],["Semiring"]],"valueReferences":[["Module","toDistribMulAction"],["IsScalarTower","left"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"]]},{"isProp":true,"kind":"theorem","name":["Submodule","sub_mem_iff_right"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1364694498._hygCtx._hyg.7 : Ring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1364694498._hygCtx._hyg.10 : AddCommGroup.{v} M] {module_M : Module.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1364694498._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1364694498._hygCtx._hyg.10)} (p : Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1364694498._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1364694498._hygCtx._hyg.10) module_M) {x : M} {y : M}, (Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1364694498._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1364694498._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1364694498._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1364694498._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1364694498._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1364694498._hygCtx._hyg.10) module_M)) p x) -> (Iff (Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1364694498._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1364694498._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1364694498._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1364694498._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1364694498._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1364694498._hygCtx._hyg.10) module_M)) p (HSub.hSub.{v, v, v} M M M (instHSub.{v} M (SubNegMonoid.toSub.{v} M (AddGroup.toSubNegMonoid.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1364694498._hygCtx._hyg.10)))) x y)) (Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1364694498._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1364694498._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1364694498._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1364694498._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1364694498._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1364694498._hygCtx._hyg.10) module_M)) p y))","typeFull":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M} (p : Submodule R M)\n  {x y : M}, x ∈ p → (x - y ∈ p ↔ y ∈ p)","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M} (p : Submodule R M)\n  {x y : M}, x ∈ p → (x - y ∈ p ↔ y ∈ p)","typeReferences":[["SetLike","instMembership"],["Module"],["AddCommGroup","toAddGroup"],["Membership","mem"],["AddCommGroup"],["Ring","toSemiring"],["Submodule"],["Submodule","setLike"],["Iff"],["SubNegMonoid","toSub"],["AddCommGroup","toAddCommMonoid"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["instHSub"],["Ring"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["AddCommGroup","toAddGroup"],["Membership","mem"],["SubtractionCommMonoid","toSubtractionMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["Submodule"],["congrArg"],["sub_eq_add_neg"],["SubNegMonoid","toSub"],["HSub","hSub"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["AddGroup","toSubNegMonoid"],["Eq"],["propext"],["SetLike","instMembership"],["instHAdd"],["Neg","neg"],["SubNegMonoid","toNeg"],["Iff","rfl"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["HAdd","hAdd"],["NegZeroClass","toNeg"],["AddCommGroup","toDivisionAddCommMonoid"],["Submodule","add_mem_iff_right"],["AddCommMonoid","toAddCommSemigroup"],["SubNegMonoid","toAddMonoid"],["Submodule","neg_mem_iff"],["Submodule","setLike"],["Iff"],["AddCommGroup","toAddCommMonoid"],["id"],["Eq","mpr"],["instHSub"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","isScalarTower"],"typeFallback":"forall {S : Type.{u'}} {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.10} (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.10 module_M) [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.25 : SMul.{u', u} S R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.29 : SMul.{u', v} S M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.33 : IsScalarTower.{u', u, v} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.25 (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.10 module_M)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.29], IsScalarTower.{u', u, v} S R (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.10 module_M)) p x)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.25 (Submodule.smul.{u, u, v} R R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.10 module_M p (SMulZeroClass.toSMul.{u, u} R R (AddZero.toZero.{u} R (AddZeroClass.toAddZero.{u} R (AddMonoid.toAddZeroClass.{u} R (AddMonoidWithOne.toAddMonoid.{u} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u} R (NonAssocSemiring.toAddCommMonoidWithOne.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.7))))))) (DistribSMul.toSMulZeroClass.{u, u} R R (AddMonoid.toAddZeroClass.{u} R (AddMonoidWithOne.toAddMonoid.{u} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u} R (NonAssocSemiring.toAddCommMonoidWithOne.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.7))))) (instDistribSMul.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.7))))) (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.10 module_M)))) (IsScalarTower.left.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.7)) (DistribMulAction.toMulAction.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.10 module_M)))) (Submodule.smul.{u', u, v} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1890278916._hygCtx._hyg.33)","typeFull":"∀ {S : Type u'} {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) [inst_2 : SMul S R] [inst_3 : SMul S M] [inst_4 : IsScalarTower S R M], IsScalarTower S R ↥p","typeReadable":"∀ {S : Type u'} {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) [inst_2 : SMul S R] [inst_3 : SMul S M] [inst_4 : IsScalarTower S R M], IsScalarTower S R ↥p","typeReferences":[["Subtype"],["IsScalarTower"],["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["instDistribSMul"],["AddMonoidWithOne","toAddMonoid"],["Submodule"],["SMul"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["IsScalarTower","left"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["Submodule","setLike"],["Submodule","smul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Submodule","toSubMulAction"],["Module","toDistribMulAction"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"],["SubMulAction","isScalarTower"]]},{"isProp":true,"kind":"theorem","name":["Submodule","coe_toAddSubgroup"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1958704647._hygCtx._hyg.7 : Ring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1958704647._hygCtx._hyg.10 : AddCommGroup.{v} M] {module_M : Module.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1958704647._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1958704647._hygCtx._hyg.10)} (p : Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1958704647._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1958704647._hygCtx._hyg.10) module_M), Eq.{succ v} (Set.{v} M) (SetLike.coe.{v, v} (AddSubgroup.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1958704647._hygCtx._hyg.10)) M (AddSubgroup.instSetLike.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1958704647._hygCtx._hyg.10)) (Submodule.toAddSubgroup.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1958704647._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1958704647._hygCtx._hyg.10 module_M p)) (SetLike.coe.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1958704647._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1958704647._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1958704647._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1958704647._hygCtx._hyg.10) module_M) p)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M} (p : Submodule R M),\n  ↑p.toAddSubgroup = ↑p","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M} (p : Submodule R M),\n  ↑p.toAddSubgroup = ↑p","typeReferences":[["Module"],["Set"],["AddCommGroup","toAddGroup"],["AddSubgroup","instSetLike"],["AddCommGroup"],["Submodule","toAddSubgroup"],["Ring","toSemiring"],["Submodule"],["AddSubgroup"],["SetLike","coe"],["Submodule","setLike"],["AddCommGroup","toAddCommMonoid"],["Eq"],["Ring"]],"valueReferences":[["rfl"],["AddSubgroup"],["SetLike","coe"],["Set"],["AddCommGroup","toAddGroup"],["AddSubgroup","instSetLike"],["Submodule","toAddSubgroup"]]},{"isProp":true,"kind":"theorem","name":["Submodule","coe_eq_zero"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10} {p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M} {x : Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M)) p x)}, Iff (Eq.{succ v} M (Subtype.val.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M)) p x) x) (OfNat.ofNat.{v} M 0 (Zero.toOfNat0.{v} M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10))))))) (Eq.{succ v} (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M)) p x)) x (OfNat.ofNat.{v} (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M)) p x)) 0 (Zero.toOfNat0.{v} (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M)) p x)) (Submodule.zero.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M p))))","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} {p : Submodule R M}\n  {x : ↥p}, ↑x = 0 ↔ x = 0","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} {p : Submodule R M}\n  {x : ↥p}, ↑x = 0 ↔ x = 0","typeReferences":[["Subtype"],["SetLike","instMembership"],["Module"],["Membership","mem"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Subtype","val"],["OfNat","ofNat"],["Submodule"],["AddCommMonoid"],["Submodule","setLike"],["Iff"],["Zero","toOfNat0"],["Submodule","zero"],["Eq"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["SetLike","instMembership"],["Subtype"],["Submodule","setLike"],["SetLike","coe_eq_coe"],["Membership","mem"],["Zero","toOfNat0"],["Submodule","zero"],["OfNat","ofNat"],["Submodule"]]},{"isProp":false,"kind":"definition","name":["Submodule","copy"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10] (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.13) (s : Set.{v} M), (Eq.{succ v} (Set.{v} M) s (SetLike.coe.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.13) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.13) p)) -> (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.13)","typeFull":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] →\n      [inst_1 : AddCommMonoid M] → [inst_2 : Module R M] → (p : Submodule R M) → (s : Set M) → s = ↑p → Submodule R M","typeReadable":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] →\n      [inst_1 : AddCommMonoid M] → [inst_2 : Module R M] → (p : Submodule R M) → (s : Set M) → s = ↑p → Submodule R M","typeReferences":[["AddCommMonoid"],["SetLike","coe"],["Submodule","setLike"],["Set"],["Module"],["Eq"],["Submodule"],["Semiring"]],"valueReferences":[["AddSubsemigroup","mk"],["Submodule","mk"],["Submodule","copy","_proof_2"],["AddSubmonoid","mk"],["Submodule","copy","_proof_1"],["Submodule","copy","_proof_3"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","module'","_proof_1"],"typeFallback":"forall {S : Type.{u_3}} {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] {module_M : Module.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10} (p : Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 : Semiring.{u_3} S] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 : SMul.{u_3, u_2} S R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 : Module.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36 : IsScalarTower.{u_3, u_2, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (SMulZeroClass.toSMul.{u_2, u_1} R 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)))) (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32))))] (a : S), Eq.{succ u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (SemigroupAction.toSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))) (MulAction.toSemigroupAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (have this : MulAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) := SubMulAction.mulAction'.{u_2, u_3, u_1} S R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7)) (DistribMulAction.toMulAction.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (DistribMulAction.toMulAction.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36 (Submodule.toSubMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p); this)))) a (OfNat.ofNat.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) 0 (Zero.toOfNat0.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddZero.toZero.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddZeroClass.toAddZero.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddMonoid.toAddZeroClass.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Submodule.addCommMonoid.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p)))))))) (OfNat.ofNat.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) 0 (Zero.toOfNat0.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddZero.toZero.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddZeroClass.toAddZero.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddMonoid.toAddZeroClass.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Submodule.addCommMonoid.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p)))))))","typeFull":"∀ {S : Type u_3} {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) [inst_2 : Semiring S] [inst_3 : SMul S R] [inst_4 : Module S M] [inst_5 : IsScalarTower S R M]\n  (a : S), a • 0 = 0","typeReadable":"∀ {S : Type u_3} {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) [inst_2 : Semiring S] [inst_3 : SMul S R] [inst_4 : Module S M] [inst_5 : IsScalarTower S R M]\n  (a : S), a • 0 = 0","typeReferences":[["Subtype"],["IsScalarTower"],["Module"],["Membership","mem"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["SubMulAction","mulAction'"],["Submodule"],["SMul"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["Zero","toOfNat0"],["Monoid","toSemigroup"],["Eq"],["DistribSMul","toSMulZeroClass"],["Submodule","toSubMulAction"],["SetLike","instMembership"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Module","toDistribMulAction"],["Submodule","setLike"],["HSMul","hSMul"],["Submodule","addCommMonoid"],["MulAction","toSemigroupAction"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["smul_zero"],["Subtype"],["Eq","trans"],["Membership","mem"],["SemigroupAction","toSMul"],["AddCommMonoid","toAddMonoid"],["Subtype","val"],["SubMulAction","mulAction'"],["congrArg"],["Submodule"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["congrFun'"],["Zero","toOfNat0"],["Monoid","toSemigroup"],["Eq"],["DistribSMul","toSMulZeroClass"],["Submodule","toSubMulAction"],["SetLike","instMembership"],["True"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["Subtype","ext"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["eq_self"],["Module","toDistribMulAction"],["of_eq_true"],["Submodule","setLike"],["HSMul","hSMul"],["Submodule","addCommMonoid"],["MulAction","toSemigroupAction"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","carrier_eq_coe"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3150621351._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3150621351._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3150621351._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3150621351._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3150621351._hygCtx._hyg.10] (s : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3150621351._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3150621351._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3150621351._hygCtx._hyg.13), Eq.{succ v} (Set.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3150621351._hygCtx._hyg.10)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3150621351._hygCtx._hyg.10)) (Submodule.toAddSubmonoid.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3150621351._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3150621351._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3150621351._hygCtx._hyg.13 s))) (SetLike.coe.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3150621351._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3150621351._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3150621351._hygCtx._hyg.13) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3150621351._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3150621351._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3150621351._hygCtx._hyg.13) s)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (s : Submodule R M),\n  s.carrier = ↑s","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (s : Submodule R M),\n  s.carrier = ↑s","typeReferences":[["Submodule","toAddSubmonoid"],["Module"],["Set"],["AddCommMonoid","toAddMonoid"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["Submodule"],["AddSubsemigroup","carrier"],["AddCommMonoid"],["SetLike","coe"],["Submodule","setLike"],["AddSubmonoid","toAddSubsemigroup"],["Eq"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["AddSubsemigroup","carrier"],["rfl"],["Submodule","toAddSubmonoid"],["Set"],["AddSubmonoid","toAddSubsemigroup"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","module'","_proof_7"],"typeFallback":"forall {S : Type.{u_3}} {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] {module_M : Module.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10} (p : Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 : Semiring.{u_3} S] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 : SMul.{u_3, u_2} S R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 : Module.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36 : IsScalarTower.{u_3, u_2, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (SMulZeroClass.toSMul.{u_2, u_1} R 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)))) (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32))))] (a : S), Eq.{succ u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (SemigroupAction.toSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))) (MulAction.toSemigroupAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (MulAction.mk.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (SemigroupAction.mk.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))) (Submodule.smul.{u_3, u_2, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36) (Submodule.module'._proof_5.{u_1, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36)) (Submodule.module'._proof_6.{u_1, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36))))) a (OfNat.ofNat.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) 0 (Zero.toOfNat0.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddZero.toZero.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddZeroClass.toAddZero.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddMonoid.toAddZeroClass.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Submodule.addCommMonoid.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p)))))))) (OfNat.ofNat.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) 0 (Zero.toOfNat0.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddZero.toZero.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddZeroClass.toAddZero.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddMonoid.toAddZeroClass.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Submodule.addCommMonoid.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p)))))))","typeFull":"∀ {S : Type u_3} {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) [inst_2 : Semiring S] [inst_3 : SMul S R] [inst_4 : Module S M] [inst_5 : IsScalarTower S R M]\n  (a : S), a • 0 = 0","typeReadable":"∀ {S : Type u_3} {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) [inst_2 : Semiring S] [inst_3 : SMul S R] [inst_4 : Module S M] [inst_5 : IsScalarTower S R M]\n  (a : S), a • 0 = 0","typeReferences":[["Submodule","module'","_proof_6"],["Subtype"],["SemigroupAction","mk"],["IsScalarTower"],["Module"],["Membership","mem"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["SMul"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["Zero","toOfNat0"],["Monoid","toSemigroup"],["Eq"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Module","toDistribMulAction"],["Submodule","setLike"],["HSMul","hSMul"],["Submodule","smul"],["Submodule","addCommMonoid"],["MulAction","toSemigroupAction"],["AddZero","toZero"],["Submodule","module'","_proof_5"],["MulAction","mk"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Submodule","module'","_proof_1"]]},{"isProp":true,"kind":"theorem","name":["Submodule","coe_smul"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10} {p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M} (r : R) (x : Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M)) p x)), Eq.{succ v} M (Subtype.val.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M)) p x) (HSMul.hSMul.{u, v, v} R (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M)) p x)) (instHSMul.{u, v} R (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M)) p x)) (Submodule.smul.{u, u, v} R R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M p (SMulZeroClass.toSMul.{u, u} R R (AddZero.toZero.{u} R (AddZeroClass.toAddZero.{u} R (AddMonoid.toAddZeroClass.{u} R (AddMonoidWithOne.toAddMonoid.{u} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u} R (NonAssocSemiring.toAddCommMonoidWithOne.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7))))))) (DistribSMul.toSMulZeroClass.{u, u} R R (AddMonoid.toAddZeroClass.{u} R (AddMonoidWithOne.toAddMonoid.{u} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u} R (NonAssocSemiring.toAddCommMonoidWithOne.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7))))) (instDistribSMul.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7))))) (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M)))) (IsScalarTower.left.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7)) (DistribMulAction.toMulAction.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M))))) r x)) (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M))))) r (Subtype.val.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M)) p x) x))","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} {p : Submodule R M}\n  (r : R) (x : ↥p), ↑(r • x) = r • ↑x","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} {p : Submodule R M}\n  (r : R) (x : ↥p), ↑(r • x) = r • ↑x","typeReferences":[["Subtype"],["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddMonoidWithOne","toAddMonoid"],["instDistribSMul"],["Subtype","val"],["Submodule"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["IsScalarTower","left"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["Submodule","setLike"],["HSMul","hSMul"],["Submodule","smul"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Subtype"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddMonoidWithOne","toAddMonoid"],["instDistribSMul"],["Subtype","val"],["Submodule"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["rfl"],["SetLike","instMembership"],["IsScalarTower","left"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["Submodule","setLike"],["HSMul","hSMul"],["Submodule","smul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","ofClass","_proof_3"],"typeFallback":"forall {S : Type.{u_2}} {R : Type.{u_3}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.20 : Semiring.{u_3} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.26 : Module.{u_3, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.30 : SetLike.{u_2, u_1} S M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.38 : SMulMemClass.{u_2, u_3, u_1} S R M (SMulZeroClass.toSMul.{u_3, u_1} R 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.Submodule.Defs.99125656._hygCtx._hyg.23)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23)) (DistribMulAction.toDistribSMul.{u_3, u_1} R M (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.20)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23) (Module.toDistribMulAction.{u_3, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.26)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.30] (s : S) (r : R) {m : M}, (Membership.mem.{u_1, u_2} M S (SetLike.instMembership.{u_2, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.30) s m) -> (Membership.mem.{u_1, u_2} M S (SetLike.instMembership.{u_2, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.30) s (HSMul.hSMul.{u_3, u_1, u_1} R M M (instHSMul.{u_3, u_1} R M (SMulZeroClass.toSMul.{u_3, u_1} R 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.Submodule.Defs.99125656._hygCtx._hyg.23)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23)) (DistribMulAction.toDistribSMul.{u_3, u_1} R M (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.20)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23) (Module.toDistribMulAction.{u_3, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.26))))) r m))","typeFull":"∀ {S : Type u_2} {R : Type u_3} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : SetLike S M] [SMulMemClass S R M] (s : S) (r : R) {m : M}, m ∈ s → r • m ∈ s","typeReadable":"∀ {S : Type u_2} {R : Type u_3} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : SetLike S M] [SMulMemClass S R M] (s : S) (r : R) {m : M}, m ∈ s → r • m ∈ s","typeReferences":[["SMulMemClass"],["SetLike","instMembership"],["Module"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["AddCommMonoid"],["HSMul","hSMul"],["MonoidWithZero","toMonoid"],["instHSMul"],["SetLike"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["Module","toDistribMulAction"],["MonoidWithZero","toMonoid"],["SMulMemClass","smul_mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","smul_mem'"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17] (self : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20) (c : R) {x : M}, (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) (Submodule.toAddSubmonoid.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 self))) x) -> (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) (Submodule.toAddSubmonoid.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 self))) (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20))))) c x))","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (self : Submodule R M)\n  (c : R) {x : M}, x ∈ self.carrier → c • x ∈ self.carrier","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (self : Submodule R M)\n  (c : R) {x : M}, x ∈ self.carrier → c • x ∈ self.carrier","typeReferences":[["Submodule","toAddSubmonoid"],["Module"],["Set"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["Submodule"],["Set","instMembership"],["AddSubsemigroup","carrier"],["Module","toDistribMulAction"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["instHSMul"],["AddSubmonoid","toAddSubsemigroup"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[]},{"isProp":false,"kind":"constructor","name":["Submodule","mk"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17] (toAddSubmonoid : AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17))), (forall (c : R) {x : M}, (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) toAddSubmonoid)) x) -> (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) toAddSubmonoid)) (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20))))) c x))) -> (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20)","typeFull":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] →\n      [inst_1 : AddCommMonoid M] →\n        [inst_2 : Module R M] →\n          (toAddSubmonoid : AddSubmonoid M) →\n            (∀ (c : R) {x : M}, x ∈ toAddSubmonoid.carrier → c • x ∈ toAddSubmonoid.carrier) → Submodule R M","typeReadable":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] →\n      [inst_1 : AddCommMonoid M] →\n        [inst_2 : Module R M] →\n          (toAddSubmonoid : AddSubmonoid M) →\n            (∀ (c : R) {x : M}, x ∈ toAddSubmonoid.carrier → c • x ∈ toAddSubmonoid.carrier) → Submodule R M","typeReferences":[["Module"],["Set"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["Submodule"],["Set","instMembership"],["AddSubsemigroup","carrier"],["Module","toDistribMulAction"],["AddCommMonoid"],["AddSubmonoid"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["instHSMul"],["AddSubmonoid","toAddSubsemigroup"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["Submodule","add_mem_iff_left"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1511919823._hygCtx._hyg.7 : Ring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1511919823._hygCtx._hyg.10 : AddCommGroup.{v} M] {module_M : Module.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1511919823._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1511919823._hygCtx._hyg.10)} (p : Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1511919823._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1511919823._hygCtx._hyg.10) module_M) {x : M} {y : M}, (Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1511919823._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1511919823._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1511919823._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1511919823._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1511919823._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1511919823._hygCtx._hyg.10) module_M)) p y) -> (Iff (Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1511919823._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1511919823._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1511919823._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1511919823._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1511919823._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1511919823._hygCtx._hyg.10) module_M)) p (HAdd.hAdd.{v, v, v} M M M (instHAdd.{v} M (AddCommMagma.toAdd.{v} M (AddCommSemigroup.toAddCommMagma.{v} M (AddCommMonoid.toAddCommSemigroup.{v} M (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1511919823._hygCtx._hyg.10))))) x y)) (Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1511919823._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1511919823._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1511919823._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1511919823._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1511919823._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1511919823._hygCtx._hyg.10) module_M)) p x))","typeFull":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M} (p : Submodule R M)\n  {x y : M}, y ∈ p → (x + y ∈ p ↔ x ∈ p)","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M} (p : Submodule R M)\n  {x y : M}, y ∈ p → (x + y ∈ p ↔ x ∈ p)","typeReferences":[["SetLike","instMembership"],["Module"],["instHAdd"],["Membership","mem"],["AddCommGroup"],["Ring","toSemiring"],["Submodule"],["HAdd","hAdd"],["AddCommMonoid","toAddCommSemigroup"],["Submodule","setLike"],["Iff"],["AddCommGroup","toAddCommMonoid"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Ring"]],"valueReferences":[["Submodule","setLike"],["AddCommGroup","toAddGroup"],["Submodule","addSubgroupClass"],["AddCommGroup","toAddCommMonoid"],["add_mem_cancel_right"],["Ring","toSemiring"],["Submodule"]]},{"isProp":false,"kind":"definition","name":["Submodule","noConfusionType"],"typeFallback":"Sort.{u_1} -> (forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17], (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20) -> (forall {R' : Type.{u}} {M' : Type.{v}} [inst'._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 : Semiring.{u} R'] [inst'._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 : AddCommMonoid.{v} M'] [inst'._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 : Module.{u, v} R' M' inst'._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst'._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17], (Submodule.{u, v} R' M' inst'._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst'._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst'._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20) -> Sort.{u_1}))","typeFull":"Sort u_1 →\n  {R : Type u} →\n    {M : Type v} →\n      [inst : Semiring R] →\n        [inst_1 : AddCommMonoid M] →\n          [inst_2 : Module R M] →\n            Submodule R M →\n              {R' : Type u} →\n                {M' : Type v} →\n                  [inst' : Semiring R'] →\n                    [inst'_1 : AddCommMonoid M'] → [inst'_2 : Module R' M'] → Submodule R' M' → Sort u_1","typeReadable":"Sort u_1 →\n  {R : Type u} →\n    {M : Type v} →\n      [inst : Semiring R] →\n        [inst_1 : AddCommMonoid M] →\n          [inst_2 : Module R M] →\n            Submodule R M →\n              {R' : Type u} →\n                {M' : Type v} →\n                  [inst' : Semiring R'] →\n                    [inst'_1 : AddCommMonoid M'] → [inst'_2 : Module R' M'] → Submodule R' M' → Sort u_1","typeReferences":[["AddCommMonoid"],["Module"],["Submodule"],["Semiring"]],"valueReferences":[["AddSubmonoid"],["HEq"],["Submodule","casesOn"],["AddCommMonoid","toAddMonoid"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["SubmoduleClass","module'","_proof_3"],"typeFallback":"forall {S : Type.{u_3}} {R : Type.{u_4}} {M : Type.{u_1}} {T : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 : Semiring.{u_4} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 : Semiring.{u_3} S] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17 : Module.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.21 : SMul.{u_3, u_4} S R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.25 : Module.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.29 : IsScalarTower.{u_3, u_4, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.21 (SMulZeroClass.toSMul.{u_4, u_1} R 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_4, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_4, u_1} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17)))) (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.25))))] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34 : SetLike.{u_2, u_1} T M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.38 : AddSubmonoidClass.{u_2, u_1} T M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.42 : SMulMemClass.{u_2, u_4, u_1} T R M (SMulZeroClass.toSMul.{u_4, u_1} R 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_4, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_4, u_1} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34] (t : T) (x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.74 : S), Eq.{succ u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (SetLike.smul'.{u_2, u_3, u_4, u_1} T S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.21 (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.25)))) (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (DistribMulAction.toMulAction.{u_4, u_1} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.42 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.29 t)) x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.74 (OfNat.ofNat.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) 0 (Zero.toOfNat0.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (AddZero.toZero.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (AddZeroClass.toAddZero.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (AddMonoid.toAddZeroClass.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (AddSubmonoidClass.toAddCommMonoid.{u_2, u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 T inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.38 t)))))))) (OfNat.ofNat.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) 0 (Zero.toOfNat0.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (AddZero.toZero.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (AddZeroClass.toAddZero.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (AddMonoid.toAddZeroClass.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (AddSubmonoidClass.toAddCommMonoid.{u_2, u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 T inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.38 t)))))))","typeFull":"∀ {S : Type u_3} {R : Type u_4} {M : Type u_1} {T : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : Semiring S] [inst_3 : Module R M] [inst_4 : SMul S R] [inst_5 : Module S M] [inst_6 : IsScalarTower S R M]\n  [inst_7 : SetLike T M] [inst_8 : AddSubmonoidClass T M] [inst_9 : SMulMemClass T R M] (t : T) (x : S), x • 0 = 0","typeReadable":"∀ {S : Type u_3} {R : Type u_4} {M : Type u_1} {T : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : Semiring S] [inst_3 : Module R M] [inst_4 : SMul S R] [inst_5 : Module S M] [inst_6 : IsScalarTower S R M]\n  [inst_7 : SetLike T M] [inst_8 : AddSubmonoidClass T M] [inst_9 : SMulMemClass T R M] (t : T) (x : S), x • 0 = 0","typeReferences":[["Subtype"],["SetLike","smul'"],["IsScalarTower"],["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["SMul"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["Zero","toOfNat0"],["SetLike"],["Eq"],["AddSubmonoidClass"],["DistribSMul","toSMulZeroClass"],["SMulMemClass"],["SetLike","instMembership"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["AddSubmonoidClass","toAddCommMonoid"],["OfNat","ofNat"],["Module","toDistribMulAction"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["smul_zero"],["Subtype"],["SetLike","smul'"],["Eq","trans"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Subtype","val"],["congrArg"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["congrFun'"],["Zero","toOfNat0"],["Eq"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["True"],["DistribMulAction","toDistribSMul"],["Subtype","ext"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["AddSubmonoidClass","toAddCommMonoid"],["OfNat","ofNat"],["eq_self"],["Module","toDistribMulAction"],["of_eq_true"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","smul_mem"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.10} (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.10 module_M) {x : M} (r : R), (Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.10 module_M)) p x) -> (Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.10 module_M)) p (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.10 module_M))))) r x))","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M)\n  {x : M} (r : R), x ∈ p → r • x ∈ p","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M)\n  {x : M} (r : R), x ∈ p → r • x ∈ p","typeReferences":[["SetLike","instMembership"],["Module"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submodule"],["Module","toDistribMulAction"],["AddCommMonoid"],["Submodule","setLike"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["instHSMul"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Submodule","smul_mem'"]]},{"isProp":false,"kind":"definition","name":["Submodule","instPartialOrder"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983714632._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983714632._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983714632._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983714632._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983714632._hygCtx._hyg.10], PartialOrder.{v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983714632._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983714632._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983714632._hygCtx._hyg.13)","typeFull":"{R : Type u} →\n  {M : Type v} → [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : Module R M] → PartialOrder (Submodule R M)","typeReadable":"{R : Type u} →\n  {M : Type v} → [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : Module R M] → PartialOrder (Submodule R M)","typeReferences":[["AddCommMonoid"],["Module"],["PartialOrder"],["Submodule"],["Semiring"]],"valueReferences":[["Submodule","setLike"],["PartialOrder","ofSetLike"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","mk","inj"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} {inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 : Semiring.{u} R} {inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 : AddCommMonoid.{v} M} {inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17} {toAddSubmonoid : AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17))} {smul_mem' : forall (c : R) {x : M}, (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) toAddSubmonoid)) x) -> (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) toAddSubmonoid)) (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20))))) c x))} {toAddSubmonoid_1 : AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17))} {smul_mem'_1 : forall (c : R) {x : M}, (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) toAddSubmonoid_1)) x) -> (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) toAddSubmonoid_1)) (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20))))) c x))}, (Eq.{succ v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20) (Submodule.mk.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 toAddSubmonoid smul_mem') (Submodule.mk.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 toAddSubmonoid_1 smul_mem'_1)) -> (Eq.{succ v} (AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17))) toAddSubmonoid toAddSubmonoid_1)","typeFull":"∀ {R : Type u} {M : Type v} {inst : Semiring R} {inst_1 : AddCommMonoid M} {inst_2 : Module R M}\n  {toAddSubmonoid : AddSubmonoid M}\n  {smul_mem' : ∀ (c : R) {x : M}, x ∈ toAddSubmonoid.carrier → c • x ∈ toAddSubmonoid.carrier}\n  {toAddSubmonoid_1 : AddSubmonoid M}\n  {smul_mem'_1 : ∀ (c : R) {x : M}, x ∈ toAddSubmonoid_1.carrier → c • x ∈ toAddSubmonoid_1.carrier},\n  { toAddSubmonoid := toAddSubmonoid, smul_mem' := smul_mem' } =\n      { toAddSubmonoid := toAddSubmonoid_1, smul_mem' := smul_mem'_1 } →\n    toAddSubmonoid = toAddSubmonoid_1","typeReadable":"∀ {R : Type u} {M : Type v} {inst : Semiring R} {inst_1 : AddCommMonoid M} {inst_2 : Module R M}\n  {toAddSubmonoid : AddSubmonoid M}\n  {smul_mem' : ∀ (c : R) {x : M}, x ∈ toAddSubmonoid.carrier → c • x ∈ toAddSubmonoid.carrier}\n  {toAddSubmonoid_1 : AddSubmonoid M}\n  {smul_mem'_1 : ∀ (c : R) {x : M}, x ∈ toAddSubmonoid_1.carrier → c • x ∈ toAddSubmonoid_1.carrier},\n  { toAddSubmonoid := toAddSubmonoid, smul_mem' := smul_mem' } =\n      { toAddSubmonoid := toAddSubmonoid_1, smul_mem' := smul_mem'_1 } →\n    toAddSubmonoid = toAddSubmonoid_1","typeReferences":[["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddSubsemigroup","carrier"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["DistribSMul","toSMulZeroClass"],["Set"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["Set","instMembership"],["Module","toDistribMulAction"],["AddSubmonoid"],["Submodule","mk"],["HSMul","hSMul"],["AddSubmonoid","toAddSubsemigroup"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Submodule","mk","noConfusion"],["AddSubmonoid"],["eq_of_heq"],["AddCommMonoid","toAddMonoid"],["Eq"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","mem_mk","_simp_1"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10] {S : AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10))} {x : M} (h : forall (c : R) {x : M}, (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10)) S)) x) -> (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10)) S)) (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.13))))) c x))), Eq.{1} Prop (Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.13) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.13) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.13)) (Submodule.mk.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.13 S h) x) (Membership.mem.{v, v} M (AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10))) (SetLike.instMembership.{v, v} (AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10))) M (AddSubmonoid.instSetLike.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10)))) S x)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : AddSubmonoid M}\n  {x : M} (h : ∀ (c : R) {x : M}, x ∈ S.carrier → c • x ∈ S.carrier),\n  (x ∈ { toAddSubmonoid := S, smul_mem' := h }) = (x ∈ S)","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : AddSubmonoid M}\n  {x : M} (h : ∀ (c : R) {x : M}, x ∈ S.carrier → c • x ∈ S.carrier),\n  (x ∈ { toAddSubmonoid := S, smul_mem' := h }) = (x ∈ S)","typeReferences":[["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddSubsemigroup","carrier"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["Set"],["AddSubmonoid","instSetLike"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["Set","instMembership"],["Module","toDistribMulAction"],["Submodule","mk"],["Submodule","setLike"],["AddSubmonoid"],["HSMul","hSMul"],["AddSubmonoid","toAddSubsemigroup"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["SetLike","instMembership"],["AddSubmonoid"],["Submodule","mk"],["Submodule","setLike"],["Submodule","mem_mk"],["AddSubmonoid","instSetLike"],["Membership","mem"],["AddCommMonoid","toAddMonoid"],["propext"],["AddMonoid","toAddZeroClass"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","instCanLiftSetCoeAndMemOfNatForallForallForallForallHAddForallForallForallHSMul"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.385143627._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.385143627._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.385143627._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.385143627._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.385143627._hygCtx._hyg.10], CanLift.{succ v, succ v} (Set.{v} M) (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.385143627._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.385143627._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.385143627._hygCtx._hyg.13) (SetLike.coe.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.385143627._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.385143627._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.385143627._hygCtx._hyg.13) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.385143627._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.385143627._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.385143627._hygCtx._hyg.13)) (fun (s : Set.{v} M) => And (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) s (OfNat.ofNat.{v} M 0 (Zero.toOfNat0.{v} M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.385143627._hygCtx._hyg.10))))))) (And (forall {x : M} {y : M}, (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) s x) -> (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) s y) -> (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) s (HAdd.hAdd.{v, v, v} M M M (instHAdd.{v} M (AddCommMagma.toAdd.{v} M (AddCommSemigroup.toAddCommMagma.{v} M (AddCommMonoid.toAddCommSemigroup.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.385143627._hygCtx._hyg.10)))) x y))) (forall (r : R) {x : M}, (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) s x) -> (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) s (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.385143627._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.385143627._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.385143627._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.385143627._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.385143627._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.385143627._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.385143627._hygCtx._hyg.13))))) r x)))))","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n  CanLift (Set M) (Submodule R M) SetLike.coe fun s =>\n    0 ∈ s ∧ (∀ {x y : M}, x ∈ s → y ∈ s → x + y ∈ s) ∧ ∀ (r : R) {x : M}, x ∈ s → r • x ∈ s","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n  CanLift (Set M) (Submodule R M) SetLike.coe fun s =>\n    0 ∈ s ∧ (∀ {x y : M}, x ∈ s → y ∈ s → x + y ∈ s) ∧ ∀ (r : R) {x : M}, x ∈ s → r • x ∈ s","typeReferences":[["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["Zero","toOfNat0"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["DistribSMul","toSMulZeroClass"],["Set"],["instHAdd"],["DistribMulAction","toDistribSMul"],["And"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Set","instMembership"],["HAdd","hAdd"],["Module","toDistribMulAction"],["SetLike","coe"],["AddCommMonoid","toAddCommSemigroup"],["Submodule","setLike"],["HSMul","hSMul"],["CanLift"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Membership","mem"],["Exists","intro"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddSubsemigroup","mk"],["MonoidWithZero","toMonoid"],["instHSMul"],["Zero","toOfNat0"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Eq"],["And","left"],["DistribSMul","toSMulZeroClass"],["rfl"],["CanLift","mk"],["Set"],["instHAdd"],["And","right"],["And"],["DistribMulAction","toDistribSMul"],["AddSubmonoid","mk"],["Semiring","toMonoidWithZero"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Set","instMembership"],["HAdd","hAdd"],["Module","toDistribMulAction"],["SetLike","coe"],["AddCommMonoid","toAddCommSemigroup"],["Submodule","mk"],["Submodule","setLike"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","module'","_proof_9"],"typeFallback":"forall {S : Type.{u_3}} {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] {module_M : Module.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10} (p : Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 : Semiring.{u_3} S] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 : SMul.{u_3, u_2} S R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 : Module.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36 : IsScalarTower.{u_3, u_2, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (SMulZeroClass.toSMul.{u_2, u_1} R 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)))) (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32))))] (r : S) (s : S) (x : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)), Eq.{succ u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (SemigroupAction.toSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))) (MulAction.toSemigroupAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (DistribMulAction.toMulAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Submodule.addCommMonoid.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p)) (DistribMulAction.mk.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Submodule.addCommMonoid.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p)) (MulAction.mk.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (SemigroupAction.mk.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))) (Submodule.smul.{u_3, u_2, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36) (Submodule.module'._proof_5.{u_1, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36)) (Submodule.module'._proof_6.{u_1, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36)) (Submodule.module'._proof_7.{u_1, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36) (Submodule.module'._proof_8.{u_1, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36)))))) (HAdd.hAdd.{u_3, u_3, u_3} S S S (instHAdd.{u_3} S (Distrib.toAdd.{u_3} S (NonUnitalNonAssocSemiring.toDistrib.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))))) r s) x) (HAdd.hAdd.{u_1, u_1, u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (instHAdd.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddCommMagma.toAdd.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddCommSemigroup.toAddCommMagma.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddCommMonoid.toAddCommSemigroup.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Submodule.addCommMonoid.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p))))) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (SemigroupAction.toSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))) (MulAction.toSemigroupAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (DistribMulAction.toMulAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Submodule.addCommMonoid.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p)) (DistribMulAction.mk.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Submodule.addCommMonoid.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p)) (MulAction.mk.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (SemigroupAction.mk.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))) (Submodule.smul.{u_3, u_2, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36) (Submodule.module'._proof_5.{u_1, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36)) (Submodule.module'._proof_6.{u_1, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36)) (Submodule.module'._proof_7.{u_1, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36) (Submodule.module'._proof_8.{u_1, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36)))))) r x) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (SemigroupAction.toSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))) (MulAction.toSemigroupAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (DistribMulAction.toMulAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Submodule.addCommMonoid.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p)) (DistribMulAction.mk.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Submodule.addCommMonoid.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p)) (MulAction.mk.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (SemigroupAction.mk.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))) (Submodule.smul.{u_3, u_2, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36) (Submodule.module'._proof_5.{u_1, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36)) (Submodule.module'._proof_6.{u_1, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36)) (Submodule.module'._proof_7.{u_1, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36) (Submodule.module'._proof_8.{u_1, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36)))))) s x))","typeFull":"∀ {S : Type u_3} {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) [inst_2 : Semiring S] [inst_3 : SMul S R] [inst_4 : Module S M] [inst_5 : IsScalarTower S R M]\n  (r s : S) (x : ↥p), (r + s) • x = r • x + s • x","typeReadable":"∀ {S : Type u_3} {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) [inst_2 : Semiring S] [inst_3 : SMul S R] [inst_4 : Module S M] [inst_5 : IsScalarTower S R M]\n  (r s : S) (x : ↥p), (r + s) • x = r • x + s • x","typeReferences":[["Submodule","module'","_proof_6"],["Subtype"],["SemigroupAction","mk"],["IsScalarTower"],["Module"],["Membership","mem"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["SMul"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Monoid","toSemigroup"],["Eq"],["DistribSMul","toSMulZeroClass"],["Distrib","toAdd"],["SetLike","instMembership"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","mk"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["HAdd","hAdd"],["Submodule","module'","_proof_7"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["Submodule","setLike"],["HSMul","hSMul"],["Submodule","smul"],["Submodule","addCommMonoid"],["Submodule","module'","_proof_8"],["MulAction","toSemigroupAction"],["AddZero","toZero"],["Submodule","module'","_proof_5"],["MulAction","mk"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Submodule","module'","_proof_3"]]},{"isProp":true,"kind":"theorem","name":["Submodule","smul_mem_iff'"],"typeFallback":"forall {G : Type.{u''}} {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.10} (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.10 module_M) {x : M} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.25 : Group.{u''} G] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.28 : MulAction.{u'', v} G M (DivInvMonoid.toMonoid.{u''} G (Group.toDivInvMonoid.{u''} G inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.25))] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.32 : SMul.{u'', u} G R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.36 : IsScalarTower.{u'', u, v} G R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.32 (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.10 module_M)))) (SemigroupAction.toSMul.{u'', v} G M (Monoid.toSemigroup.{u''} G (DivInvMonoid.toMonoid.{u''} G (Group.toDivInvMonoid.{u''} G inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.25))) (MulAction.toSemigroupAction.{u'', v} G M (DivInvMonoid.toMonoid.{u''} G (Group.toDivInvMonoid.{u''} G inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.25)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.28))] (g : G), Iff (Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.10 module_M)) p (HSMul.hSMul.{u'', v, v} G M M (instHSMul.{u'', v} G M (SemigroupAction.toSMul.{u'', v} G M (Monoid.toSemigroup.{u''} G (DivInvMonoid.toMonoid.{u''} G (Group.toDivInvMonoid.{u''} G inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.25))) (MulAction.toSemigroupAction.{u'', v} G M (DivInvMonoid.toMonoid.{u''} G (Group.toDivInvMonoid.{u''} G inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.25)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.28))) g x)) (Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.10 module_M)) p x)","typeFull":"∀ {G : Type u''} {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) {x : M} [inst_2 : Group G] [inst_3 : MulAction G M] [inst_4 : SMul G R] [IsScalarTower G R M]\n  (g : G), g • x ∈ p ↔ x ∈ p","typeReadable":"∀ {G : Type u''} {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) {x : M} [inst_2 : Group G] [inst_3 : MulAction G M] [inst_4 : SMul G R] [IsScalarTower G R M]\n  (g : G), g • x ∈ p ↔ x ∈ p","typeReferences":[["IsScalarTower"],["Module"],["Membership","mem"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["SMul"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["Monoid","toSemigroup"],["Group","toDivInvMonoid"],["DistribSMul","toSMulZeroClass"],["Group"],["SetLike","instMembership"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["DivInvMonoid","toMonoid"],["MulAction"],["Submodule","setLike"],["Iff"],["HSMul","hSMul"],["AddZero","toZero"],["MulAction","toSemigroupAction"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Submodule","toSubMulAction"],["Module","toDistribMulAction"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Semiring","toMonoidWithZero"],["SubMulAction","smul_mem_iff'"],["AddCommMonoid","toAddMonoid"]]},{"isProp":true,"kind":"theorem","name":["Submodule","add","_proof_1"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.7 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] {module_M : Module.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.10} (p : Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.10 module_M) (x : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.10 module_M)) p x)) (y : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.10 module_M)) p x)), Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.10 module_M)) p (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddCommMagma.toAdd.{u_1} M (AddCommSemigroup.toAddCommMagma.{u_1} M (AddCommMonoid.toAddCommSemigroup.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.10)))) (Subtype.val.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.10 module_M)) p x) x) (Subtype.val.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.10 module_M)) p x) y))","typeFull":"∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) (x y : ↥p), ↑x + ↑y ∈ p","typeReadable":"∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) (x y : ↥p), ↑x + ↑y ∈ p","typeReferences":[["Subtype"],["SetLike","instMembership"],["Module"],["instHAdd"],["Membership","mem"],["Subtype","val"],["Submodule"],["HAdd","hAdd"],["AddCommMonoid"],["AddCommMonoid","toAddCommSemigroup"],["Submodule","setLike"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Semiring"]],"valueReferences":[["SetLike","instMembership"],["Membership","mem"],["AddCommMonoid","toAddMonoid"],["Subtype","val"],["Submodule"],["AddSubmonoidClass","toAddMemClass"],["AddCommMonoid","toAddCommSemigroup"],["Submodule","setLike"],["Subtype","property"],["AddMemClass","add_mem"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Submodule","addSubmonoidClass"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","coe_toSubMulAction"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.10] (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.13), Eq.{succ v} (Set.{v} M) (SetLike.coe.{v, v} (SubMulAction.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.13))))) M (SubMulAction.instSetLike.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.13))))) (Submodule.toSubMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.13 p)) (SetLike.coe.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.13) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4219822720._hygCtx._hyg.13) p)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (p : Submodule R M),\n  ↑p.toSubMulAction = ↑p","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (p : Submodule R M),\n  ↑p.toSubMulAction = ↑p","typeReferences":[["Submodule","toSubMulAction"],["SubMulAction","instSetLike"],["Module"],["Set"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["SubMulAction"],["AddZeroClass","toAddZero"],["Submodule"],["Module","toDistribMulAction"],["AddCommMonoid"],["SetLike","coe"],["Submodule","setLike"],["MonoidWithZero","toMonoid"],["Eq"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["rfl"],["Submodule","toSubMulAction"],["SubMulAction","instSetLike"],["Set"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["SubMulAction"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["SetLike","coe"],["MonoidWithZero","toMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","module'","_proof_2"],"typeFallback":"forall {S : Type.{u_3}} {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] {module_M : Module.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10} (p : Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 : Semiring.{u_3} S] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 : SMul.{u_3, u_2} S R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 : Module.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36 : IsScalarTower.{u_3, u_2, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (SMulZeroClass.toSMul.{u_2, u_1} R 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)))) (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32))))] (a : S) (x : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (y : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)), Eq.{succ u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (SemigroupAction.toSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))) (MulAction.toSemigroupAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (have this : MulAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) := SubMulAction.mulAction'.{u_2, u_3, u_1} S R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7)) (DistribMulAction.toMulAction.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (DistribMulAction.toMulAction.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36 (Submodule.toSubMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p); this)))) a (HAdd.hAdd.{u_1, u_1, u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (instHAdd.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddSemigroup.toAdd.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddMonoid.toAddSemigroup.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Submodule.addCommMonoid.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p))))) x y)) (HAdd.hAdd.{u_1, u_1, u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (instHAdd.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddSemigroup.toAdd.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddMonoid.toAddSemigroup.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Submodule.addCommMonoid.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p))))) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (SemigroupAction.toSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))) (MulAction.toSemigroupAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (have this : MulAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) := SubMulAction.mulAction'.{u_2, u_3, u_1} S R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7)) (DistribMulAction.toMulAction.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (DistribMulAction.toMulAction.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36 (Submodule.toSubMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p); this)))) a x) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (SemigroupAction.toSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))) (MulAction.toSemigroupAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (have this : MulAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) := SubMulAction.mulAction'.{u_2, u_3, u_1} S R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7)) (DistribMulAction.toMulAction.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (DistribMulAction.toMulAction.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36 (Submodule.toSubMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p); this)))) a y))","typeFull":"∀ {S : Type u_3} {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) [inst_2 : Semiring S] [inst_3 : SMul S R] [inst_4 : Module S M] [inst_5 : IsScalarTower S R M]\n  (a : S) (x y : ↥p), a • (x + y) = a • x + a • y","typeReadable":"∀ {S : Type u_3} {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) [inst_2 : Semiring S] [inst_3 : SMul S R] [inst_4 : Module S M] [inst_5 : IsScalarTower S R M]\n  (a : S) (x y : ↥p), a • (x + y) = a • x + a • y","typeReferences":[["Subtype"],["IsScalarTower"],["Module"],["Membership","mem"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["SubMulAction","mulAction'"],["Submodule"],["SMul"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["Monoid","toSemigroup"],["Eq"],["AddSemigroup","toAdd"],["DistribSMul","toSMulZeroClass"],["Submodule","toSubMulAction"],["SetLike","instMembership"],["instHAdd"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["HAdd","hAdd"],["Module","toDistribMulAction"],["Submodule","setLike"],["AddMonoid","toAddSemigroup"],["HSMul","hSMul"],["Submodule","addCommMonoid"],["MulAction","toSemigroupAction"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Subtype"],["Eq","trans"],["Membership","mem"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Subtype","val"],["SubMulAction","mulAction'"],["congrArg"],["Submodule"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["congrFun'"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Monoid","toSemigroup"],["Eq"],["DistribSMul","toSMulZeroClass"],["AddSemigroup","toAdd"],["Submodule","toSubMulAction"],["SetLike","instMembership"],["True"],["instHAdd"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["Subtype","ext"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["HAdd","hAdd"],["eq_self"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["of_eq_true"],["Submodule","setLike"],["AddMonoid","toAddSemigroup"],["HSMul","hSMul"],["smul_add"],["Submodule","addCommMonoid"],["AddZero","toZero"],["MulAction","toSemigroupAction"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"recursor","name":["Submodule","rec"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17] {motive : (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20) -> Sort.{u_1}}, (forall (toAddSubmonoid : AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17))) (smul_mem' : forall (c : R) {x : M}, (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) toAddSubmonoid)) x) -> (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) toAddSubmonoid)) (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20))))) c x))), motive (Submodule.mk.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 toAddSubmonoid smul_mem')) -> (forall (t : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20), motive t)","typeFull":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] →\n      [inst_1 : AddCommMonoid M] →\n        [inst_2 : Module R M] →\n          {motive : Submodule R M → Sort u_1} →\n            ((toAddSubmonoid : AddSubmonoid M) →\n                (smul_mem' : ∀ (c : R) {x : M}, x ∈ toAddSubmonoid.carrier → c • x ∈ toAddSubmonoid.carrier) →\n                  motive { toAddSubmonoid := toAddSubmonoid, smul_mem' := smul_mem' }) →\n              (t : Submodule R M) → motive t","typeReadable":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] →\n      [inst_1 : AddCommMonoid M] →\n        [inst_2 : Module R M] →\n          {motive : Submodule R M → Sort u_1} →\n            ((toAddSubmonoid : AddSubmonoid M) →\n                (smul_mem' : ∀ (c : R) {x : M}, x ∈ toAddSubmonoid.carrier → c • x ∈ toAddSubmonoid.carrier) →\n                  motive { toAddSubmonoid := toAddSubmonoid, smul_mem' := smul_mem' }) →\n              (t : Submodule R M) → motive t","typeReferences":[["Module"],["Set"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["Submodule"],["Set","instMembership"],["AddSubsemigroup","carrier"],["Module","toDistribMulAction"],["AddCommMonoid"],["Submodule","mk"],["AddSubmonoid"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["instHSMul"],["AddSubmonoid","toAddSubsemigroup"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":null},{"isProp":false,"kind":"definition","name":["Submodule","mk","noConfusion"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} {inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 : Semiring.{u} R} {inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 : AddCommMonoid.{v} M} {inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17} {P : Sort.{u_1}} {toAddSubmonoid : AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17))} {smul_mem' : forall (c : R) {x : M}, (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) toAddSubmonoid)) x) -> (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) toAddSubmonoid)) (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20))))) c x))} {toAddSubmonoid' : AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17))} {smul_mem'' : forall (c : R) {x : M}, (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) toAddSubmonoid')) x) -> (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) toAddSubmonoid')) (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20))))) c x))}, (Eq.{succ v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20) (Submodule.mk.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 toAddSubmonoid smul_mem') (Submodule.mk.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 toAddSubmonoid' smul_mem'')) -> ((HEq.{succ v} (AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17))) toAddSubmonoid (AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17))) toAddSubmonoid') -> P) -> P","typeFull":"{R : Type u} →\n  {M : Type v} →\n    {inst : Semiring R} →\n      {inst_1 : AddCommMonoid M} →\n        {inst_2 : Module R M} →\n          {P : Sort u_1} →\n            {toAddSubmonoid : AddSubmonoid M} →\n              {smul_mem' : ∀ (c : R) {x : M}, x ∈ toAddSubmonoid.carrier → c • x ∈ toAddSubmonoid.carrier} →\n                {toAddSubmonoid' : AddSubmonoid M} →\n                  {smul_mem'' : ∀ (c : R) {x : M}, x ∈ toAddSubmonoid'.carrier → c • x ∈ toAddSubmonoid'.carrier} →\n                    { toAddSubmonoid := toAddSubmonoid, smul_mem' := smul_mem' } =\n                        { toAddSubmonoid := toAddSubmonoid', smul_mem' := smul_mem'' } →\n                      (toAddSubmonoid ≍ toAddSubmonoid' → P) → P","typeReadable":"{R : Type u} →\n  {M : Type v} →\n    {inst : Semiring R} →\n      {inst_1 : AddCommMonoid M} →\n        {inst_2 : Module R M} →\n          {P : Sort u_1} →\n            {toAddSubmonoid : AddSubmonoid M} →\n              {smul_mem' : ∀ (c : R) {x : M}, x ∈ toAddSubmonoid.carrier → c • x ∈ toAddSubmonoid.carrier} →\n                {toAddSubmonoid' : AddSubmonoid M} →\n                  {smul_mem'' : ∀ (c : R) {x : M}, x ∈ toAddSubmonoid'.carrier → c • x ∈ toAddSubmonoid'.carrier} →\n                    { toAddSubmonoid := toAddSubmonoid, smul_mem' := smul_mem' } =\n                        { toAddSubmonoid := toAddSubmonoid', smul_mem' := smul_mem'' } →\n                      (toAddSubmonoid ≍ toAddSubmonoid' → P) → P","typeReferences":[["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddSubsemigroup","carrier"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["DistribSMul","toSMulZeroClass"],["Set"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["Set","instMembership"],["Module","toDistribMulAction"],["AddSubmonoid"],["Submodule","mk"],["HSMul","hSMul"],["HEq"],["AddSubmonoid","toAddSubsemigroup"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["AddCommMonoid"],["HEq","refl"],["Module"],["Submodule","mk"],["Eq","refl"],["Submodule","noConfusion"],["id"],["heq_of_eq"],["Submodule"],["Semiring"]]},{"isProp":true,"kind":"theorem","name":["SubmoduleClass","module'","_proof_1"],"typeFallback":"forall {S : Type.{u_3}} {R : Type.{u_4}} {M : Type.{u_1}} {T : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 : Semiring.{u_4} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 : Semiring.{u_3} S] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17 : Module.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.21 : SMul.{u_3, u_4} S R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.25 : Module.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.29 : IsScalarTower.{u_3, u_4, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.21 (SMulZeroClass.toSMul.{u_4, u_1} R 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_4, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_4, u_1} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17)))) (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.25))))] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34 : SetLike.{u_2, u_1} T M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.42 : SMulMemClass.{u_2, u_4, u_1} T R M (SMulZeroClass.toSMul.{u_4, u_1} R 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_4, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_4, u_1} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34] (t : T) (x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.60 : S) (x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.62 : S) (x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.64 : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)), Eq.{succ u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (SetLike.smul'.{u_2, u_3, u_4, u_1} T S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.21 (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.25)))) (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (DistribMulAction.toMulAction.{u_4, u_1} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.42 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.29 t)) (HMul.hMul.{u_3, u_3, u_3} S S S (instHMul.{u_3} S (Semigroup.toMul.{u_3} S (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14))))) x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.60 x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.62) x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.64) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (SetLike.smul'.{u_2, u_3, u_4, u_1} T S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.21 (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.25)))) (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (DistribMulAction.toMulAction.{u_4, u_1} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.42 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.29 t)) x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.60 (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (SetLike.smul'.{u_2, u_3, u_4, u_1} T S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.21 (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.25)))) (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (DistribMulAction.toMulAction.{u_4, u_1} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.42 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.29 t)) x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.62 x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.64))","typeFull":"∀ {S : Type u_3} {R : Type u_4} {M : Type u_1} {T : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : Semiring S] [inst_3 : Module R M] [inst_4 : SMul S R] [inst_5 : Module S M] [inst_6 : IsScalarTower S R M]\n  [inst_7 : SetLike T M] [inst_8 : SMulMemClass T R M] (t : T) (x x_1 : S) (x_2 : ↥t), (x * x_1) • x_2 = x • x_1 • x_2","typeReadable":"∀ {S : Type u_3} {R : Type u_4} {M : Type u_1} {T : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : Semiring S] [inst_3 : Module R M] [inst_4 : SMul S R] [inst_5 : Module S M] [inst_6 : IsScalarTower S R M]\n  [inst_7 : SetLike T M] [inst_8 : SMulMemClass T R M] (t : T) (x x_1 : S) (x_2 : ↥t), (x * x_1) • x_2 = x • x_1 • x_2","typeReferences":[["Subtype"],["SetLike","smul'"],["IsScalarTower"],["Module"],["Membership","mem"],["HMul","hMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Semigroup","toMul"],["SMul"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["SetLike"],["Monoid","toSemigroup"],["Eq"],["DistribSMul","toSMulZeroClass"],["SMulMemClass"],["SetLike","instMembership"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["HSMul","hSMul"],["instHMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Subtype"],["Eq","trans"],["SetLike","smul'"],["Membership","mem"],["NonUnitalSemiring","toSemigroupWithZero"],["SemigroupAction","toSMul"],["HMul","hMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Subtype","val"],["congrArg"],["Semigroup","toMul"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["congrFun'"],["Monoid","toSemigroup"],["Eq"],["Semiring","toNonUnitalSemiring"],["SemigroupAction","mul_smul"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["True"],["DistribMulAction","toDistribSMul"],["Subtype","ext"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["eq_self"],["Module","toDistribMulAction"],["of_eq_true"],["HSMul","hSMul"],["instHMul"],["SemigroupWithZero","toSemigroup"],["MulAction","toSemigroupAction"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"definition","name":["SubmoduleClass","module'"],"typeFallback":"forall {S : Type.{u'}} {R : Type.{u}} {M : Type.{v}} {T : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 : Semiring.{u'} S] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.21 : SMul.{u', u} S R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.25 : Module.{u', v} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.29 : IsScalarTower.{u', u, v} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.21 (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17)))) (SMulZeroClass.toSMul.{u', v} S M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u', v} S M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u', v} S M (MonoidWithZero.toMonoid.{u'} S (Semiring.toMonoidWithZero.{u'} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u', v} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.25))))] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34 : SetLike.{u_1, v} T M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.38 : AddSubmonoidClass.{u_1, v} T M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.42 : SMulMemClass.{u_1, u, v} T R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34] (t : T), Module.{u', v} S (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, u_1} M T (SetLike.instMembership.{u_1, v} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 (AddSubmonoidClass.toAddCommMonoid.{u_1, v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 T inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.38 t)","typeFull":"{S : Type u'} →\n  {R : Type u} →\n    {M : Type v} →\n      {T : Type u_1} →\n        [inst : Semiring R] →\n          [inst_1 : AddCommMonoid M] →\n            [inst_2 : Semiring S] →\n              [inst_3 : Module R M] →\n                [inst_4 : SMul S R] →\n                  [inst_5 : Module S M] →\n                    [IsScalarTower S R M] →\n                      [inst_7 : SetLike T M] →\n                        [inst_8 : AddSubmonoidClass T M] → [SMulMemClass T R M] → (t : T) → Module S ↥t","typeReadable":"{S : Type u'} →\n  {R : Type u} →\n    {M : Type v} →\n      {T : Type u_1} →\n        [inst : Semiring R] →\n          [inst_1 : AddCommMonoid M] →\n            [inst_2 : Semiring S] →\n              [inst_3 : Module R M] →\n                [inst_4 : SMul S R] →\n                  [inst_5 : Module S M] →\n                    [IsScalarTower S R M] →\n                      [inst_7 : SetLike T M] →\n                        [inst_8 : AddSubmonoidClass T M] → [SMulMemClass T R M] → (t : T) → Module S ↥t","typeReferences":[["SMulMemClass"],["SetLike","instMembership"],["Subtype"],["Module"],["IsScalarTower"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddSubmonoidClass","toAddCommMonoid"],["Module","toDistribMulAction"],["AddCommMonoid"],["SMul"],["MonoidWithZero","toMonoid"],["SetLike"],["AddSubmonoidClass"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["Subtype"],["SetLike","smul'"],["SemigroupAction","mk"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["SubmoduleClass","module'","_proof_2"],["SubmoduleClass","module'","_proof_5"],["SubmoduleClass","module'","_proof_6"],["Module","mk"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Monoid","toSemigroup"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["DistribMulAction","mk"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["AddSubmonoidClass","toAddCommMonoid"],["Module","toDistribMulAction"],["SubmoduleClass","module'","_proof_4"],["SubmoduleClass","module'","_proof_3"],["AddZero","toZero"],["MulAction","mk"],["AddMonoid","toAddZeroClass"],["SubmoduleClass","module'","_proof_1"]]},{"isProp":true,"kind":"theorem","name":["Submodule","mk","injEq"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17] (toAddSubmonoid : AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17))) (smul_mem' : forall (c : R) {x : M}, (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) toAddSubmonoid)) x) -> (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) toAddSubmonoid)) (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20))))) c x))) (toAddSubmonoid_1 : AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17))) (smul_mem'_1 : forall (c : R) {x : M}, (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) toAddSubmonoid_1)) x) -> (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) toAddSubmonoid_1)) (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20))))) c x))), Eq.{1} Prop (Eq.{succ v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20) (Submodule.mk.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 toAddSubmonoid smul_mem') (Submodule.mk.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 toAddSubmonoid_1 smul_mem'_1)) (Eq.{succ v} (AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17))) toAddSubmonoid toAddSubmonoid_1)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  (toAddSubmonoid : AddSubmonoid M)\n  (smul_mem' : ∀ (c : R) {x : M}, x ∈ toAddSubmonoid.carrier → c • x ∈ toAddSubmonoid.carrier)\n  (toAddSubmonoid_1 : AddSubmonoid M)\n  (smul_mem'_1 : ∀ (c : R) {x : M}, x ∈ toAddSubmonoid_1.carrier → c • x ∈ toAddSubmonoid_1.carrier),\n  ({ toAddSubmonoid := toAddSubmonoid, smul_mem' := smul_mem' } =\n      { toAddSubmonoid := toAddSubmonoid_1, smul_mem' := smul_mem'_1 }) =\n    (toAddSubmonoid = toAddSubmonoid_1)","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  (toAddSubmonoid : AddSubmonoid M)\n  (smul_mem' : ∀ (c : R) {x : M}, x ∈ toAddSubmonoid.carrier → c • x ∈ toAddSubmonoid.carrier)\n  (toAddSubmonoid_1 : AddSubmonoid M)\n  (smul_mem'_1 : ∀ (c : R) {x : M}, x ∈ toAddSubmonoid_1.carrier → c • x ∈ toAddSubmonoid_1.carrier),\n  ({ toAddSubmonoid := toAddSubmonoid, smul_mem' := smul_mem' } =\n      { toAddSubmonoid := toAddSubmonoid_1, smul_mem' := smul_mem'_1 }) =\n    (toAddSubmonoid = toAddSubmonoid_1)","typeReferences":[["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddSubsemigroup","carrier"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["DistribSMul","toSMulZeroClass"],["Set"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["Set","instMembership"],["Module","toDistribMulAction"],["AddSubmonoid"],["Submodule","mk"],["HSMul","hSMul"],["AddSubmonoid","toAddSubsemigroup"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Membership","mem"],["Submodule","mk","inj"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddSubsemigroup","carrier"],["Eq","propIntro"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq","ndrec"],["Eq"],["DistribSMul","toSMulZeroClass"],["Set"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["Set","instMembership"],["Module","toDistribMulAction"],["AddSubmonoid"],["Submodule","mk"],["Eq","refl"],["HSMul","hSMul"],["AddSubmonoid","toAddSubsemigroup"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","coe_add","_simp_1"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10} {p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M} (x : Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M)) p x)) (y : Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M)) p x)), Eq.{succ v} M (HAdd.hAdd.{v, v, v} M M M (instHAdd.{v} M (AddCommMagma.toAdd.{v} M (AddCommSemigroup.toAddCommMagma.{v} M (AddCommMonoid.toAddCommSemigroup.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10)))) (Subtype.val.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M)) p x) x) (Subtype.val.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M)) p x) y)) (Subtype.val.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M)) p x) (HAdd.hAdd.{v, v, v} (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M)) p x)) (instHAdd.{v} (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M)) p x)) (Submodule.add.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M p)) x y))","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} {p : Submodule R M}\n  (x y : ↥p), ↑x + ↑y = ↑(x + y)","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} {p : Submodule R M}\n  (x y : ↥p), ↑x + ↑y = ↑(x + y)","typeReferences":[["Subtype"],["SetLike","instMembership"],["Submodule","add"],["Module"],["instHAdd"],["Membership","mem"],["Subtype","val"],["Submodule"],["HAdd","hAdd"],["AddCommMonoid"],["AddCommMonoid","toAddCommSemigroup"],["Submodule","setLike"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Eq"],["Semiring"]],"valueReferences":[["SetLike","instMembership"],["Subtype"],["Submodule","add"],["instHAdd"],["Membership","mem"],["Subtype","val"],["Submodule"],["HAdd","hAdd"],["AddCommMonoid","toAddCommSemigroup"],["Submodule","setLike"],["Eq","symm"],["Submodule","coe_add"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"]]},{"isProp":false,"kind":"definition","name":["Submodule","decidableEq"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.774292894._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.774292894._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.774292894._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.774292894._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.774292894._hygCtx._hyg.10], DecidableEq.{succ v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.774292894._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.774292894._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.774292894._hygCtx._hyg.13)","typeFull":"{R : Type u} →\n  {M : Type v} → [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : Module R M] → DecidableEq (Submodule R M)","typeReadable":"{R : Type u} →\n  {M : Type v} → [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : Module R M] → DecidableEq (Submodule R M)","typeReferences":[["AddCommMonoid"],["Module"],["DecidableEq"],["Submodule"],["Semiring"]],"valueReferences":[["Classical","typeDecidableEq"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","smul_mem_iff_of_isUnit"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.63354477._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.63354477._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.63354477._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.63354477._hygCtx._hyg.10} (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.63354477._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.63354477._hygCtx._hyg.10 module_M) {r : R} {x : M}, (IsUnit.{u} R (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.63354477._hygCtx._hyg.7)) r) -> (Iff (Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.63354477._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.63354477._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.63354477._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.63354477._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.63354477._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.63354477._hygCtx._hyg.10 module_M)) p (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.63354477._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.63354477._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.63354477._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.63354477._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.63354477._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.63354477._hygCtx._hyg.10 module_M))))) r x)) (Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.63354477._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.63354477._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.63354477._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.63354477._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.63354477._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.63354477._hygCtx._hyg.10 module_M)) p x))","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M)\n  {r : R} {x : M}, IsUnit r → (r • x ∈ p ↔ x ∈ p)","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M)\n  {r : R} {x : M}, IsUnit r → (r • x ∈ p ↔ x ∈ p)","typeReferences":[["SetLike","instMembership"],["Module"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["IsUnit"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submodule"],["Module","toDistribMulAction"],["AddCommMonoid"],["Submodule","setLike"],["Iff"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["instHSMul"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["IsUnit","invertible"],["MonoidWithZero","toMonoid"],["Submodule","smul_mem_iff''"],["Semiring","toMonoidWithZero"]]},{"isProp":true,"kind":"theorem","name":["Submodule","mk_le_mk","_simp_1"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10] {S : AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10))} {S' : AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10))} (h : forall (c : R) {x : M}, (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10)) S)) x) -> (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10)) S)) (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.13))))) c x))) (h' : forall (c : R) {x : M}, (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10)) S')) x) -> (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10)) S')) (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.13))))) c x))), Eq.{1} Prop (LE.le.{v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.13) (Preorder.toLE.{v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.13) (PartialOrder.toPreorder.{v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.13) (Submodule.instPartialOrder.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.13))) (Submodule.mk.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.13 S h) (Submodule.mk.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.13 S' h')) (LE.le.{v} (AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10))) (Preorder.toLE.{v} (AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10))) (PartialOrder.toPreorder.{v} (AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10))) (AddSubmonoid.instPartialOrder.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10))))) S S')","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S S' : AddSubmonoid M}\n  (h : ∀ (c : R) {x : M}, x ∈ S.carrier → c • x ∈ S.carrier)\n  (h' : ∀ (c : R) {x : M}, x ∈ S'.carrier → c • x ∈ S'.carrier),\n  ({ toAddSubmonoid := S, smul_mem' := h } ≤ { toAddSubmonoid := S', smul_mem' := h' }) = (S ≤ S')","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S S' : AddSubmonoid M}\n  (h : ∀ (c : R) {x : M}, x ∈ S.carrier → c • x ∈ S.carrier)\n  (h' : ∀ (c : R) {x : M}, x ∈ S'.carrier → c • x ∈ S'.carrier),\n  ({ toAddSubmonoid := S, smul_mem' := h } ≤ { toAddSubmonoid := S', smul_mem' := h' }) = (S ≤ S')","typeReferences":[["AddSubmonoid","instPartialOrder"],["PartialOrder","toPreorder"],["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddSubsemigroup","carrier"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["Preorder","toLE"],["Eq"],["DistribSMul","toSMulZeroClass"],["Set"],["Submodule","instPartialOrder"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["Set","instMembership"],["Module","toDistribMulAction"],["Submodule","mk"],["AddSubmonoid"],["HSMul","hSMul"],["LE","le"],["AddSubmonoid","toAddSubsemigroup"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Submodule","mk_le_mk"],["AddSubmonoid","instPartialOrder"],["PartialOrder","toPreorder"],["AddSubmonoid"],["Submodule","mk"],["Submodule","instPartialOrder"],["LE","le"],["AddCommMonoid","toAddMonoid"],["Preorder","toLE"],["propext"],["AddMonoid","toAddZeroClass"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","add_mem_iff_right"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.738535602._hygCtx._hyg.7 : Ring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.738535602._hygCtx._hyg.10 : AddCommGroup.{v} M] {module_M : Module.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.738535602._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.738535602._hygCtx._hyg.10)} (p : Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.738535602._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.738535602._hygCtx._hyg.10) module_M) {x : M} {y : M}, (Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.738535602._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.738535602._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.738535602._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.738535602._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.738535602._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.738535602._hygCtx._hyg.10) module_M)) p x) -> (Iff (Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.738535602._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.738535602._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.738535602._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.738535602._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.738535602._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.738535602._hygCtx._hyg.10) module_M)) p (HAdd.hAdd.{v, v, v} M M M (instHAdd.{v} M (AddCommMagma.toAdd.{v} M (AddCommSemigroup.toAddCommMagma.{v} M (AddCommMonoid.toAddCommSemigroup.{v} M (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.738535602._hygCtx._hyg.10))))) x y)) (Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.738535602._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.738535602._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.738535602._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.738535602._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.738535602._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.738535602._hygCtx._hyg.10) module_M)) p y))","typeFull":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M} (p : Submodule R M)\n  {x y : M}, x ∈ p → (x + y ∈ p ↔ y ∈ p)","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M} (p : Submodule R M)\n  {x y : M}, x ∈ p → (x + y ∈ p ↔ y ∈ p)","typeReferences":[["SetLike","instMembership"],["Module"],["instHAdd"],["Membership","mem"],["AddCommGroup"],["Ring","toSemiring"],["Submodule"],["HAdd","hAdd"],["AddCommMonoid","toAddCommSemigroup"],["Submodule","setLike"],["Iff"],["AddCommGroup","toAddCommMonoid"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Ring"]],"valueReferences":[["Submodule","setLike"],["AddCommGroup","toAddGroup"],["add_mem_cancel_left"],["Submodule","addSubgroupClass"],["AddCommGroup","toAddCommMonoid"],["Ring","toSemiring"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","coe_mk"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4209750549._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4209750549._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4209750549._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4209750549._hygCtx._hyg.10} {p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4209750549._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4209750549._hygCtx._hyg.10 module_M} (x : M) (hx : Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4209750549._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4209750549._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4209750549._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4209750549._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4209750549._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4209750549._hygCtx._hyg.10 module_M)) p x), Eq.{succ v} M (Subtype.val.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4209750549._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4209750549._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4209750549._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4209750549._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4209750549._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4209750549._hygCtx._hyg.10 module_M)) p x) (Subtype.mk.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4209750549._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4209750549._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4209750549._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4209750549._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4209750549._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4209750549._hygCtx._hyg.10 module_M)) p x) x hx)) x","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} {p : Submodule R M}\n  (x : M) (hx : x ∈ p), ↑⟨x, hx⟩ = x","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} {p : Submodule R M}\n  (x : M) (hx : x ∈ p), ↑⟨x, hx⟩ = x","typeReferences":[["AddCommMonoid"],["SetLike","instMembership"],["Submodule","setLike"],["Module"],["Membership","mem"],["Subtype","mk"],["Eq"],["Subtype","val"],["Submodule"],["Semiring"]],"valueReferences":[["rfl"],["SetLike","instMembership"],["Submodule","setLike"],["Membership","mem"],["Subtype","mk"],["Subtype","val"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","coe_mem"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4222170203._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4222170203._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4222170203._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4222170203._hygCtx._hyg.10} {p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4222170203._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4222170203._hygCtx._hyg.10 module_M} (x : Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4222170203._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4222170203._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4222170203._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4222170203._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4222170203._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4222170203._hygCtx._hyg.10 module_M)) p x)), Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4222170203._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4222170203._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4222170203._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4222170203._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4222170203._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4222170203._hygCtx._hyg.10 module_M)) p (Subtype.val.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4222170203._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4222170203._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4222170203._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4222170203._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4222170203._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4222170203._hygCtx._hyg.10 module_M)) p x) x)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} {p : Submodule R M}\n  (x : ↥p), ↑x ∈ p","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} {p : Submodule R M}\n  (x : ↥p), ↑x ∈ p","typeReferences":[["AddCommMonoid"],["SetLike","instMembership"],["Subtype"],["Submodule","setLike"],["Module"],["Membership","mem"],["Subtype","val"],["Submodule"],["Semiring"]],"valueReferences":[["SetLike","instMembership"],["Submodule","setLike"],["Membership","mem"],["Subtype","property"],["Submodule"]]},{"isProp":false,"kind":"definition","name":["Submodule","toSubMulAction"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17], (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20) -> (SubMulAction.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20)))))","typeFull":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : Module R M] → Submodule R M → SubMulAction R M","typeReadable":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : Module R M] → Submodule R M → SubMulAction R M","typeReferences":[["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["SubMulAction"],["AddZeroClass","toAddZero"],["Submodule"],["Module","toDistribMulAction"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["Submodule","toAddSubmonoid"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["AddSubsemigroup","carrier"],["Module","toDistribMulAction"],["Submodule","smul_mem'"],["MonoidWithZero","toMonoid"],["SubMulAction","mk"],["AddSubmonoid","toAddSubsemigroup"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","coe_smul","_simp_1"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10} {p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M} (r : R) (x : Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M)) p x)), Eq.{succ v} M (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M))))) r (Subtype.val.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M)) p x) x)) (Subtype.val.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M)) p x) (HSMul.hSMul.{u, v, v} R (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M)) p x)) (instHSMul.{u, v} R (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M)) p x)) (Submodule.smul.{u, u, v} R R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M p (SMulZeroClass.toSMul.{u, u} R R (AddZero.toZero.{u} R (AddZeroClass.toAddZero.{u} R (AddMonoid.toAddZeroClass.{u} R (AddMonoidWithOne.toAddMonoid.{u} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u} R (NonAssocSemiring.toAddCommMonoidWithOne.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7))))))) (DistribSMul.toSMulZeroClass.{u, u} R R (AddMonoid.toAddZeroClass.{u} R (AddMonoidWithOne.toAddMonoid.{u} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u} R (NonAssocSemiring.toAddCommMonoidWithOne.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7))))) (instDistribSMul.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7))))) (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M)))) (IsScalarTower.left.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7)) (DistribMulAction.toMulAction.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1393146857._hygCtx._hyg.10 module_M))))) r x))","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} {p : Submodule R M}\n  (r : R) (x : ↥p), r • ↑x = ↑(r • x)","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} {p : Submodule R M}\n  (r : R) (x : ↥p), r • ↑x = ↑(r • x)","typeReferences":[["Subtype"],["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["instDistribSMul"],["AddMonoidWithOne","toAddMonoid"],["Subtype","val"],["Submodule"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["IsScalarTower","left"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["Submodule","setLike"],["HSMul","hSMul"],["Submodule","smul"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Subtype"],["Membership","mem"],["Submodule","coe_smul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddMonoidWithOne","toAddMonoid"],["instDistribSMul"],["Subtype","val"],["Submodule"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Eq","symm"],["instHSMul"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["IsScalarTower","left"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["Submodule","setLike"],["HSMul","hSMul"],["Submodule","smul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","module'","_proof_6"],"typeFallback":"forall {S : Type.{u_3}} {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] {module_M : Module.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10} (p : Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 : Semiring.{u_3} S] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 : SMul.{u_3, u_2} S R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 : Module.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36 : IsScalarTower.{u_3, u_2, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (SMulZeroClass.toSMul.{u_2, u_1} R 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)))) (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32))))] (b : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)), Eq.{succ u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (SemigroupAction.toSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))) (SemigroupAction.mk.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))) (Submodule.smul.{u_3, u_2, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36) (Submodule.module'._proof_5.{u_1, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36)))) (OfNat.ofNat.{u_3} S 1 (One.toOfNat1.{u_3} S (MulOne.toOne.{u_3} S (MulOneClass.toMulOne.{u_3} S (Monoid.toMulOneClass.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))))))) b) b","typeFull":"∀ {S : Type u_3} {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) [inst_2 : Semiring S] [inst_3 : SMul S R] [inst_4 : Module S M] [inst_5 : IsScalarTower S R M]\n  (b : ↥p), 1 • b = b","typeReadable":"∀ {S : Type u_3} {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) [inst_2 : Semiring S] [inst_3 : SMul S R] [inst_4 : Module S M] [inst_5 : IsScalarTower S R M]\n  (b : ↥p), 1 • b = b","typeReferences":[["MulOneClass","toMulOne"],["Subtype"],["SemigroupAction","mk"],["IsScalarTower"],["Module"],["Membership","mem"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["SMul"],["AddCommMonoid"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["instHSMul"],["Monoid","toSemigroup"],["Eq"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["MulOne","toOne"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Module","toDistribMulAction"],["One","toOfNat1"],["Submodule","setLike"],["HSMul","hSMul"],["Submodule","smul"],["AddZero","toZero"],["Submodule","module'","_proof_5"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Submodule","toSubMulAction"],["Module","toDistribMulAction"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"],["SubMulAction","mulAction'","_proof_2"]]},{"isProp":true,"kind":"theorem","name":["Submodule","mem_toAddSubmonoid","_simp_1"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.10] (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.13) (x : M), Eq.{1} Prop (Membership.mem.{v, v} M (AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.10))) (SetLike.instMembership.{v, v} (AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.10))) M (AddSubmonoid.instSetLike.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.10)))) (Submodule.toAddSubmonoid.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.13 p) x) (Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.13) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.13) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.13)) p x)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (p : Submodule R M)\n  (x : M), (x ∈ p.toAddSubmonoid) = (x ∈ p)","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (p : Submodule R M)\n  (x : M), (x ∈ p.toAddSubmonoid) = (x ∈ p)","typeReferences":[["SetLike","instMembership"],["Submodule","toAddSubmonoid"],["Module"],["Membership","mem"],["AddSubmonoid","instSetLike"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["AddSubmonoid"],["Submodule","setLike"],["Eq"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Submodule","toAddSubmonoid"],["SetLike","instMembership"],["Submodule","setLike"],["AddSubmonoid"],["AddSubmonoid","instSetLike"],["Membership","mem"],["Submodule","mem_toAddSubmonoid"],["AddCommMonoid","toAddMonoid"],["propext"],["Submodule"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","copy","_proof_2"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.13 : Module.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10] (p : Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.13) (s : Set.{u_1} M), (Eq.{succ u_1} (Set.{u_1} M) s (SetLike.coe.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.13) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.13) p)) -> (forall (c : R) {x : M}, (Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) s x) -> (Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) s (HSMul.hSMul.{u_2, u_1, u_1} R M M (instHSMul.{u_2, u_1} R M (SMulZeroClass.toSMul.{u_2, u_1} R 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.Submodule.Defs.2619604494._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.13))))) c x)))","typeFull":"∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (p : Submodule R M)\n  (s : Set M), s = ↑p → ∀ (c : R) {x : M}, x ∈ s → c • x ∈ s","typeReadable":"∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (p : Submodule R M)\n  (s : Set M), s = ↑p → ∀ (c : R) {x : M}, x ∈ s → c • x ∈ s","typeReferences":[["Module"],["Set"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submodule"],["Set","instMembership"],["Module","toDistribMulAction"],["AddCommMonoid"],["SetLike","coe"],["Submodule","setLike"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["instHSMul"],["Eq"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["implies_congr"],["Submodule","toAddSubmonoid"],["Eq","trans"],["Eq","mp"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["congrArg"],["Submodule"],["AddSubsemigroup","carrier"],["Submodule","smul_mem'"],["MonoidWithZero","toMonoid"],["forall_congr"],["instHSMul"],["congrFun'"],["Eq"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["Set"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["Set","instMembership"],["Module","toDistribMulAction"],["SetLike","coe"],["SetLike","mem_coe","_simp_1"],["Submodule","setLike"],["HSMul","hSMul"],["id"],["AddSubmonoid","toAddSubsemigroup"],["Eq","mpr"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","toAddSubmonoid_injective"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.411564490._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.411564490._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.411564490._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.411564490._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.411564490._hygCtx._hyg.10], Function.Injective.{succ v, succ v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.411564490._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.411564490._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.411564490._hygCtx._hyg.13) (AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.411564490._hygCtx._hyg.10))) (Submodule.toAddSubmonoid.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.411564490._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.411564490._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.411564490._hygCtx._hyg.13)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n  Function.Injective Submodule.toAddSubmonoid","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n  Function.Injective Submodule.toAddSubmonoid","typeReferences":[["AddCommMonoid"],["Submodule","toAddSubmonoid"],["AddSubmonoid"],["Module"],["AddCommMonoid","toAddMonoid"],["AddMonoid","toAddZeroClass"],["Submodule"],["Function","Injective"],["Semiring"]],"valueReferences":[["SetLike","ext'_iff"],["Submodule","toAddSubmonoid"],["Set"],["AddSubmonoid","instSetLike"],["Iff","mp"],["AddCommMonoid","toAddMonoid"],["Submodule"],["SetLike","coe"],["Submodule","setLike"],["AddSubmonoid"],["Iff","mpr"],["Eq"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","mk_le_mk"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10] {S : AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10))} {S' : AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10))} (h : forall (c : R) {x : M}, (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10)) S)) x) -> (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10)) S)) (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.13))))) c x))) (h' : forall (c : R) {x : M}, (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10)) S')) x) -> (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10)) S')) (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.13))))) c x))), Iff (LE.le.{v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.13) (Preorder.toLE.{v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.13) (PartialOrder.toPreorder.{v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.13) (Submodule.instPartialOrder.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.13))) (Submodule.mk.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.13 S h) (Submodule.mk.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.13 S' h')) (LE.le.{v} (AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10))) (Preorder.toLE.{v} (AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10))) (PartialOrder.toPreorder.{v} (AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10))) (AddSubmonoid.instPartialOrder.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3610184693._hygCtx._hyg.10))))) S S')","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S S' : AddSubmonoid M}\n  (h : ∀ (c : R) {x : M}, x ∈ S.carrier → c • x ∈ S.carrier)\n  (h' : ∀ (c : R) {x : M}, x ∈ S'.carrier → c • x ∈ S'.carrier),\n  { toAddSubmonoid := S, smul_mem' := h } ≤ { toAddSubmonoid := S', smul_mem' := h' } ↔ S ≤ S'","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S S' : AddSubmonoid M}\n  (h : ∀ (c : R) {x : M}, x ∈ S.carrier → c • x ∈ S.carrier)\n  (h' : ∀ (c : R) {x : M}, x ∈ S'.carrier → c • x ∈ S'.carrier),\n  { toAddSubmonoid := S, smul_mem' := h } ≤ { toAddSubmonoid := S', smul_mem' := h' } ↔ S ≤ S'","typeReferences":[["AddSubmonoid","instPartialOrder"],["PartialOrder","toPreorder"],["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddSubsemigroup","carrier"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["Preorder","toLE"],["DistribSMul","toSMulZeroClass"],["Set"],["Submodule","instPartialOrder"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["Set","instMembership"],["Module","toDistribMulAction"],["Submodule","mk"],["AddSubmonoid"],["Iff"],["HSMul","hSMul"],["LE","le"],["AddSubmonoid","toAddSubsemigroup"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["PartialOrder","toPreorder"],["Submodule","mk"],["Submodule","instPartialOrder"],["LE","le"],["Iff","rfl"],["Preorder","toLE"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","coe_ofLinearComb"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10] (C : Set.{v} M) (nonempty : Set.Nonempty.{v} M C) (linearComb : forall (x : M), (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) C x) -> (forall (y : M), (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) C y) -> (forall (a : R) (b : R), Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) C (HAdd.hAdd.{v, v, v} M M M (instHAdd.{v} M (AddCommMagma.toAdd.{v} M (AddCommSemigroup.toAddCommMagma.{v} M (AddCommMonoid.toAddCommSemigroup.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10)))) (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.13))))) a x) (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.13))))) b y))))), Eq.{succ v} (Set.{v} M) (SetLike.coe.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.13) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.13) (Submodule.ofLinearComb.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.13 C nonempty linearComb)) C","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (C : Set M)\n  (nonempty : C.Nonempty) (linearComb : ∀ x ∈ C, ∀ y ∈ C, ∀ (a b : R), a • x + b • y ∈ C),\n  ↑(Submodule.ofLinearComb C nonempty linearComb) = C","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (C : Set M)\n  (nonempty : C.Nonempty) (linearComb : ∀ x ∈ C, ∀ y ∈ C, ∀ (a b : R), a • x + b • y ∈ C),\n  ↑(Submodule.ofLinearComb C nonempty linearComb) = C","typeReferences":[["Submodule","ofLinearComb"],["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Eq"],["DistribSMul","toSMulZeroClass"],["Set","Nonempty"],["instHAdd"],["Set"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Set","instMembership"],["HAdd","hAdd"],["Module","toDistribMulAction"],["SetLike","coe"],["AddCommMonoid","toAddCommSemigroup"],["Submodule","setLike"],["HSMul","hSMul"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["SetLike","coe"],["Submodule","ofLinearComb"],["Submodule","setLike"],["Set"],["Eq","refl"],["Submodule"]]},{"isProp":false,"kind":"definition","name":["Submodule","noConfusion"],"typeFallback":"forall {P : Sort.{u_1}} {R : Type.{u}} {M : Type.{v}} {inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 : Semiring.{u} R} {inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 : AddCommMonoid.{v} M} {inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17} {t : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20} {R' : Type.{u}} {M' : Type.{v}} {inst'._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 : Semiring.{u} R'} {inst'._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 : AddCommMonoid.{v} M'} {inst'._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 : Module.{u, v} R' M' inst'._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst'._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17} {t' : Submodule.{u, v} R' M' inst'._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst'._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst'._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20}, (Eq.{succ (succ u)} Type.{u} R R') -> (Eq.{succ (succ v)} Type.{v} M M') -> (HEq.{succ u} (Semiring.{u} R) inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 (Semiring.{u} R') inst'._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14) -> (HEq.{succ v} (AddCommMonoid.{v} M) inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 (AddCommMonoid.{v} M') inst'._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17) -> (HEq.{max (succ u) (succ v)} (Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17) inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 (Module.{u, v} R' M' inst'._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst'._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17) inst'._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20) -> (HEq.{succ v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20) t (Submodule.{u, v} R' M' inst'._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst'._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst'._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20) t') -> (Submodule.noConfusionType.{u_1, u, v} P R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 t R' M' inst'._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst'._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst'._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 t')","typeFull":"{P : Sort u_1} →\n  {R : Type u} →\n    {M : Type v} →\n      {inst : Semiring R} →\n        {inst_1 : AddCommMonoid M} →\n          {inst_2 : Module R M} →\n            {t : Submodule R M} →\n              {R' : Type u} →\n                {M' : Type v} →\n                  {inst' : Semiring R'} →\n                    {inst'_1 : AddCommMonoid M'} →\n                      {inst'_2 : Module R' M'} →\n                        {t' : Submodule R' M'} →\n                          R = R' →\n                            M = M' →\n                              inst ≍ inst' →\n                                inst_1 ≍ inst'_1 → inst_2 ≍ inst'_2 → t ≍ t' → Submodule.noConfusionType P t t'","typeReadable":"{P : Sort u_1} →\n  {R : Type u} →\n    {M : Type v} →\n      {inst : Semiring R} →\n        {inst_1 : AddCommMonoid M} →\n          {inst_2 : Module R M} →\n            {t : Submodule R M} →\n              {R' : Type u} →\n                {M' : Type v} →\n                  {inst' : Semiring R'} →\n                    {inst'_1 : AddCommMonoid M'} →\n                      {inst'_2 : Module R' M'} →\n                        {t' : Submodule R' M'} →\n                          R = R' →\n                            M = M' →\n                              inst ≍ inst' →\n                                inst_1 ≍ inst'_1 → inst_2 ≍ inst'_2 → t ≍ t' → Submodule.noConfusionType P t t'","typeReferences":[["AddCommMonoid"],["Module"],["HEq"],["Submodule","noConfusionType"],["Eq"],["Submodule"],["Semiring"]],"valueReferences":[["HEq","refl"],["Module"],["Submodule","casesOn"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["AddSubmonoid"],["eq_of_heq"],["HEq"],["Submodule","noConfusionType"],["Eq","ndrec"],["Semiring"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","smul_of_tower_mem"],"typeFallback":"forall {S : Type.{u'}} {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.10} (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.10 module_M) {x : M} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.25 : SMul.{u', u} S R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.29 : SMul.{u', v} S M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.33 : IsScalarTower.{u', u, v} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.25 (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.10 module_M)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.29] (r : S), (Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.10 module_M)) p x) -> (Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.10 module_M)) p (HSMul.hSMul.{u', v, v} S M M (instHSMul.{u', v} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1388556321._hygCtx._hyg.29) r x))","typeFull":"∀ {S : Type u'} {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) {x : M} [inst_2 : SMul S R] [inst_3 : SMul S M] [IsScalarTower S R M] (r : S), x ∈ p → r • x ∈ p","typeReadable":"∀ {S : Type u'} {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) {x : M} [inst_2 : SMul S R] [inst_3 : SMul S M] [IsScalarTower S R M] (r : S), x ∈ p → r • x ∈ p","typeReferences":[["SetLike","instMembership"],["Module"],["IsScalarTower"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submodule"],["Module","toDistribMulAction"],["AddCommMonoid"],["SMul"],["Submodule","setLike"],["HSMul","hSMul"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["Submodule","toSubMulAction"],["SubMulAction","smul_of_tower_mem"],["Module","toDistribMulAction"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"]]},{"isProp":true,"kind":"theorem","name":["Submodule","module'","_proof_3"],"typeFallback":"forall {S : Type.{u_3}} {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] {module_M : Module.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10} (p : Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 : Semiring.{u_3} S] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 : SMul.{u_3, u_2} S R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 : Module.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36 : IsScalarTower.{u_3, u_2, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (SMulZeroClass.toSMul.{u_2, u_1} R 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)))) (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32))))] (a : S) (b : S) (x : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)), Eq.{succ u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (SemigroupAction.toSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))) (MulAction.toSemigroupAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (have this : MulAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) := SubMulAction.mulAction'.{u_2, u_3, u_1} S R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7)) (DistribMulAction.toMulAction.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (DistribMulAction.toMulAction.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36 (Submodule.toSubMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p); this)))) (HAdd.hAdd.{u_3, u_3, u_3} S S S (instHAdd.{u_3} S (Distrib.toAdd.{u_3} S (NonUnitalNonAssocSemiring.toDistrib.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))))) a b) x) (HAdd.hAdd.{u_1, u_1, u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (instHAdd.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddCommMagma.toAdd.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddCommSemigroup.toAddCommMagma.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddCommMonoid.toAddCommSemigroup.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Submodule.addCommMonoid.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p))))) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (SemigroupAction.toSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))) (MulAction.toSemigroupAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (have this : MulAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) := SubMulAction.mulAction'.{u_2, u_3, u_1} S R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7)) (DistribMulAction.toMulAction.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (DistribMulAction.toMulAction.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36 (Submodule.toSubMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p); this)))) a x) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (SemigroupAction.toSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))) (MulAction.toSemigroupAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (have this : MulAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) := SubMulAction.mulAction'.{u_2, u_3, u_1} S R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7)) (DistribMulAction.toMulAction.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (DistribMulAction.toMulAction.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36 (Submodule.toSubMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p); this)))) b x))","typeFull":"∀ {S : Type u_3} {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) [inst_2 : Semiring S] [inst_3 : SMul S R] [inst_4 : Module S M] [inst_5 : IsScalarTower S R M]\n  (a b : S) (x : ↥p), (a + b) • x = a • x + b • x","typeReadable":"∀ {S : Type u_3} {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) [inst_2 : Semiring S] [inst_3 : SMul S R] [inst_4 : Module S M] [inst_5 : IsScalarTower S R M]\n  (a b : S) (x : ↥p), (a + b) • x = a • x + b • x","typeReferences":[["Subtype"],["IsScalarTower"],["Module"],["Membership","mem"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["SubMulAction","mulAction'"],["Submodule"],["SMul"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Monoid","toSemigroup"],["Eq"],["DistribSMul","toSMulZeroClass"],["Submodule","toSubMulAction"],["Distrib","toAdd"],["SetLike","instMembership"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["HAdd","hAdd"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["Submodule","setLike"],["HSMul","hSMul"],["Submodule","addCommMonoid"],["MulAction","toSemigroupAction"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Subtype"],["Eq","trans"],["Membership","mem"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Subtype","val"],["SubMulAction","mulAction'"],["congrArg"],["Submodule"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["congrFun'"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Monoid","toSemigroup"],["Eq"],["DistribSMul","toSMulZeroClass"],["Submodule","toSubMulAction"],["Distrib","toAdd"],["SetLike","instMembership"],["True"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["Subtype","ext"],["AddZeroClass","toAddZero"],["HAdd","hAdd"],["eq_self"],["Module","toDistribMulAction"],["of_eq_true"],["AddCommMonoid","toAddCommSemigroup"],["Submodule","setLike"],["HSMul","hSMul"],["Submodule","addCommMonoid"],["AddZero","toZero"],["MulAction","toSemigroupAction"],["add_smul"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"definition","name":["Submodule","addCommMonoid"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4245081335._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4245081335._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4245081335._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4245081335._hygCtx._hyg.10} (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4245081335._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4245081335._hygCtx._hyg.10 module_M), AddCommMonoid.{v} (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4245081335._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4245081335._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4245081335._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4245081335._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4245081335._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4245081335._hygCtx._hyg.10 module_M)) p x))","typeFull":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] → [inst_1 : AddCommMonoid M] → {module_M : Module R M} → (p : Submodule R M) → AddCommMonoid ↥p","typeReadable":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] → [inst_1 : AddCommMonoid M] → {module_M : Module R M} → (p : Submodule R M) → AddCommMonoid ↥p","typeReferences":[["AddCommMonoid"],["SetLike","instMembership"],["Subtype"],["Submodule","setLike"],["Module"],["Membership","mem"],["Submodule"],["Semiring"]],"valueReferences":[["Submodule","setLike"],["AddSubmonoidClass","toAddCommMonoid"],["Submodule","addSubmonoidClass"],["Submodule"]]},{"isProp":false,"kind":"definition","name":["Submodule","module'"],"typeFallback":"forall {S : Type.{u'}} {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10} (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 : Semiring.{u'} S] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 : SMul.{u', u} S R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 : Module.{u', v} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36 : IsScalarTower.{u', u, v} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)))) (SMulZeroClass.toSMul.{u', v} S M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u', v} S M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u', v} S M (MonoidWithZero.toMonoid.{u'} S (Semiring.toMonoidWithZero.{u'} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u', v} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32))))], Module.{u', v} S (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 (Submodule.addCommMonoid.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p)","typeFull":"{S : Type u'} →\n  {R : Type u} →\n    {M : Type v} →\n      [inst : Semiring R] →\n        [inst_1 : AddCommMonoid M] →\n          {module_M : Module R M} →\n            (p : Submodule R M) →\n              [inst_2 : Semiring S] → [inst_3 : SMul S R] → [inst_4 : Module S M] → [IsScalarTower S R M] → Module S ↥p","typeReadable":"{S : Type u'} →\n  {R : Type u} →\n    {M : Type v} →\n      [inst : Semiring R] →\n        [inst_1 : AddCommMonoid M] →\n          {module_M : Module R M} →\n            (p : Submodule R M) →\n              [inst_2 : Semiring S] → [inst_3 : SMul S R] → [inst_4 : Module S M] → [IsScalarTower S R M] → Module S ↥p","typeReferences":[["SetLike","instMembership"],["Subtype"],["Module"],["IsScalarTower"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submodule"],["Module","toDistribMulAction"],["AddCommMonoid"],["SMul"],["Submodule","setLike"],["MonoidWithZero","toMonoid"],["Submodule","addCommMonoid"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["Submodule","module'","_proof_6"],["Submodule","module'","_proof_10"],["Subtype"],["SemigroupAction","mk"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["Submodule","module'","_proof_9"],["Module","mk"],["MonoidWithZero","toMonoid"],["Monoid","toSemigroup"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["DistribMulAction","mk"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["Submodule","module'","_proof_7"],["Submodule","setLike"],["Submodule","smul"],["Submodule","addCommMonoid"],["Submodule","module'","_proof_8"],["AddZero","toZero"],["Submodule","module'","_proof_5"],["MulAction","mk"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","mk_eq_zero","_simp_1"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10} (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M) {x : M} (h : Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M)) p x), Eq.{1} Prop (Eq.{succ v} (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M)) p x)) (Subtype.mk.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M)) p x) x h) (OfNat.ofNat.{v} (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M)) p x)) 0 (Zero.toOfNat0.{v} (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M)) p x)) (Submodule.zero.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M p)))) (Eq.{succ v} M x (OfNat.ofNat.{v} M 0 (Zero.toOfNat0.{v} M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10)))))))","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M)\n  {x : M} (h : x ∈ p), (⟨x, h⟩ = 0) = (x = 0)","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M)\n  {x : M} (h : x ∈ p), (⟨x, h⟩ = 0) = (x = 0)","typeReferences":[["SetLike","instMembership"],["Subtype"],["Module"],["Membership","mem"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Submodule"],["AddCommMonoid"],["Submodule","setLike"],["Zero","toOfNat0"],["Subtype","mk"],["Submodule","zero"],["Eq"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Subtype"],["SetLike","instMembership"],["Membership","mem"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Submodule"],["Submodule","mk_eq_zero"],["Submodule","setLike"],["Zero","toOfNat0"],["Subtype","mk"],["Submodule","zero"],["Eq"],["AddZero","toZero"],["propext"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","eta"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.10] {p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.13} (h : forall (c : R) {x : M}, (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.10)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.10)) (Submodule.toAddSubmonoid.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.13 p))) x) -> (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.10)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.10)) (Submodule.toAddSubmonoid.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.13 p))) (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.13))))) c x))), Eq.{succ v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.13) (Submodule.mk.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.13 (Submodule.toAddSubmonoid.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3983214305._hygCtx._hyg.13 p) h) p","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {p : Submodule R M}\n  (h : ∀ (c : R) {x : M}, x ∈ p.carrier → c • x ∈ p.carrier), { toAddSubmonoid := p.toAddSubmonoid, smul_mem' := h } = p","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {p : Submodule R M}\n  (h : ∀ (c : R) {x : M}, x ∈ p.carrier → c • x ∈ p.carrier), { toAddSubmonoid := p.toAddSubmonoid, smul_mem' := h } = p","typeReferences":[["Submodule","toAddSubmonoid"],["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddSubsemigroup","carrier"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["DistribSMul","toSMulZeroClass"],["Set"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["Set","instMembership"],["Module","toDistribMulAction"],["Submodule","mk"],["HSMul","hSMul"],["AddSubmonoid","toAddSubsemigroup"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["rfl"],["Submodule","toAddSubmonoid"],["Submodule","mk"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","toSubMulAction_injective"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3010045949._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3010045949._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3010045949._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3010045949._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3010045949._hygCtx._hyg.10], Function.Injective.{succ v, succ v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3010045949._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3010045949._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3010045949._hygCtx._hyg.13) (SubMulAction.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3010045949._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3010045949._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3010045949._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3010045949._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3010045949._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3010045949._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3010045949._hygCtx._hyg.13))))) (Submodule.toSubMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3010045949._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3010045949._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3010045949._hygCtx._hyg.13)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n  Function.Injective Submodule.toSubMulAction","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n  Function.Injective Submodule.toSubMulAction","typeReferences":[["Submodule","toSubMulAction"],["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["SubMulAction"],["AddZeroClass","toAddZero"],["Submodule"],["Module","toDistribMulAction"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["AddZero","toZero"],["Semiring"],["Function","Injective"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["Submodule","toSubMulAction"],["SetLike","ext'_iff"],["SubMulAction","instSetLike"],["Set"],["DistribMulAction","toDistribSMul"],["Iff","mp"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["SubMulAction"],["AddZeroClass","toAddZero"],["Submodule"],["Module","toDistribMulAction"],["SetLike","coe"],["Submodule","setLike"],["Iff","mpr"],["MonoidWithZero","toMonoid"],["Eq"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"definition","name":["Submodule","_sizeOf_1"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} {inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 : Semiring.{u} R} {inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 : AddCommMonoid.{v} M} {inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.32 : SizeOf.{succ u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.33 : SizeOf.{succ v} M], (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20) -> Nat","typeFull":"{R : Type u} →\n  {M : Type v} →\n    {inst : Semiring R} →\n      {inst_1 : AddCommMonoid M} → {inst_2 : Module R M} → [SizeOf R] → [SizeOf M] → Submodule R M → ℕ","typeReadable":"{R : Type u} →\n  {M : Type v} →\n    {inst : Semiring R} →\n      {inst_1 : AddCommMonoid M} → {inst_2 : Module R M} → [SizeOf R] → [SizeOf M] → Submodule R M → ℕ","typeReferences":[["Nat"],["AddCommMonoid"],["Module"],["SizeOf"],["Submodule"],["Semiring"]],"valueReferences":[["instAddNat"],["HAdd","hAdd"],["SizeOf","sizeOf"],["Nat"],["AddSubmonoid","_sizeOf_inst"],["AddSubmonoid"],["instOfNatNat"],["instHAdd"],["Submodule","rec"],["AddCommMonoid","toAddMonoid"],["OfNat","ofNat"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"definition","name":["Submodule","toAddSubmonoid"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17], (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20) -> (AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))","typeFull":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : Module R M] → Submodule R M → AddSubmonoid M","typeReadable":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : Module R M] → Submodule R M → AddSubmonoid M","typeReferences":[["AddCommMonoid"],["AddSubmonoid"],["Module"],["AddCommMonoid","toAddMonoid"],["AddMonoid","toAddZeroClass"],["Submodule"],["Semiring"]],"valueReferences":[]},{"isProp":false,"kind":"definition","name":["Submodule","inhabited"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2802996212._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2802996212._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2802996212._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2802996212._hygCtx._hyg.10} (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2802996212._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2802996212._hygCtx._hyg.10 module_M), Inhabited.{succ v} (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2802996212._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2802996212._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2802996212._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2802996212._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2802996212._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2802996212._hygCtx._hyg.10 module_M)) p x))","typeFull":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] → [inst_1 : AddCommMonoid M] → {module_M : Module R M} → (p : Submodule R M) → Inhabited ↥p","typeReadable":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] → [inst_1 : AddCommMonoid M] → {module_M : Module R M} → (p : Submodule R M) → Inhabited ↥p","typeReferences":[["AddCommMonoid"],["SetLike","instMembership"],["Subtype"],["Submodule","setLike"],["Module"],["Membership","mem"],["Inhabited"],["Submodule"],["Semiring"]],"valueReferences":[["SetLike","instMembership"],["Subtype"],["Inhabited","mk"],["Submodule","setLike"],["Membership","mem"],["Zero","toOfNat0"],["Submodule","zero"],["OfNat","ofNat"],["Submodule"]]},{"isProp":false,"kind":"definition","name":["SMulMemClass","toModule'"],"typeFallback":"forall (S : Type.{u_2}) (R' : Type.{u_3}) (R : Type.{u_4}) (A : Type.{u_5}) [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.36 : Semiring.{u_4} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39 : NonUnitalNonAssocSemiring.{u_5} A] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.42 : Module.{u_4, u_5} R A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.36 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39)] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.46 : Semiring.{u_3} R'] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.49 : SMul.{u_3, u_4} R' R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.53 : Module.{u_3, u_5} R' A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.46 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39)] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.57 : IsScalarTower.{u_3, u_4, u_5} R' R A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.49 (SMulZeroClass.toSMul.{u_4, u_5} R A (AddZero.toZero.{u_5} A (AddZeroClass.toAddZero.{u_5} A (AddMonoid.toAddZeroClass.{u_5} A (AddCommMonoid.toAddMonoid.{u_5} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39))))) (DistribSMul.toSMulZeroClass.{u_4, u_5} R A (AddMonoid.toAddZeroClass.{u_5} A (AddCommMonoid.toAddMonoid.{u_5} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39))) (DistribMulAction.toDistribSMul.{u_4, u_5} R A (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.36)) (AddCommMonoid.toAddMonoid.{u_5} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39)) (Module.toDistribMulAction.{u_4, u_5} R A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.36 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39) inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.42)))) (SMulZeroClass.toSMul.{u_3, u_5} R' A (AddZero.toZero.{u_5} A (AddZeroClass.toAddZero.{u_5} A (AddMonoid.toAddZeroClass.{u_5} A (AddCommMonoid.toAddMonoid.{u_5} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39))))) (DistribSMul.toSMulZeroClass.{u_3, u_5} R' A (AddMonoid.toAddZeroClass.{u_5} A (AddCommMonoid.toAddMonoid.{u_5} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39))) (DistribMulAction.toDistribSMul.{u_3, u_5} R' A (MonoidWithZero.toMonoid.{u_3} R' (Semiring.toMonoidWithZero.{u_3} R' inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.46)) (AddCommMonoid.toAddMonoid.{u_5} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39)) (Module.toDistribMulAction.{u_3, u_5} R' A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.46 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39) inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.53))))] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.62 : SetLike.{u_2, u_5} S A] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.66 : AddSubmonoidClass.{u_2, u_5} S A (AddMonoid.toAddZeroClass.{u_5} A (AddCommMonoid.toAddMonoid.{u_5} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.62] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.70 : SMulMemClass.{u_2, u_4, u_5} S R A (SMulZeroClass.toSMul.{u_4, u_5} R A (AddZero.toZero.{u_5} A (AddZeroClass.toAddZero.{u_5} A (AddMonoid.toAddZeroClass.{u_5} A (AddCommMonoid.toAddMonoid.{u_5} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39))))) (DistribSMul.toSMulZeroClass.{u_4, u_5} R A (AddMonoid.toAddZeroClass.{u_5} A (AddCommMonoid.toAddMonoid.{u_5} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39))) (DistribMulAction.toDistribSMul.{u_4, u_5} R A (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.36)) (AddCommMonoid.toAddMonoid.{u_5} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39)) (Module.toDistribMulAction.{u_4, u_5} R A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.36 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39) inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.42)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.62] (s : S), Module.{u_3, u_5} R' (Subtype.{succ u_5} A (fun (x : A) => Membership.mem.{u_5, u_2} A S (SetLike.instMembership.{u_2, u_5} S A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.62) s x)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.46 (AddSubmonoidClass.toAddCommMonoid.{u_2, u_5} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_5} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39) S inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.62 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.66 s)","typeFull":"(S : Type u_2) →\n  (R' : Type u_3) →\n    (R : Type u_4) →\n      (A : Type u_5) →\n        [inst : Semiring R] →\n          [inst_1 : NonUnitalNonAssocSemiring A] →\n            [inst_2 : Module R A] →\n              [inst_3 : Semiring R'] →\n                [inst_4 : SMul R' R] →\n                  [inst_5 : Module R' A] →\n                    [IsScalarTower R' R A] →\n                      [inst_7 : SetLike S A] →\n                        [inst_8 : AddSubmonoidClass S A] → [SMulMemClass S R A] → (s : S) → Module R' ↥s","typeReadable":"(S : Type u_2) →\n  (R' : Type u_3) →\n    (R : Type u_4) →\n      (A : Type u_5) →\n        [inst : Semiring R] →\n          [inst_1 : NonUnitalNonAssocSemiring A] →\n            [inst_2 : Module R A] →\n              [inst_3 : Semiring R'] →\n                [inst_4 : SMul R' R] →\n                  [inst_5 : Module R' A] →\n                    [IsScalarTower R' R A] →\n                      [inst_7 : SetLike S A] →\n                        [inst_8 : AddSubmonoidClass S A] → [SMulMemClass S R A] → (s : S) → Module R' ↥s","typeReferences":[["SMulMemClass"],["SetLike","instMembership"],["Subtype"],["Module"],["IsScalarTower"],["Membership","mem"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddSubmonoidClass","toAddCommMonoid"],["Module","toDistribMulAction"],["SMul"],["MonoidWithZero","toMonoid"],["SetLike"],["AddSubmonoidClass"],["AddZero","toZero"],["NonUnitalNonAssocSemiring"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["NonUnitalNonAssocSemiring","toAddCommMonoid"],["SMulMemClass","toModule"],["SMulMemClass","toModule'","_proof_1"]]},{"isProp":true,"kind":"theorem","name":["Submodule","mem_toAddSubgroup"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3553177071._hygCtx._hyg.7 : Ring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3553177071._hygCtx._hyg.10 : AddCommGroup.{v} M] {module_M : Module.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3553177071._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3553177071._hygCtx._hyg.10)} (p : Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3553177071._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3553177071._hygCtx._hyg.10) module_M) {x : M}, Iff (Membership.mem.{v, v} M (AddSubgroup.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3553177071._hygCtx._hyg.10)) (SetLike.instMembership.{v, v} (AddSubgroup.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3553177071._hygCtx._hyg.10)) M (AddSubgroup.instSetLike.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3553177071._hygCtx._hyg.10))) (Submodule.toAddSubgroup.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3553177071._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3553177071._hygCtx._hyg.10 module_M p) x) (Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3553177071._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3553177071._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3553177071._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3553177071._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3553177071._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3553177071._hygCtx._hyg.10) module_M)) p x)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M} (p : Submodule R M)\n  {x : M}, x ∈ p.toAddSubgroup ↔ x ∈ p","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M} (p : Submodule R M)\n  {x : M}, x ∈ p.toAddSubgroup ↔ x ∈ p","typeReferences":[["SetLike","instMembership"],["Module"],["AddCommGroup","toAddGroup"],["Membership","mem"],["AddSubgroup","instSetLike"],["AddCommGroup"],["Submodule","toAddSubgroup"],["Ring","toSemiring"],["Submodule"],["AddSubgroup"],["Submodule","setLike"],["Iff"],["AddCommGroup","toAddCommMonoid"],["Ring"]],"valueReferences":[["AddSubgroup"],["SetLike","instMembership"],["AddCommGroup","toAddGroup"],["AddSubgroup","instSetLike"],["Membership","mem"],["Submodule","toAddSubgroup"],["Iff","rfl"]]},{"isProp":false,"kind":"definition","name":["Submodule","addCommGroup"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.405133445._hygCtx._hyg.7 : Ring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.405133445._hygCtx._hyg.10 : AddCommGroup.{v} M] {module_M : Module.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.405133445._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.405133445._hygCtx._hyg.10)} (p : Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.405133445._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.405133445._hygCtx._hyg.10) module_M), AddCommGroup.{v} (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.405133445._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.405133445._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.405133445._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.405133445._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.405133445._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.405133445._hygCtx._hyg.10) module_M)) p x))","typeFull":"{R : Type u} →\n  {M : Type v} →\n    [inst : Ring R] → [inst_1 : AddCommGroup M] → {module_M : Module R M} → (p : Submodule R M) → AddCommGroup ↥p","typeReadable":"{R : Type u} →\n  {M : Type v} →\n    [inst : Ring R] → [inst_1 : AddCommGroup M] → {module_M : Module R M} → (p : Submodule R M) → AddCommGroup ↥p","typeReferences":[["SetLike","instMembership"],["Subtype"],["Submodule","setLike"],["Module"],["Membership","mem"],["AddCommGroup","toAddCommMonoid"],["AddCommGroup"],["Submodule"],["Ring","toSemiring"],["Ring"]],"valueReferences":[["Submodule","setLike"],["Submodule","addSubgroupClass"],["AddCommGroup","toAddCommMonoid"],["AddSubgroupClass","toAddCommGroup"],["Ring","toSemiring"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["SubmoduleClass","module'","_proof_5"],"typeFallback":"forall {S : Type.{u_3}} {R : Type.{u_4}} {M : Type.{u_1}} {T : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 : Semiring.{u_4} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 : Semiring.{u_3} S] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17 : Module.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.21 : SMul.{u_3, u_4} S R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.25 : Module.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.29 : IsScalarTower.{u_3, u_4, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.21 (SMulZeroClass.toSMul.{u_4, u_1} R 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_4, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_4, u_1} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17)))) (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.25))))] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34 : SetLike.{u_2, u_1} T M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.38 : AddSubmonoidClass.{u_2, u_1} T M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.42 : SMulMemClass.{u_2, u_4, u_1} T R M (SMulZeroClass.toSMul.{u_4, u_1} R 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_4, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_4, u_1} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34] (t : T) (x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.88 : S) (x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.90 : S) (x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.92 : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)), Eq.{succ u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (SetLike.smul'.{u_2, u_3, u_4, u_1} T S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.21 (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.25)))) (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (DistribMulAction.toMulAction.{u_4, u_1} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.42 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.29 t)) (HAdd.hAdd.{u_3, u_3, u_3} S S S (instHAdd.{u_3} S (Distrib.toAdd.{u_3} S (NonUnitalNonAssocSemiring.toDistrib.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14))))) x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.88 x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.90) x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.92) (HAdd.hAdd.{u_1, u_1, u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (instHAdd.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (AddCommMagma.toAdd.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (AddCommSemigroup.toAddCommMagma.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (AddCommMonoid.toAddCommSemigroup.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (AddSubmonoidClass.toAddCommMonoid.{u_2, u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 T inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.38 t))))) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (SetLike.smul'.{u_2, u_3, u_4, u_1} T S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.21 (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.25)))) (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (DistribMulAction.toMulAction.{u_4, u_1} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.42 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.29 t)) x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.88 x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.92) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (SetLike.smul'.{u_2, u_3, u_4, u_1} T S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.21 (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.25)))) (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (DistribMulAction.toMulAction.{u_4, u_1} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.42 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.29 t)) x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.90 x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.92))","typeFull":"∀ {S : Type u_3} {R : Type u_4} {M : Type u_1} {T : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : Semiring S] [inst_3 : Module R M] [inst_4 : SMul S R] [inst_5 : Module S M] [inst_6 : IsScalarTower S R M]\n  [inst_7 : SetLike T M] [inst_8 : AddSubmonoidClass T M] [inst_9 : SMulMemClass T R M] (t : T) (x x_1 : S) (x_2 : ↥t),\n  (x + x_1) • x_2 = x • x_2 + x_1 • x_2","typeReadable":"∀ {S : Type u_3} {R : Type u_4} {M : Type u_1} {T : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : Semiring S] [inst_3 : Module R M] [inst_4 : SMul S R] [inst_5 : Module S M] [inst_6 : IsScalarTower S R M]\n  [inst_7 : SetLike T M] [inst_8 : AddSubmonoidClass T M] [inst_9 : SMulMemClass T R M] (t : T) (x x_1 : S) (x_2 : ↥t),\n  (x + x_1) • x_2 = x • x_2 + x_1 • x_2","typeReferences":[["Subtype"],["SetLike","smul'"],["IsScalarTower"],["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["SMul"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["SetLike"],["Eq"],["AddSubmonoidClass"],["DistribSMul","toSMulZeroClass"],["SMulMemClass"],["Distrib","toAdd"],["SetLike","instMembership"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["AddSubmonoidClass","toAddCommMonoid"],["HAdd","hAdd"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Subtype"],["Eq","trans"],["SetLike","smul'"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Subtype","val"],["congrArg"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["congrFun'"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Eq"],["AddSemigroup","toAdd"],["DistribSMul","toSMulZeroClass"],["Distrib","toAdd"],["SetLike","instMembership"],["True"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Subtype","ext"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["AddSubmonoidClass","toAddCommMonoid"],["HAdd","hAdd"],["eq_self"],["Module","toDistribMulAction"],["of_eq_true"],["AddCommMonoid","toAddCommSemigroup"],["AddMonoid","toAddSemigroup"],["HSMul","hSMul"],["AddZero","toZero"],["add_smul"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","copy_eq"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4016340264._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4016340264._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4016340264._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4016340264._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4016340264._hygCtx._hyg.10] (S : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4016340264._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4016340264._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4016340264._hygCtx._hyg.13) (s : Set.{v} M) (hs : Eq.{succ v} (Set.{v} M) s (SetLike.coe.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4016340264._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4016340264._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4016340264._hygCtx._hyg.13) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4016340264._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4016340264._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4016340264._hygCtx._hyg.13) S)), Eq.{succ v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4016340264._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4016340264._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4016340264._hygCtx._hyg.13) (Submodule.copy.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4016340264._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4016340264._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4016340264._hygCtx._hyg.13 S s hs) S","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (S : Submodule R M)\n  (s : Set M) (hs : s = ↑S), S.copy s hs = S","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (S : Submodule R M)\n  (s : Set M) (hs : s = ↑S), S.copy s hs = S","typeReferences":[["AddCommMonoid"],["SetLike","coe"],["Submodule","copy"],["Submodule","setLike"],["Set"],["Module"],["Eq"],["Submodule"],["Semiring"]],"valueReferences":[["SetLike","coe_injective"],["Submodule","copy"],["Submodule","setLike"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","copy","_proof_3"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.13 : Module.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10] (p : Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.13) (s : Set.{u_1} M), (Eq.{succ u_1} (Set.{u_1} M) s (SetLike.coe.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.13) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.13) p)) -> (forall {a._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.45 : M} {b._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.46 : M}, (Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) s a._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.45) -> (Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) s b._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.46) -> (Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) s (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10))))) a._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.45 b._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.46)))","typeFull":"∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (p : Submodule R M)\n  (s : Set M), s = ↑p → ∀ {a b : M}, a ∈ s → b ∈ s → a + b ∈ s","typeReadable":"∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (p : Submodule R M)\n  (s : Set M), s = ↑p → ∀ {a b : M}, a ∈ s → b ∈ s → a + b ∈ s","typeReferences":[["Module"],["Set"],["instHAdd"],["Membership","mem"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["Submodule"],["Set","instMembership"],["HAdd","hAdd"],["AddCommMonoid"],["SetLike","coe"],["Submodule","setLike"],["Eq"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Submodule","toAddSubmonoid"],["Set"],["instHAdd"],["Membership","mem"],["AddSubsemigroup","add_mem'"],["AddCommMonoid","toAddMonoid"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["Submodule"],["Set","instMembership"],["HAdd","hAdd"],["SetLike","coe"],["Submodule","setLike"],["Eq","symm"],["AddSubmonoid","toAddSubsemigroup"],["Eq","rec"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"definition","name":["Submodule","ofLinearComb"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10] (C : Set.{v} M), (Set.Nonempty.{v} M C) -> (forall (x : M), (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) C x) -> (forall (y : M), (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) C y) -> (forall (a : R) (b : R), Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) C (HAdd.hAdd.{v, v, v} M M M (instHAdd.{v} M (AddCommMagma.toAdd.{v} M (AddCommSemigroup.toAddCommMagma.{v} M (AddCommMonoid.toAddCommSemigroup.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10)))) (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.13))))) a x) (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.13))))) b y))))) -> (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.13)","typeFull":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] →\n      [inst_1 : AddCommMonoid M] →\n        [inst_2 : Module R M] →\n          (C : Set M) → C.Nonempty → (∀ x ∈ C, ∀ y ∈ C, ∀ (a b : R), a • x + b • y ∈ C) → Submodule R M","typeReadable":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] →\n      [inst_1 : AddCommMonoid M] →\n        [inst_2 : Module R M] →\n          (C : Set M) → C.Nonempty → (∀ x ∈ C, ∀ y ∈ C, ∀ (a b : R), a • x + b • y ∈ C) → Submodule R M","typeReferences":[["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["DistribSMul","toSMulZeroClass"],["Set","Nonempty"],["instHAdd"],["Set"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Set","instMembership"],["HAdd","hAdd"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["HSMul","hSMul"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["AddSubsemigroup","mk"],["Submodule","ofLinearComb","_proof_3"],["Submodule","ofLinearComb","_proof_1"],["Submodule","ofLinearComb","_proof_2"],["Submodule","mk"],["AddSubmonoid","mk"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","mem_mk"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10] {S : AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10))} {x : M} (h : forall (c : R) {x : M}, (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10)) S)) x) -> (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10)) S)) (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.13))))) c x))), Iff (Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.13) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.13) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.13)) (Submodule.mk.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.13 S h) x) (Membership.mem.{v, v} M (AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10))) (SetLike.instMembership.{v, v} (AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10))) M (AddSubmonoid.instSetLike.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3030241945._hygCtx._hyg.10)))) S x)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : AddSubmonoid M}\n  {x : M} (h : ∀ (c : R) {x : M}, x ∈ S.carrier → c • x ∈ S.carrier),\n  x ∈ { toAddSubmonoid := S, smul_mem' := h } ↔ x ∈ S","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : AddSubmonoid M}\n  {x : M} (h : ∀ (c : R) {x : M}, x ∈ S.carrier → c • x ∈ S.carrier),\n  x ∈ { toAddSubmonoid := S, smul_mem' := h } ↔ x ∈ S","typeReferences":[["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddSubsemigroup","carrier"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["Set"],["AddSubmonoid","instSetLike"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["Set","instMembership"],["Module","toDistribMulAction"],["Submodule","mk"],["Submodule","setLike"],["AddSubmonoid"],["Iff"],["HSMul","hSMul"],["AddSubmonoid","toAddSubsemigroup"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["SetLike","instMembership"],["Submodule","mk"],["Submodule","setLike"],["Membership","mem"],["Iff","rfl"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","add_mem"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2448538266._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2448538266._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2448538266._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2448538266._hygCtx._hyg.10} (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2448538266._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2448538266._hygCtx._hyg.10 module_M) {x : M} {y : M}, (Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2448538266._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2448538266._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2448538266._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2448538266._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2448538266._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2448538266._hygCtx._hyg.10 module_M)) p x) -> (Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2448538266._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2448538266._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2448538266._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2448538266._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2448538266._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2448538266._hygCtx._hyg.10 module_M)) p y) -> (Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2448538266._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2448538266._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2448538266._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2448538266._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2448538266._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2448538266._hygCtx._hyg.10 module_M)) p (HAdd.hAdd.{v, v, v} M M M (instHAdd.{v} M (AddCommMagma.toAdd.{v} M (AddCommSemigroup.toAddCommMagma.{v} M (AddCommMonoid.toAddCommSemigroup.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2448538266._hygCtx._hyg.10)))) x y))","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M)\n  {x y : M}, x ∈ p → y ∈ p → x + y ∈ p","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M)\n  {x y : M}, x ∈ p → y ∈ p → x + y ∈ p","typeReferences":[["HAdd","hAdd"],["AddCommMonoid"],["SetLike","instMembership"],["AddCommMonoid","toAddCommSemigroup"],["instHAdd"],["Submodule","setLike"],["Module"],["Membership","mem"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Submodule"],["Semiring"]],"valueReferences":[["AddSubmonoidClass","toAddMemClass"],["AddCommMonoid","toAddCommSemigroup"],["Submodule","setLike"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["AddMemClass","add_mem"],["AddCommMonoid","toAddMonoid"],["Submodule","addSubmonoidClass"],["AddMonoid","toAddZeroClass"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","addSubmonoidClass"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1805911574._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1805911574._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1805911574._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1805911574._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1805911574._hygCtx._hyg.10], AddSubmonoidClass.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1805911574._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1805911574._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1805911574._hygCtx._hyg.13) M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1805911574._hygCtx._hyg.10)) (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1805911574._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1805911574._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1805911574._hygCtx._hyg.13)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n  AddSubmonoidClass (Submodule R M) M","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n  AddSubmonoidClass (Submodule R M) M","typeReferences":[["AddCommMonoid"],["Submodule","setLike"],["Module"],["AddCommMonoid","toAddMonoid"],["AddSubmonoidClass"],["AddMonoid","toAddZeroClass"],["Submodule"],["Semiring"]],"valueReferences":[["Submodule","toAddSubmonoid"],["AddSubsemigroup","add_mem'"],["AddCommMonoid","toAddMonoid"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["Submodule"],["Submodule","setLike"],["AddSubmonoid","zero_mem'"],["ZeroMemClass","mk"],["AddMemClass","mk"],["AddSubmonoid","toAddSubsemigroup"],["AddSubmonoidClass","mk"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","module'","_proof_4"],"typeFallback":"forall {S : Type.{u_3}} {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] {module_M : Module.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10} (p : Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 : Semiring.{u_3} S] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 : SMul.{u_3, u_2} S R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 : Module.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36 : IsScalarTower.{u_3, u_2, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (SMulZeroClass.toSMul.{u_2, u_1} R 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)))) (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32))))] (a : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)), Eq.{succ u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (SemigroupAction.toSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))) (MulAction.toSemigroupAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (have this : MulAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) := SubMulAction.mulAction'.{u_2, u_3, u_1} S R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7)) (DistribMulAction.toMulAction.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (DistribMulAction.toMulAction.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36 (Submodule.toSubMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p); this)))) (OfNat.ofNat.{u_3} S 0 (Zero.toOfNat0.{u_3} S (MulZeroClass.toZero.{u_3} S (NonUnitalNonAssocSemiring.toMulZeroClass.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)))))) a) (OfNat.ofNat.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) 0 (Zero.toOfNat0.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddZero.toZero.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddZeroClass.toAddZero.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddMonoid.toAddZeroClass.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Submodule.addCommMonoid.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p)))))))","typeFull":"∀ {S : Type u_3} {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) [inst_2 : Semiring S] [inst_3 : SMul S R] [inst_4 : Module S M] [inst_5 : IsScalarTower S R M]\n  (a : ↥p), 0 • a = 0","typeReadable":"∀ {S : Type u_3} {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) [inst_2 : Semiring S] [inst_3 : SMul S R] [inst_4 : Module S M] [inst_5 : IsScalarTower S R M]\n  (a : ↥p), 0 • a = 0","typeReferences":[["Subtype"],["IsScalarTower"],["Module"],["Membership","mem"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["SubMulAction","mulAction'"],["Submodule"],["SMul"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["Zero","toOfNat0"],["Monoid","toSemigroup"],["Eq"],["DistribSMul","toSMulZeroClass"],["Submodule","toSubMulAction"],["SetLike","instMembership"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Module","toDistribMulAction"],["MulZeroClass","toZero"],["Submodule","setLike"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["HSMul","hSMul"],["Submodule","addCommMonoid"],["MulAction","toSemigroupAction"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Subtype"],["Eq","trans"],["Membership","mem"],["SemigroupAction","toSMul"],["AddCommMonoid","toAddMonoid"],["Subtype","val"],["SubMulAction","mulAction'"],["congrArg"],["Submodule"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["congrFun'"],["Zero","toOfNat0"],["Monoid","toSemigroup"],["Eq"],["MulActionWithZero","toSMulWithZero"],["Submodule","toSubMulAction"],["SetLike","instMembership"],["True"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Semiring","toMonoidWithZero"],["Subtype","ext"],["Module","toMulActionWithZero"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["eq_self"],["Module","toDistribMulAction"],["of_eq_true"],["Submodule","setLike"],["MulZeroClass","toZero"],["HSMul","hSMul"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["zero_smul"],["Submodule","addCommMonoid"],["MulAction","toSemigroupAction"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["SMulMemClass","toModule","_proof_4"],"typeFallback":"forall {R : Type.{u_3}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 : Semiring.{u_3} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.13 : Module.{u_3, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 : SetLike.{u_2, u_1} A M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 : AddSubmonoidClass.{u_2, u_1} A M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.26 : SMulMemClass.{u_2, u_3, u_1} A R M (SMulZeroClass.toSMul.{u_3, u_1} R 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.Submodule.Defs.1426575160._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_3, u_1} R M (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.13)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18] (S' : A) (x : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)), Eq.{succ u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (HSMul.hSMul.{u_3, u_1, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (instHSMul.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (SemigroupAction.toSMul.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (Monoid.toSemigroup.{u_3} R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (DistribMulAction.toMulAction.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddSubmonoidClass.toAddCommMonoid.{u_2, u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 S')) (DistribMulAction.mk.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddSubmonoidClass.toAddCommMonoid.{u_2, u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 S')) (SubMulAction.SMulMemClass.toMulAction.{u_3, u_1, u_2} R M (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (DistribMulAction.toMulAction.{u_3, u_1} R M (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.13)) A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.26 S') (SMulMemClass.toModule._proof_1.{u_1, u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.13 A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.26 S') (SMulMemClass.toModule._proof_2.{u_1, u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.13 A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.26 S')))))) (OfNat.ofNat.{u_3} R 0 (Zero.toOfNat0.{u_3} R (MulZeroClass.toZero.{u_3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)))))) x) (OfNat.ofNat.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) 0 (Zero.toOfNat0.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddZero.toZero.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddZeroClass.toAddZero.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddMonoid.toAddZeroClass.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddSubmonoidClass.toAddCommMonoid.{u_2, u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 S')))))))","typeFull":"∀ {R : Type u_3} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {A : Type u_2}\n  [inst_3 : SetLike A M] [inst_4 : AddSubmonoidClass A M] [inst_5 : SMulMemClass A R M] (S' : A) (x : ↥S'), 0 • x = 0","typeReadable":"∀ {R : Type u_3} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {A : Type u_2}\n  [inst_3 : SetLike A M] [inst_4 : AddSubmonoidClass A M] [inst_5 : SMulMemClass A R M] (S' : A) (x : ↥S'), 0 • x = 0","typeReferences":[["Subtype"],["Module"],["Membership","mem"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["SMulMemClass","toModule","_proof_2"],["AddCommMonoid","toAddMonoid"],["SubMulAction","SMulMemClass","toMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["SMulMemClass","toModule","_proof_1"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["Zero","toOfNat0"],["SetLike"],["Monoid","toSemigroup"],["Eq"],["AddSubmonoidClass"],["DistribSMul","toSMulZeroClass"],["SMulMemClass"],["SetLike","instMembership"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","mk"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["AddSubmonoidClass","toAddCommMonoid"],["OfNat","ofNat"],["Module","toDistribMulAction"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["HSMul","hSMul"],["AddZero","toZero"],["MulAction","toSemigroupAction"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Subtype"],["SetLike","instMembership"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["SemigroupAction","toSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Function","Injective","module","_proof_4"],["SetLike","val_smul"],["AddSubmonoidClass","toAddCommMonoid"],["SubMulAction","SMulMemClass","toMulAction"],["Module","toDistribMulAction"],["Subtype","coe_injective"],["AddSubmonoidClass","subtype"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Monoid","toSemigroup"],["AddZero","toZero"],["MulAction","toSemigroupAction"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","coe_set_mk"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.10] (S : AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.10))) (h : forall (c : R) {x : M}, (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.10)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.10)) S)) x) -> (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.10)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.10)) S)) (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.13))))) c x))), Eq.{succ v} (Set.{v} M) (SetLike.coe.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.13) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.13) (Submodule.mk.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.13 S h)) (SetLike.coe.{v, v} (AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.10))) M (AddSubmonoid.instSetLike.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.67814220._hygCtx._hyg.10))) S)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (S : AddSubmonoid M)\n  (h : ∀ (c : R) {x : M}, x ∈ S.carrier → c • x ∈ S.carrier), ↑{ toAddSubmonoid := S, smul_mem' := h } = ↑S","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (S : AddSubmonoid M)\n  (h : ∀ (c : R) {x : M}, x ∈ S.carrier → c • x ∈ S.carrier), ↑{ toAddSubmonoid := S, smul_mem' := h } = ↑S","typeReferences":[["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddSubsemigroup","carrier"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["DistribSMul","toSMulZeroClass"],["Set"],["AddSubmonoid","instSetLike"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["Set","instMembership"],["Module","toDistribMulAction"],["SetLike","coe"],["Submodule","mk"],["Submodule","setLike"],["AddSubmonoid"],["HSMul","hSMul"],["AddSubmonoid","toAddSubsemigroup"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["rfl"],["SetLike","coe"],["Submodule","mk"],["Submodule","setLike"],["Set"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","neg_mem_iff"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3865998448._hygCtx._hyg.7 : Ring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3865998448._hygCtx._hyg.10 : AddCommGroup.{v} M] {module_M : Module.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3865998448._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3865998448._hygCtx._hyg.10)} (p : Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3865998448._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3865998448._hygCtx._hyg.10) module_M) {x : M}, Iff (Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3865998448._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3865998448._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3865998448._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3865998448._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3865998448._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3865998448._hygCtx._hyg.10) module_M)) p (Neg.neg.{v} M (NegZeroClass.toNeg.{v} M (SubNegZeroMonoid.toNegZeroClass.{v} M (SubtractionMonoid.toSubNegZeroMonoid.{v} M (SubtractionCommMonoid.toSubtractionMonoid.{v} M (AddCommGroup.toDivisionAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3865998448._hygCtx._hyg.10))))) x)) (Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3865998448._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3865998448._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3865998448._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3865998448._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3865998448._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3865998448._hygCtx._hyg.10) module_M)) p x)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M} (p : Submodule R M)\n  {x : M}, -x ∈ p ↔ x ∈ p","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M} (p : Submodule R M)\n  {x : M}, -x ∈ p ↔ x ∈ p","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["SetLike","instMembership"],["Module"],["Neg","neg"],["Membership","mem"],["SubtractionCommMonoid","toSubtractionMonoid"],["AddCommGroup"],["SubNegZeroMonoid","toNegZeroClass"],["Ring","toSemiring"],["Submodule"],["NegZeroClass","toNeg"],["AddCommGroup","toDivisionAddCommMonoid"],["Submodule","setLike"],["Iff"],["AddCommGroup","toAddCommMonoid"],["Ring"]],"valueReferences":[["AddSubgroupClass","toNegMemClass"],["AddCommGroup","toDivisionAddCommMonoid"],["neg_mem_iff"],["Submodule","setLike"],["AddCommGroup","toAddGroup"],["Submodule","addSubgroupClass"],["AddCommGroup","toAddCommMonoid"],["SubtractionCommMonoid","toSubtractionMonoid"],["AddGroup","toSubNegMonoid"],["SubtractionMonoid","toInvolutiveNeg"],["Ring","toSemiring"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","ofClass","_proof_2"],"typeFallback":"forall {S : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.30 : SetLike.{u_2, u_1} S M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.34 : AddSubmonoidClass.{u_2, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.30] (s : S), Membership.mem.{u_1, u_2} M S (SetLike.instMembership.{u_2, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.30) s (OfNat.ofNat.{u_1} M 0 (Zero.toOfNat0.{u_1} 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.Submodule.Defs.99125656._hygCtx._hyg.23))))))","typeFull":"∀ {S : Type u_2} {M : Type u_1} [inst : AddCommMonoid M] [inst_1 : SetLike S M] [AddSubmonoidClass S M] (s : S), 0 ∈ s","typeReadable":"∀ {S : Type u_2} {M : Type u_1} [inst : AddCommMonoid M] [inst_1 : SetLike S M] [AddSubmonoidClass S M] (s : S), 0 ∈ s","typeReferences":[["AddCommMonoid"],["SetLike","instMembership"],["Membership","mem"],["Zero","toOfNat0"],["SetLike"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toZero"],["AddSubmonoidClass"],["OfNat","ofNat"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["ZeroMemClass","zero_mem"],["AddSubmonoidClass","toZeroMemClass"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","ofClass","_proof_1"],"typeFallback":"forall {S : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.30 : SetLike.{u_2, u_1} S M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.34 : AddSubmonoidClass.{u_2, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.30] (s : S) {a._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.53 : M} {b._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.54 : M}, (Membership.mem.{u_1, u_2} M S (SetLike.instMembership.{u_2, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.30) s a._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.53) -> (Membership.mem.{u_1, u_2} M S (SetLike.instMembership.{u_2, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.30) s b._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.54) -> (Membership.mem.{u_1, u_2} M S (SetLike.instMembership.{u_2, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.30) s (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23))))) a._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.53 b._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.54))","typeFull":"∀ {S : Type u_2} {M : Type u_1} [inst : AddCommMonoid M] [inst_1 : SetLike S M] [AddSubmonoidClass S M] (s : S)\n  {a b : M}, a ∈ s → b ∈ s → a + b ∈ s","typeReadable":"∀ {S : Type u_2} {M : Type u_1} [inst : AddCommMonoid M] [inst_1 : SetLike S M] [AddSubmonoidClass S M] (s : S)\n  {a b : M}, a ∈ s → b ∈ s → a + b ∈ s","typeReferences":[["HAdd","hAdd"],["AddCommMonoid"],["SetLike","instMembership"],["instHAdd"],["Membership","mem"],["SetLike"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["AddSubmonoidClass"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["AddSubmonoidClass","toAddMemClass"],["AddMemClass","add_mem"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","mem_carrier"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3236929189._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3236929189._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3236929189._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3236929189._hygCtx._hyg.10} (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3236929189._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3236929189._hygCtx._hyg.10 module_M) {x : M}, Iff (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3236929189._hygCtx._hyg.10)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3236929189._hygCtx._hyg.10)) (Submodule.toAddSubmonoid.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3236929189._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3236929189._hygCtx._hyg.10 module_M p))) x) (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (SetLike.coe.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3236929189._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3236929189._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3236929189._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3236929189._hygCtx._hyg.10 module_M) p) x)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M)\n  {x : M}, x ∈ p.carrier ↔ x ∈ ↑p","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M)\n  {x : M}, x ∈ p.carrier ↔ x ∈ ↑p","typeReferences":[["Submodule","toAddSubmonoid"],["Module"],["Set"],["Membership","mem"],["AddCommMonoid","toAddMonoid"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["Submodule"],["Set","instMembership"],["AddSubsemigroup","carrier"],["AddCommMonoid"],["SetLike","coe"],["Submodule","setLike"],["Iff"],["AddSubmonoid","toAddSubsemigroup"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["AddSubsemigroup","carrier"],["Submodule","toAddSubmonoid"],["Set"],["Membership","mem"],["AddSubmonoid","toAddSubsemigroup"],["AddCommMonoid","toAddMonoid"],["Iff","rfl"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["AddMonoid","toAddZeroClass"],["Set","instMembership"]]},{"isProp":true,"kind":"theorem","name":["SubmoduleClass","module'","_proof_6"],"typeFallback":"forall {S : Type.{u_3}} {R : Type.{u_4}} {M : Type.{u_1}} {T : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 : Semiring.{u_4} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 : Semiring.{u_3} S] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17 : Module.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.21 : SMul.{u_3, u_4} S R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.25 : Module.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.29 : IsScalarTower.{u_3, u_4, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.21 (SMulZeroClass.toSMul.{u_4, u_1} R 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_4, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_4, u_1} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17)))) (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.25))))] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34 : SetLike.{u_2, u_1} T M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.38 : AddSubmonoidClass.{u_2, u_1} T M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.42 : SMulMemClass.{u_2, u_4, u_1} T R M (SMulZeroClass.toSMul.{u_4, u_1} R 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_4, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_4, u_1} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34] (t : T) (x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.97 : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)), Eq.{succ u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (SetLike.smul'.{u_2, u_3, u_4, u_1} T S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.21 (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.25)))) (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (DistribMulAction.toMulAction.{u_4, u_1} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.42 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.29 t)) (OfNat.ofNat.{u_3} S 0 (Zero.toOfNat0.{u_3} S (MulZeroClass.toZero.{u_3} S (NonUnitalNonAssocSemiring.toMulZeroClass.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14)))))) x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.97) (OfNat.ofNat.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) 0 (Zero.toOfNat0.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (AddZero.toZero.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (AddZeroClass.toAddZero.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (AddMonoid.toAddZeroClass.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (AddSubmonoidClass.toAddCommMonoid.{u_2, u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 T inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.38 t)))))))","typeFull":"∀ {S : Type u_3} {R : Type u_4} {M : Type u_1} {T : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : Semiring S] [inst_3 : Module R M] [inst_4 : SMul S R] [inst_5 : Module S M] [inst_6 : IsScalarTower S R M]\n  [inst_7 : SetLike T M] [inst_8 : AddSubmonoidClass T M] [inst_9 : SMulMemClass T R M] (t : T) (x : ↥t), 0 • x = 0","typeReadable":"∀ {S : Type u_3} {R : Type u_4} {M : Type u_1} {T : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : Semiring S] [inst_3 : Module R M] [inst_4 : SMul S R] [inst_5 : Module S M] [inst_6 : IsScalarTower S R M]\n  [inst_7 : SetLike T M] [inst_8 : AddSubmonoidClass T M] [inst_9 : SMulMemClass T R M] (t : T) (x : ↥t), 0 • x = 0","typeReferences":[["Subtype"],["SetLike","smul'"],["IsScalarTower"],["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["SMul"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["Zero","toOfNat0"],["SetLike"],["Eq"],["AddSubmonoidClass"],["DistribSMul","toSMulZeroClass"],["SMulMemClass"],["SetLike","instMembership"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["AddSubmonoidClass","toAddCommMonoid"],["OfNat","ofNat"],["Module","toDistribMulAction"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Subtype"],["Eq","trans"],["SetLike","smul'"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Subtype","val"],["congrArg"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["congrFun'"],["Zero","toOfNat0"],["Eq"],["MulActionWithZero","toSMulWithZero"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["True"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Subtype","ext"],["Semiring","toMonoidWithZero"],["Module","toMulActionWithZero"],["AddZeroClass","toAddZero"],["AddSubmonoidClass","toAddCommMonoid"],["OfNat","ofNat"],["eq_self"],["Module","toDistribMulAction"],["of_eq_true"],["MulZeroClass","toZero"],["HSMul","hSMul"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["zero_smul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"definition","name":["Submodule","module"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.10} (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.10 module_M), Module.{u, v} R (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.10 module_M)) p x)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.7 (Submodule.addCommMonoid.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340096._hygCtx._hyg.10 module_M p)","typeFull":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] → [inst_1 : AddCommMonoid M] → {module_M : Module R M} → (p : Submodule R M) → Module R ↥p","typeReadable":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] → [inst_1 : AddCommMonoid M] → {module_M : Module R M} → (p : Submodule R M) → Module R ↥p","typeReferences":[["AddCommMonoid"],["SetLike","instMembership"],["Subtype"],["Submodule","setLike"],["Module"],["Membership","mem"],["Submodule","addCommMonoid"],["Submodule"],["Semiring"]],"valueReferences":[["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Submodule","module","_proof_1"],["SMulZeroClass","toSMul"],["AddZeroClass","toAddZero"],["AddMonoidWithOne","toAddMonoid"],["instDistribSMul"],["Submodule","module'"],["Semiring","toNonAssocSemiring"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["AddZero","toZero"],["NonAssocSemiring","toAddCommMonoidWithOne"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","mk","sizeOf_spec"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.36 : SizeOf.{succ u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.37 : SizeOf.{succ v} M] (toAddSubmonoid : AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17))) (smul_mem' : forall (c : R) {x : M}, (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) toAddSubmonoid)) x) -> (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) toAddSubmonoid)) (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20))))) c x))), Eq.{1} Nat (SizeOf.sizeOf.{succ v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20) (Submodule._sizeOf_inst.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.37) (Submodule.mk.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 toAddSubmonoid smul_mem')) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (SizeOf.sizeOf.{succ v} (AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17))) (AddSubmonoid._sizeOf_inst.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.37) toAddSubmonoid))","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [inst_3 : SizeOf R]\n  [inst_4 : SizeOf M] (toAddSubmonoid : AddSubmonoid M)\n  (smul_mem' : ∀ (c : R) {x : M}, x ∈ toAddSubmonoid.carrier → c • x ∈ toAddSubmonoid.carrier),\n  sizeOf { toAddSubmonoid := toAddSubmonoid, smul_mem' := smul_mem' } = 1 + sizeOf toAddSubmonoid","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [inst_3 : SizeOf R]\n  [inst_4 : SizeOf M] (toAddSubmonoid : AddSubmonoid M)\n  (smul_mem' : ∀ (c : R) {x : M}, x ∈ toAddSubmonoid.carrier → c • x ∈ toAddSubmonoid.carrier),\n  sizeOf { toAddSubmonoid := toAddSubmonoid, smul_mem' := smul_mem' } = 1 + sizeOf toAddSubmonoid","typeReferences":[["instAddNat"],["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddSubsemigroup","carrier"],["AddCommMonoid"],["instOfNatNat"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["DistribSMul","toSMulZeroClass"],["instHAdd"],["Set"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["SizeOf"],["OfNat","ofNat"],["Submodule","_sizeOf_inst"],["Set","instMembership"],["HAdd","hAdd"],["SizeOf","sizeOf"],["Module","toDistribMulAction"],["Nat"],["AddSubmonoid","_sizeOf_inst"],["Submodule","mk"],["AddSubmonoid"],["HSMul","hSMul"],["AddSubmonoid","toAddSubsemigroup"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["instAddNat"],["HAdd","hAdd"],["SizeOf","sizeOf"],["Nat"],["AddSubmonoid","_sizeOf_inst"],["AddSubmonoid"],["instOfNatNat"],["instHAdd"],["Eq","refl"],["AddCommMonoid","toAddMonoid"],["OfNat","ofNat"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"definition","name":["Submodule","ofClass"],"typeFallback":"forall {S : Type.{u_1}} {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.20 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.26 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.30 : SetLike.{u_1, u_3} S M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.34 : AddSubmonoidClass.{u_1, u_3} S M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.30] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.38 : SMulMemClass.{u_1, u_2, u_3} S R M (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.20)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.26)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.30], S -> (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.26)","typeFull":"{S : Type u_1} →\n  {R : Type u_2} →\n    {M : Type u_3} →\n      [inst : Semiring R] →\n        [inst_1 : AddCommMonoid M] →\n          [inst_2 : Module R M] →\n            [inst_3 : SetLike S M] → [AddSubmonoidClass S M] → [SMulMemClass S R M] → S → Submodule R M","typeReadable":"{S : Type u_1} →\n  {R : Type u_2} →\n    {M : Type u_3} →\n      [inst : Semiring R] →\n        [inst_1 : AddCommMonoid M] →\n          [inst_2 : Module R M] →\n            [inst_3 : SetLike S M] → [AddSubmonoidClass S M] → [SMulMemClass S R M] → S → Submodule R M","typeReferences":[["SMulMemClass"],["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submodule"],["Module","toDistribMulAction"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["SetLike"],["AddSubmonoidClass"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["AddSubsemigroup","mk"],["SetLike","coe"],["Submodule","mk"],["Submodule","ofClass","_proof_2"],["AddSubmonoid","mk"],["AddCommMonoid","toAddMonoid"],["Submodule","ofClass","_proof_3"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["Submodule","ofClass","_proof_1"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","smul_mem_iff''","_simp_1"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.10} (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.10 module_M) {r : R} {x : M} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.25 : Invertible.{u} R (Distrib.toMul.{u} R (NonUnitalNonAssocSemiring.toDistrib.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.7)))) (AddMonoidWithOne.toOne.{u} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u} R (NonAssocSemiring.toAddCommMonoidWithOne.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.7)))) r], Eq.{1} Prop (Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.10 module_M)) p (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.10 module_M))))) r x)) (Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.10 module_M)) p x)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M)\n  {r : R} {x : M} [Invertible r], (r • x ∈ p) = (x ∈ p)","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M)\n  {r : R} {x : M} [Invertible r], (r • x ∈ p) = (x ∈ p)","typeReferences":[["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["Invertible"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["NonUnitalNonAssocSemiring","toDistrib"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Distrib","toMul"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["Submodule","setLike"],["AddMonoidWithOne","toOne"],["HSMul","hSMul"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["SetLike","instMembership"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Submodule","smul_mem_iff''"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submodule"],["Module","toDistribMulAction"],["Submodule","setLike"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["instHSMul"],["AddZero","toZero"],["propext"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","mk_eq_zero"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10} (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M) {x : M} (h : Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M)) p x), Iff (Eq.{succ v} (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M)) p x)) (Subtype.mk.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M)) p x) x h) (OfNat.ofNat.{v} (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M)) p x)) 0 (Zero.toOfNat0.{v} (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M)) p x)) (Submodule.zero.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10 module_M p)))) (Eq.{succ v} M x (OfNat.ofNat.{v} M 0 (Zero.toOfNat0.{v} M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2432357411._hygCtx._hyg.10)))))))","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M)\n  {x : M} (h : x ∈ p), ⟨x, h⟩ = 0 ↔ x = 0","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M)\n  {x : M} (h : x ∈ p), ⟨x, h⟩ = 0 ↔ x = 0","typeReferences":[["SetLike","instMembership"],["Subtype"],["Module"],["Membership","mem"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Submodule"],["AddCommMonoid"],["Submodule","setLike"],["Iff"],["Zero","toOfNat0"],["Submodule","zero"],["Subtype","mk"],["Eq"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Subtype"],["SetLike","instMembership"],["Submodule","setLike"],["Subtype","ext_iff"],["Membership","mem"],["Zero","toOfNat0"],["Submodule","zero"],["Subtype","mk"],["OfNat","ofNat"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["SMulMemClass","toModule","_proof_1"],"typeFallback":"forall {R : Type.{u_3}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 : Semiring.{u_3} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.13 : Module.{u_3, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 : SetLike.{u_2, u_1} A M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 : AddSubmonoidClass.{u_2, u_1} A M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.26 : SMulMemClass.{u_2, u_3, u_1} A R M (SMulZeroClass.toSMul.{u_3, u_1} R 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.Submodule.Defs.1426575160._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_3, u_1} R M (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.13)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18] (S' : A) (a : R), Eq.{succ u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (HSMul.hSMul.{u_3, u_1, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (instHSMul.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (SemigroupAction.toSMul.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (Monoid.toSemigroup.{u_3} R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (SubMulAction.SMulMemClass.toMulAction.{u_3, u_1, u_2} R M (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (DistribMulAction.toMulAction.{u_3, u_1} R M (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.13)) A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.26 S')))) a (OfNat.ofNat.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) 0 (Zero.toOfNat0.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddZero.toZero.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddZeroClass.toAddZero.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddMonoid.toAddZeroClass.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddSubmonoidClass.toAddCommMonoid.{u_2, u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 S')))))))) (OfNat.ofNat.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) 0 (Zero.toOfNat0.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddZero.toZero.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddZeroClass.toAddZero.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddMonoid.toAddZeroClass.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddSubmonoidClass.toAddCommMonoid.{u_2, u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 S')))))))","typeFull":"∀ {R : Type u_3} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {A : Type u_2}\n  [inst_3 : SetLike A M] [inst_4 : AddSubmonoidClass A M] [inst_5 : SMulMemClass A R M] (S' : A) (a : R), a • 0 = 0","typeReadable":"∀ {R : Type u_3} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {A : Type u_2}\n  [inst_3 : SetLike A M] [inst_4 : AddSubmonoidClass A M] [inst_5 : SMulMemClass A R M] (S' : A) (a : R), a • 0 = 0","typeReferences":[["Subtype"],["Module"],["Membership","mem"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["SubMulAction","SMulMemClass","toMulAction"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["Zero","toOfNat0"],["SetLike"],["Monoid","toSemigroup"],["Eq"],["AddSubmonoidClass"],["DistribSMul","toSMulZeroClass"],["SMulMemClass"],["SetLike","instMembership"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["AddSubmonoidClass","toAddCommMonoid"],["OfNat","ofNat"],["Module","toDistribMulAction"],["HSMul","hSMul"],["AddZero","toZero"],["MulAction","toSemigroupAction"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Subtype"],["SetLike","instMembership"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["SetLike","val_smul"],["AddSubmonoidClass","toAddCommMonoid"],["SubMulAction","SMulMemClass","toMulAction"],["Module","toDistribMulAction"],["Subtype","coe_injective"],["AddSubmonoidClass","subtype"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Monoid","toSemigroup"],["AddZero","toZero"],["MulAction","toSemigroupAction"],["Function","Injective","distribMulAction","_proof_2"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","coe_copy"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10] (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.13) (s : Set.{v} M) (hs : Eq.{succ v} (Set.{v} M) s (SetLike.coe.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.13) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.13) p)), Eq.{succ v} (Set.{v} M) (SetLike.coe.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.13) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.13) (Submodule.copy.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.13 p s hs)) s","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (p : Submodule R M)\n  (s : Set M) (hs : s = ↑p), ↑(p.copy s hs) = s","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (p : Submodule R M)\n  (s : Set M) (hs : s = ↑p), ↑(p.copy s hs) = s","typeReferences":[["AddCommMonoid"],["SetLike","coe"],["Submodule","copy"],["Submodule","setLike"],["Set"],["Module"],["Eq"],["Submodule"],["Semiring"]],"valueReferences":[["SetLike","coe"],["Submodule","copy"],["Submodule","setLike"],["Set"],["Eq","refl"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","ofLinearComb","_proof_1"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.13 : Module.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10] (C : Set.{u_1} M), (forall (x : M), (Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) C x) -> (forall (y : M), (Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) C y) -> (forall (a : R) (b : R), Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) C (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddCommMagma.toAdd.{u_1} M (AddCommSemigroup.toAddCommMagma.{u_1} M (AddCommMonoid.toAddCommSemigroup.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10)))) (HSMul.hSMul.{u_2, u_1, u_1} R M M (instHSMul.{u_2, u_1} R M (SMulZeroClass.toSMul.{u_2, u_1} R 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.Submodule.Defs.2678515438._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.13))))) a x) (HSMul.hSMul.{u_2, u_1, u_1} R M M (instHSMul.{u_2, u_1} R M (SMulZeroClass.toSMul.{u_2, u_1} R 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.Submodule.Defs.2678515438._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.13))))) b y))))) -> (forall {x : M} {y : M}, (Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) C x) -> (Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) C y) -> (Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) C (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10))))) x y)))","typeFull":"∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (C : Set M),\n  (∀ x ∈ C, ∀ y ∈ C, ∀ (a b : R), a • x + b • y ∈ C) → ∀ {x y : M}, x ∈ C → y ∈ C → x + y ∈ C","typeReadable":"∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (C : Set M),\n  (∀ x ∈ C, ∀ y ∈ C, ∀ (a b : R), a • x + b • y ∈ C) → ∀ {x y : M}, x ∈ C → y ∈ C → x + y ∈ C","typeReferences":[["Module"],["Set"],["instHAdd"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["Set","instMembership"],["HAdd","hAdd"],["Module","toDistribMulAction"],["AddCommMonoid"],["AddCommMonoid","toAddCommSemigroup"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["instHSMul"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Eq","mp"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["congrArg"],["Semiring","toNonAssocSemiring"],["congr"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["instHAdd"],["Set"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Set","instMembership"],["HAdd","hAdd"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["One","toOfNat1"],["one_smul"],["HSMul","hSMul"],["AddMonoidWithOne","toOne"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","toAddSubgroup_injective"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.7 : Ring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.10 : AddCommGroup.{v} M] {module_M : Module.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.10)}, Function.Injective.{succ v, succ v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.10) module_M) (AddSubgroup.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.10)) (Submodule.toAddSubgroup.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.10 module_M)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M},\n  Function.Injective Submodule.toAddSubgroup","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M},\n  Function.Injective Submodule.toAddSubgroup","typeReferences":[["AddSubgroup"],["Module"],["AddCommGroup","toAddGroup"],["AddCommGroup","toAddCommMonoid"],["AddCommGroup"],["Submodule","toAddSubgroup"],["Submodule"],["Function","Injective"],["Ring","toSemiring"],["Ring"]],"valueReferences":[["SetLike","instMembership"],["AddCommGroup","toAddGroup"],["AddSubgroup","instSetLike"],["Membership","mem"],["Iff","mp"],["_private","Mathlib","Algebra","Module","Submodule","Defs",0,"Submodule","toAddSubgroup_injective","match_1_1"],["SetLike","ext_iff"],["Submodule","toAddSubgroup"],["Submodule"],["Ring","toSemiring"],["AddSubgroup"],["Submodule","setLike"],["Iff"],["AddCommGroup","toAddCommMonoid"],["Eq"],["SetLike","ext"]]},{"isProp":true,"kind":"theorem","name":["Submodule","neg_mem"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4136279062._hygCtx._hyg.7 : Ring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4136279062._hygCtx._hyg.10 : AddCommGroup.{v} M] {module_M : Module.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4136279062._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4136279062._hygCtx._hyg.10)} (p : Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4136279062._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4136279062._hygCtx._hyg.10) module_M) {x : M}, (Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4136279062._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4136279062._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4136279062._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4136279062._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4136279062._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4136279062._hygCtx._hyg.10) module_M)) p x) -> (Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4136279062._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4136279062._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4136279062._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4136279062._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4136279062._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4136279062._hygCtx._hyg.10) module_M)) p (Neg.neg.{v} M (NegZeroClass.toNeg.{v} M (SubNegZeroMonoid.toNegZeroClass.{v} M (SubtractionMonoid.toSubNegZeroMonoid.{v} M (SubtractionCommMonoid.toSubtractionMonoid.{v} M (AddCommGroup.toDivisionAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4136279062._hygCtx._hyg.10))))) x))","typeFull":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M} (p : Submodule R M)\n  {x : M}, x ∈ p → -x ∈ p","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M} (p : Submodule R M)\n  {x : M}, x ∈ p → -x ∈ p","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["SetLike","instMembership"],["Module"],["Neg","neg"],["Membership","mem"],["SubtractionCommMonoid","toSubtractionMonoid"],["AddCommGroup"],["SubNegZeroMonoid","toNegZeroClass"],["Ring","toSemiring"],["Submodule"],["NegZeroClass","toNeg"],["AddCommGroup","toDivisionAddCommMonoid"],["Submodule","setLike"],["AddCommGroup","toAddCommMonoid"],["Ring"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["AddSubgroupClass","toNegMemClass"],["AddCommGroup","toAddGroup"],["SubtractionCommMonoid","toSubtractionMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["Submodule"],["Ring","toSemiring"],["NegZeroClass","toNeg"],["AddCommGroup","toDivisionAddCommMonoid"],["Submodule","setLike"],["Submodule","addSubgroupClass"],["AddCommGroup","toAddCommMonoid"],["AddGroup","toSubNegMonoid"],["NegMemClass","neg_mem"]]},{"isProp":true,"kind":"theorem","name":["Submodule","smul","_proof_1"],"typeFallback":"forall {S : Type.{u_3}} {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.7 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] {module_M : Module.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.10} (p : Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.10 module_M) [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.25 : SMul.{u_3, u_2} S R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.29 : SMul.{u_3, u_1} S M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.33 : IsScalarTower.{u_3, u_2, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.25 (SMulZeroClass.toSMul.{u_2, u_1} R 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.Submodule.Defs.4232235684._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.10) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.10 module_M)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.29] (c : S) (x : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.10 module_M)) p x)), Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.10 module_M)) p (HSMul.hSMul.{u_3, u_1, u_1} S M M (instHSMul.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.29) c (Subtype.val.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.10 module_M)) p x) x))","typeFull":"∀ {S : Type u_3} {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) [inst_2 : SMul S R] [inst_3 : SMul S M] [IsScalarTower S R M] (c : S) (x : ↥p), c • ↑x ∈ p","typeReadable":"∀ {S : Type u_3} {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) [inst_2 : SMul S R] [inst_3 : SMul S M] [IsScalarTower S R M] (c : S) (x : ↥p), c • ↑x ∈ p","typeReferences":[["SetLike","instMembership"],["Subtype"],["Module"],["IsScalarTower"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Subtype","val"],["Submodule"],["Module","toDistribMulAction"],["AddCommMonoid"],["SMul"],["Submodule","setLike"],["HSMul","hSMul"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["SetLike","instMembership"],["Submodule","setLike"],["Subtype","property"],["Membership","mem"],["Submodule","smul_of_tower_mem"],["Subtype","val"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","coe_smul_of_tower"],"typeFallback":"forall {S : Type.{u'}} {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10} {p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.25 : SMul.{u', u} S R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.29 : SMul.{u', v} S M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.33 : IsScalarTower.{u', u, v} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.25 (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.29] (r : S) (x : Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M)) p x)), Eq.{succ v} M (Subtype.val.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M)) p x) (HSMul.hSMul.{u', v, v} S (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M)) p x)) (instHSMul.{u', v} S (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M)) p x)) (Submodule.smul.{u', u, v} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.33)) r x)) (HSMul.hSMul.{u', v, v} S M M (instHSMul.{u', v} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.29) r (Subtype.val.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M)) p x) x))","typeFull":"∀ {S : Type u'} {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  {p : Submodule R M} [inst_2 : SMul S R] [inst_3 : SMul S M] [inst_4 : IsScalarTower S R M] (r : S) (x : ↥p),\n  ↑(r • x) = r • ↑x","typeReadable":"∀ {S : Type u'} {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  {p : Submodule R M} [inst_2 : SMul S R] [inst_3 : SMul S M] [inst_4 : IsScalarTower S R M] (r : S) (x : ↥p),\n  ↑(r • x) = r • ↑x","typeReferences":[["Subtype"],["IsScalarTower"],["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Subtype","val"],["Submodule"],["SMul"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["Submodule","setLike"],["HSMul","hSMul"],["Submodule","smul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["rfl"],["Subtype"],["SetLike","instMembership"],["Submodule","setLike"],["HSMul","hSMul"],["Membership","mem"],["instHSMul"],["Submodule","smul"],["Subtype","val"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["SMulMemClass","toModule","_proof_3"],"typeFallback":"forall {R : Type.{u_3}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 : Semiring.{u_3} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.13 : Module.{u_3, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 : SetLike.{u_2, u_1} A M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 : AddSubmonoidClass.{u_2, u_1} A M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.26 : SMulMemClass.{u_2, u_3, u_1} A R M (SMulZeroClass.toSMul.{u_3, u_1} R 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.Submodule.Defs.1426575160._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_3, u_1} R M (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.13)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18] (S' : A) (r : R) (s : R) (x : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)), Eq.{succ u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (HSMul.hSMul.{u_3, u_1, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (instHSMul.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (SemigroupAction.toSMul.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (Monoid.toSemigroup.{u_3} R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (DistribMulAction.toMulAction.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddSubmonoidClass.toAddCommMonoid.{u_2, u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 S')) (DistribMulAction.mk.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddSubmonoidClass.toAddCommMonoid.{u_2, u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 S')) (SubMulAction.SMulMemClass.toMulAction.{u_3, u_1, u_2} R M (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (DistribMulAction.toMulAction.{u_3, u_1} R M (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.13)) A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.26 S') (SMulMemClass.toModule._proof_1.{u_1, u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.13 A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.26 S') (SMulMemClass.toModule._proof_2.{u_1, u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.13 A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.26 S')))))) (HAdd.hAdd.{u_3, u_3, u_3} R R R (instHAdd.{u_3} R (Distrib.toAdd.{u_3} R (NonUnitalNonAssocSemiring.toDistrib.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7))))) r s) x) (HAdd.hAdd.{u_1, u_1, u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (instHAdd.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddCommMagma.toAdd.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddCommSemigroup.toAddCommMagma.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddCommMonoid.toAddCommSemigroup.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddSubmonoidClass.toAddCommMonoid.{u_2, u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 S'))))) (HSMul.hSMul.{u_3, u_1, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (instHSMul.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (SemigroupAction.toSMul.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (Monoid.toSemigroup.{u_3} R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (DistribMulAction.toMulAction.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddSubmonoidClass.toAddCommMonoid.{u_2, u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 S')) (DistribMulAction.mk.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddSubmonoidClass.toAddCommMonoid.{u_2, u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 S')) (SubMulAction.SMulMemClass.toMulAction.{u_3, u_1, u_2} R M (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (DistribMulAction.toMulAction.{u_3, u_1} R M (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.13)) A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.26 S') (SMulMemClass.toModule._proof_1.{u_1, u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.13 A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.26 S') (SMulMemClass.toModule._proof_2.{u_1, u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.13 A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.26 S')))))) r x) (HSMul.hSMul.{u_3, u_1, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (instHSMul.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (SemigroupAction.toSMul.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (Monoid.toSemigroup.{u_3} R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (DistribMulAction.toMulAction.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddSubmonoidClass.toAddCommMonoid.{u_2, u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 S')) (DistribMulAction.mk.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddSubmonoidClass.toAddCommMonoid.{u_2, u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 S')) (SubMulAction.SMulMemClass.toMulAction.{u_3, u_1, u_2} R M (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (DistribMulAction.toMulAction.{u_3, u_1} R M (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.13)) A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.26 S') (SMulMemClass.toModule._proof_1.{u_1, u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.13 A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.26 S') (SMulMemClass.toModule._proof_2.{u_1, u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.13 A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.26 S')))))) s x))","typeFull":"∀ {R : Type u_3} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {A : Type u_2}\n  [inst_3 : SetLike A M] [inst_4 : AddSubmonoidClass A M] [inst_5 : SMulMemClass A R M] (S' : A) (r s : R) (x : ↥S'),\n  (r + s) • x = r • x + s • x","typeReadable":"∀ {R : Type u_3} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {A : Type u_2}\n  [inst_3 : SetLike A M] [inst_4 : AddSubmonoidClass A M] [inst_5 : SMulMemClass A R M] (S' : A) (r s : R) (x : ↥S'),\n  (r + s) • x = r • x + s • x","typeReferences":[["Subtype"],["Module"],["Membership","mem"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["SMulMemClass","toModule","_proof_2"],["AddCommMonoid","toAddMonoid"],["SubMulAction","SMulMemClass","toMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["SMulMemClass","toModule","_proof_1"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["SetLike"],["Monoid","toSemigroup"],["Eq"],["AddSubmonoidClass"],["DistribSMul","toSMulZeroClass"],["SMulMemClass"],["Distrib","toAdd"],["SetLike","instMembership"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","mk"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["AddSubmonoidClass","toAddCommMonoid"],["HAdd","hAdd"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["HSMul","hSMul"],["AddZero","toZero"],["MulAction","toSemigroupAction"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Subtype"],["SetLike","instMembership"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["SemigroupAction","toSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["SetLike","val_smul"],["AddSubmonoidClass","toAddCommMonoid"],["SubMulAction","SMulMemClass","toMulAction"],["Module","toDistribMulAction"],["Subtype","coe_injective"],["AddSubmonoidClass","subtype"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Monoid","toSemigroup"],["AddZero","toZero"],["Function","Injective","module","_proof_3"],["MulAction","toSemigroupAction"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","coe_add"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10} {p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M} (x : Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M)) p x)) (y : Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M)) p x)), Eq.{succ v} M (Subtype.val.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M)) p x) (HAdd.hAdd.{v, v, v} (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M)) p x)) (instHAdd.{v} (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M)) p x)) (Submodule.add.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M p)) x y)) (HAdd.hAdd.{v, v, v} M M M (instHAdd.{v} M (AddCommMagma.toAdd.{v} M (AddCommSemigroup.toAddCommMagma.{v} M (AddCommMonoid.toAddCommSemigroup.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10)))) (Subtype.val.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M)) p x) x) (Subtype.val.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4231069628._hygCtx._hyg.10 module_M)) p x) y))","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} {p : Submodule R M}\n  (x y : ↥p), ↑(x + y) = ↑x + ↑y","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} {p : Submodule R M}\n  (x y : ↥p), ↑(x + y) = ↑x + ↑y","typeReferences":[["Subtype"],["SetLike","instMembership"],["Submodule","add"],["Module"],["instHAdd"],["Membership","mem"],["Subtype","val"],["Submodule"],["HAdd","hAdd"],["AddCommMonoid"],["AddCommMonoid","toAddCommSemigroup"],["Submodule","setLike"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Eq"],["Semiring"]],"valueReferences":[["HAdd","hAdd"],["rfl"],["Subtype"],["SetLike","instMembership"],["Submodule","add"],["instHAdd"],["Submodule","setLike"],["Membership","mem"],["Subtype","val"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","coe_neg"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.7 : Ring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.10 : AddCommGroup.{v} M] {module_M : Module.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.10)} (p : Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.10) module_M) (x : Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.10) module_M)) p x)), Eq.{succ v} M (Subtype.val.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.10) module_M)) p x) (Neg.neg.{v} (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.10) module_M)) p x)) (NegMemClass.neg.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.10) module_M) (NegZeroClass.toNeg.{v} M (SubNegZeroMonoid.toNegZeroClass.{v} M (SubtractionMonoid.toSubNegZeroMonoid.{v} M (SubtractionCommMonoid.toSubtractionMonoid.{v} M (AddCommGroup.toDivisionAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.10))))) (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.10) module_M) (AddSubgroupClass.toNegMemClass.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.10) module_M) M (AddGroup.toSubNegMonoid.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.10)) (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.10) module_M) (Submodule.addSubgroupClass.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.10 module_M)) p) x)) (Neg.neg.{v} M (NegZeroClass.toNeg.{v} M (SubNegZeroMonoid.toNegZeroClass.{v} M (SubtractionMonoid.toSubNegZeroMonoid.{v} M (SubtractionCommMonoid.toSubtractionMonoid.{v} M (AddCommGroup.toDivisionAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.10))))) (Subtype.val.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2490041572._hygCtx._hyg.10) module_M)) p x) x))","typeFull":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M} (p : Submodule R M)\n  (x : ↥p), ↑(-x) = -↑x","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M} (p : Submodule R M)\n  (x : ↥p), ↑(-x) = -↑x","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Subtype"],["SetLike","instMembership"],["AddSubgroupClass","toNegMemClass"],["Module"],["Neg","neg"],["AddCommGroup","toAddGroup"],["Membership","mem"],["SubtractionCommMonoid","toSubtractionMonoid"],["AddCommGroup"],["SubNegZeroMonoid","toNegZeroClass"],["NegMemClass","neg"],["Subtype","val"],["Ring","toSemiring"],["Submodule"],["NegZeroClass","toNeg"],["AddCommGroup","toDivisionAddCommMonoid"],["Submodule","setLike"],["Submodule","addSubgroupClass"],["AddCommGroup","toAddCommMonoid"],["AddGroup","toSubNegMonoid"],["Eq"],["Ring"]],"valueReferences":[["Submodule","setLike"],["AddCommGroup","toAddGroup"],["NegMemClass","coe_neg"],["Submodule","addSubgroupClass"],["AddCommGroup","toAddCommMonoid"],["Ring","toSemiring"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","coe_ofClass"],"typeFallback":"forall {S : Type.{u_1}} {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.20 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.26 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.30 : SetLike.{u_1, u_3} S M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.34 : AddSubmonoidClass.{u_1, u_3} S M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.30] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.38 : SMulMemClass.{u_1, u_2, u_3} S R M (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.20)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.26)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.30] (s : S), Eq.{succ u_3} (Set.{u_3} M) (SetLike.coe.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.26) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.26) (Submodule.ofClass.{u_1, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.38 s)) (SetLike.coe.{u_1, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.99125656._hygCtx._hyg.30 s)","typeFull":"∀ {S : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : SetLike S M] [inst_4 : AddSubmonoidClass S M] [inst_5 : SMulMemClass S R M] (s : S),\n  ↑(Submodule.ofClass s) = ↑s","typeReadable":"∀ {S : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : SetLike S M] [inst_4 : AddSubmonoidClass S M] [inst_5 : SMulMemClass S R M] (s : S),\n  ↑(Submodule.ofClass s) = ↑s","typeReferences":[["SMulMemClass"],["Set"],["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submodule"],["Module","toDistribMulAction"],["AddCommMonoid"],["SetLike","coe"],["Submodule","ofClass"],["Submodule","setLike"],["MonoidWithZero","toMonoid"],["SetLike"],["Eq"],["AddSubmonoidClass"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["SetLike","coe"],["Submodule","ofClass"],["Submodule","setLike"],["Set"],["Eq","refl"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","sub_mem"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4139836142._hygCtx._hyg.7 : Ring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4139836142._hygCtx._hyg.10 : AddCommGroup.{v} M] {module_M : Module.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4139836142._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4139836142._hygCtx._hyg.10)} (p : Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4139836142._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4139836142._hygCtx._hyg.10) module_M) {x : M} {y : M}, (Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4139836142._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4139836142._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4139836142._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4139836142._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4139836142._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4139836142._hygCtx._hyg.10) module_M)) p x) -> (Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4139836142._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4139836142._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4139836142._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4139836142._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4139836142._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4139836142._hygCtx._hyg.10) module_M)) p y) -> (Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4139836142._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4139836142._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4139836142._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4139836142._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4139836142._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4139836142._hygCtx._hyg.10) module_M)) p (HSub.hSub.{v, v, v} M M M (instHSub.{v} M (SubNegMonoid.toSub.{v} M (AddGroup.toSubNegMonoid.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4139836142._hygCtx._hyg.10)))) x y))","typeFull":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M} (p : Submodule R M)\n  {x y : M}, x ∈ p → y ∈ p → x - y ∈ p","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M} (p : Submodule R M)\n  {x y : M}, x ∈ p → y ∈ p → x - y ∈ p","typeReferences":[["SetLike","instMembership"],["Module"],["AddCommGroup","toAddGroup"],["Membership","mem"],["AddCommGroup"],["Ring","toSemiring"],["Submodule"],["Submodule","setLike"],["SubNegMonoid","toSub"],["AddCommGroup","toAddCommMonoid"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["instHSub"],["Ring"]],"valueReferences":[["Submodule","setLike"],["AddCommGroup","toAddGroup"],["Submodule","addSubgroupClass"],["AddCommGroup","toAddCommMonoid"],["AddGroup","toSubNegMonoid"],["sub_mem"],["Ring","toSemiring"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","module'","_proof_5"],"typeFallback":"forall {S : Type.{u_3}} {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] {module_M : Module.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10} (p : Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 : Semiring.{u_3} S] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 : SMul.{u_3, u_2} S R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 : Module.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36 : IsScalarTower.{u_3, u_2, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (SMulZeroClass.toSMul.{u_2, u_1} R 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)))) (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32))))] (x : S) (y : S) (b : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)), Eq.{succ u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Submodule.smul.{u_3, u_2, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36)) (HMul.hMul.{u_3, u_3, u_3} S S S (instHMul.{u_3} S (Semigroup.toMul.{u_3} S (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))))) x y) b) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Submodule.smul.{u_3, u_2, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36)) x (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Submodule.smul.{u_3, u_2, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36)) y b))","typeFull":"∀ {S : Type u_3} {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) [inst_2 : Semiring S] [inst_3 : SMul S R] [inst_4 : Module S M] [inst_5 : IsScalarTower S R M]\n  (x y : S) (b : ↥p), (x * y) • b = x • y • b","typeReadable":"∀ {S : Type u_3} {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) [inst_2 : Semiring S] [inst_3 : SMul S R] [inst_4 : Module S M] [inst_5 : IsScalarTower S R M]\n  (x y : S) (b : ↥p), (x * y) • b = x • y • b","typeReferences":[["Subtype"],["IsScalarTower"],["Module"],["Membership","mem"],["HMul","hMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Semigroup","toMul"],["Submodule"],["SMul"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["Monoid","toSemigroup"],["Eq"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["Submodule","setLike"],["HSMul","hSMul"],["instHMul"],["Submodule","smul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Submodule","toSubMulAction"],["Module","toDistribMulAction"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"],["SubMulAction","mulAction'","_proof_1"]]},{"isProp":true,"kind":"theorem","name":["Submodule","coe_smul_of_tower","_simp_1"],"typeFallback":"forall {S : Type.{u'}} {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10} {p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.25 : SMul.{u', u} S R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.29 : SMul.{u', v} S M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.33 : IsScalarTower.{u', u, v} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.25 (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.29] (r : S) (x : Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M)) p x)), Eq.{succ v} M (HSMul.hSMul.{u', v, v} S M M (instHSMul.{u', v} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.29) r (Subtype.val.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M)) p x) x)) (Subtype.val.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M)) p x) (HSMul.hSMul.{u', v, v} S (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M)) p x)) (instHSMul.{u', v} S (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M)) p x)) (Submodule.smul.{u', u, v} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4004309095._hygCtx._hyg.33)) r x))","typeFull":"∀ {S : Type u'} {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  {p : Submodule R M} [inst_2 : SMul S R] [inst_3 : SMul S M] [inst_4 : IsScalarTower S R M] (r : S) (x : ↥p),\n  r • ↑x = ↑(r • x)","typeReadable":"∀ {S : Type u'} {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  {p : Submodule R M} [inst_2 : SMul S R] [inst_3 : SMul S M] [inst_4 : IsScalarTower S R M] (r : S) (x : ↥p),\n  r • ↑x = ↑(r • x)","typeReferences":[["Subtype"],["IsScalarTower"],["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Subtype","val"],["Submodule"],["SMul"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["Submodule","setLike"],["HSMul","hSMul"],["Submodule","smul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Submodule","coe_smul_of_tower"],["Subtype"],["SetLike","instMembership"],["Submodule","setLike"],["HSMul","hSMul"],["Membership","mem"],["Eq","symm"],["instHSMul"],["Submodule","smul"],["Subtype","val"],["Submodule"]]},{"isProp":false,"kind":"definition","name":["Submodule","setLike"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.10], SetLike.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.13) M","typeFull":"{R : Type u} →\n  {M : Type v} → [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : Module R M] → SetLike (Submodule R M) M","typeReadable":"{R : Type u} →\n  {M : Type v} → [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : Module R M] → SetLike (Submodule R M) M","typeReferences":[["AddCommMonoid"],["Module"],["SetLike"],["Submodule"],["Semiring"]],"valueReferences":[["AddSubsemigroup","carrier"],["SetLike","mk"],["Submodule","toAddSubmonoid"],["AddSubmonoid","toAddSubsemigroup"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["Submodule","setLike","_proof_1"],["AddMonoid","toAddZeroClass"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","coe_sub"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7 : Ring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10 : AddCommGroup.{v} M] {module_M : Module.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10)} (p : Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M) (x : Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M)) p x)) (y : Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M)) p x)), Eq.{succ v} M (Subtype.val.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M)) p x) (HSub.hSub.{v, v, v} (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M)) p x)) (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M)) p x)) (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M)) p x)) (instHSub.{v} (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M)) p x)) (AddSubgroupClass.sub.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M) (AddGroup.toSubNegMonoid.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10)) (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M) (Submodule.addSubgroupClass.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10 module_M) p)) x y)) (HSub.hSub.{v, v, v} M M M (instHSub.{v} M (SubNegMonoid.toSub.{v} M (AddGroup.toSubNegMonoid.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10)))) (Subtype.val.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M)) p x) x) (Subtype.val.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1887138232._hygCtx._hyg.10) module_M)) p x) y))","typeFull":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M} (p : Submodule R M)\n  (x y : ↥p), ↑(x - y) = ↑x - ↑y","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M} (p : Submodule R M)\n  (x y : ↥p), ↑(x - y) = ↑x - ↑y","typeReferences":[["Subtype"],["SetLike","instMembership"],["AddSubgroupClass","sub"],["Module"],["AddCommGroup","toAddGroup"],["Membership","mem"],["AddCommGroup"],["Subtype","val"],["Ring","toSemiring"],["Submodule"],["Submodule","setLike"],["SubNegMonoid","toSub"],["Submodule","addSubgroupClass"],["AddCommGroup","toAddCommMonoid"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["instHSub"],["Eq"],["Ring"]],"valueReferences":[["Submodule","setLike"],["AddCommGroup","toAddGroup"],["Submodule","addSubgroupClass"],["AddCommGroup","toAddCommMonoid"],["AddSubgroupClass","coe_sub"],["Ring","toSemiring"],["Submodule"]]},{"isProp":false,"kind":"definition","name":["Submodule","toAddSubgroup"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1166923647._hygCtx._hyg.7 : Ring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1166923647._hygCtx._hyg.10 : AddCommGroup.{v} M] {module_M : Module.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1166923647._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1166923647._hygCtx._hyg.10)}, (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1166923647._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1166923647._hygCtx._hyg.10) module_M) -> (AddSubgroup.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1166923647._hygCtx._hyg.10))","typeFull":"{R : Type u} →\n  {M : Type v} → [inst : Ring R] → [inst_1 : AddCommGroup M] → {module_M : Module R M} → Submodule R M → AddSubgroup M","typeReadable":"{R : Type u} →\n  {M : Type v} → [inst : Ring R] → [inst_1 : AddCommGroup M] → {module_M : Module R M} → Submodule R M → AddSubgroup M","typeReferences":[["AddSubgroup"],["Module"],["AddCommGroup","toAddGroup"],["AddCommGroup","toAddCommMonoid"],["AddCommGroup"],["Submodule"],["Ring","toSemiring"],["Ring"]],"valueReferences":[["Submodule","neg_mem"],["Submodule","toAddSubmonoid"],["AddCommGroup","toAddGroup"],["AddCommGroup","toAddCommMonoid"],["AddSubgroup","mk"],["Ring","toSemiring"]]},{"isProp":true,"kind":"theorem","name":["Submodule","ext"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.10] {p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.13} {q : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.13}, (forall (x : M), Iff (Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.13) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.13) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.13)) p x) (Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.13) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.13) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.13)) q x)) -> (Eq.{succ v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.13) p q)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {p q : Submodule R M},\n  (∀ (x : M), x ∈ p ↔ x ∈ q) → p = q","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {p q : Submodule R M},\n  (∀ (x : M), x ∈ p ↔ x ∈ q) → p = q","typeReferences":[["AddCommMonoid"],["SetLike","instMembership"],["Submodule","setLike"],["Module"],["Iff"],["Membership","mem"],["Eq"],["Submodule"],["Semiring"]],"valueReferences":[["Submodule","setLike"],["SetLike","ext"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","toAddSubmonoid_inj","_simp_1"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.10] {p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.13} {q : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.13}, Eq.{1} Prop (Eq.{succ v} (AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.10))) (Submodule.toAddSubmonoid.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.13 p) (Submodule.toAddSubmonoid.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.13 q)) (Eq.{succ v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.13) p q)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {p q : Submodule R M},\n  (p.toAddSubmonoid = q.toAddSubmonoid) = (p = q)","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {p q : Submodule R M},\n  (p.toAddSubmonoid = q.toAddSubmonoid) = (p = q)","typeReferences":[["AddCommMonoid"],["Submodule","toAddSubmonoid"],["AddSubmonoid"],["Module"],["AddCommMonoid","toAddMonoid"],["Eq"],["AddMonoid","toAddZeroClass"],["Submodule"],["Semiring"]],"valueReferences":[["Submodule","toAddSubmonoid_inj"],["Submodule","toAddSubmonoid"],["AddSubmonoid"],["AddCommMonoid","toAddMonoid"],["Eq"],["propext"],["Submodule"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","ofLinearComb","_proof_3"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.13 : Module.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10] (C : Set.{u_1} M), (forall (x : M), (Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) C x) -> (forall (y : M), (Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) C y) -> (forall (a : R) (b : R), Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) C (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddCommMagma.toAdd.{u_1} M (AddCommSemigroup.toAddCommMagma.{u_1} M (AddCommMonoid.toAddCommSemigroup.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10)))) (HSMul.hSMul.{u_2, u_1, u_1} R M M (instHSMul.{u_2, u_1} R M (SMulZeroClass.toSMul.{u_2, u_1} R 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.Submodule.Defs.2678515438._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.13))))) a x) (HSMul.hSMul.{u_2, u_1, u_1} R M M (instHSMul.{u_2, u_1} R M (SMulZeroClass.toSMul.{u_2, u_1} R 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.Submodule.Defs.2678515438._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.13))))) b y))))) -> (forall (c : R) (x : M), (Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) C x) -> (Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) C (HSMul.hSMul.{u_2, u_1, u_1} R M M (instHSMul.{u_2, u_1} R M (SMulZeroClass.toSMul.{u_2, u_1} R 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.Submodule.Defs.2678515438._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.13))))) c x)))","typeFull":"∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (C : Set M),\n  (∀ x ∈ C, ∀ y ∈ C, ∀ (a b : R), a • x + b • y ∈ C) → ∀ (c : R), ∀ x ∈ C, c • x ∈ C","typeReadable":"∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (C : Set M),\n  (∀ x ∈ C, ∀ y ∈ C, ∀ (a b : R), a • x + b • y ∈ C) → ∀ (c : R), ∀ x ∈ C, c • x ∈ C","typeReferences":[["Module"],["Set"],["instHAdd"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Set","instMembership"],["HAdd","hAdd"],["Module","toDistribMulAction"],["AddCommMonoid"],["AddCommMonoid","toAddCommSemigroup"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["instHSMul"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Eq","trans"],["Eq","mp"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["congrArg"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Zero","toOfNat0"],["MulActionWithZero","toSMulWithZero"],["DistribSMul","toSMulZeroClass"],["instHAdd"],["Set"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["Module","toMulActionWithZero"],["add_zero"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Set","instMembership"],["HAdd","hAdd"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["MulZeroClass","toZero"],["HSMul","hSMul"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["zero_smul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"definition","name":["Submodule","recOn"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17] {motive : (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20) -> Sort.{u_1}} (t : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20), (forall (toAddSubmonoid : AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17))) (smul_mem' : forall (c : R) {x : M}, (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) toAddSubmonoid)) x) -> (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) toAddSubmonoid)) (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20))))) c x))), motive (Submodule.mk.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 toAddSubmonoid smul_mem')) -> (motive t)","typeFull":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] →\n      [inst_1 : AddCommMonoid M] →\n        [inst_2 : Module R M] →\n          {motive : Submodule R M → Sort u_1} →\n            (t : Submodule R M) →\n              ((toAddSubmonoid : AddSubmonoid M) →\n                  (smul_mem' : ∀ (c : R) {x : M}, x ∈ toAddSubmonoid.carrier → c • x ∈ toAddSubmonoid.carrier) →\n                    motive { toAddSubmonoid := toAddSubmonoid, smul_mem' := smul_mem' }) →\n                motive t","typeReadable":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] →\n      [inst_1 : AddCommMonoid M] →\n        [inst_2 : Module R M] →\n          {motive : Submodule R M → Sort u_1} →\n            (t : Submodule R M) →\n              ((toAddSubmonoid : AddSubmonoid M) →\n                  (smul_mem' : ∀ (c : R) {x : M}, x ∈ toAddSubmonoid.carrier → c • x ∈ toAddSubmonoid.carrier) →\n                    motive { toAddSubmonoid := toAddSubmonoid, smul_mem' := smul_mem' }) →\n                motive t","typeReferences":[["Module"],["Set"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["Submodule"],["Set","instMembership"],["AddSubsemigroup","carrier"],["Module","toDistribMulAction"],["AddCommMonoid"],["Submodule","mk"],["AddSubmonoid"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["instHSMul"],["AddSubmonoid","toAddSubsemigroup"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Submodule","rec"]]},{"isProp":true,"kind":"theorem","name":["SMulMemClass","toModule'","_proof_1"],"typeFallback":"forall (S : Type.{u_1}) (R' : Type.{u_2}) (R : Type.{u_4}) (A : Type.{u_3}) [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.36 : Semiring.{u_4} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39 : NonUnitalNonAssocSemiring.{u_3} A] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.42 : Module.{u_4, u_3} R A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.36 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39)] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.46 : Semiring.{u_2} R'] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.49 : SMul.{u_2, u_4} R' R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.53 : Module.{u_2, u_3} R' A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.46 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39)] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.57 : IsScalarTower.{u_2, u_4, u_3} R' R A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.49 (SMulZeroClass.toSMul.{u_4, u_3} R A (AddZero.toZero.{u_3} A (AddZeroClass.toAddZero.{u_3} A (AddMonoid.toAddZeroClass.{u_3} A (AddCommMonoid.toAddMonoid.{u_3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39))))) (DistribSMul.toSMulZeroClass.{u_4, u_3} R A (AddMonoid.toAddZeroClass.{u_3} A (AddCommMonoid.toAddMonoid.{u_3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39))) (DistribMulAction.toDistribSMul.{u_4, u_3} R A (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.36)) (AddCommMonoid.toAddMonoid.{u_3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39)) (Module.toDistribMulAction.{u_4, u_3} R A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.36 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39) inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.42)))) (SMulZeroClass.toSMul.{u_2, u_3} R' A (AddZero.toZero.{u_3} A (AddZeroClass.toAddZero.{u_3} A (AddMonoid.toAddZeroClass.{u_3} A (AddCommMonoid.toAddMonoid.{u_3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39))))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R' A (AddMonoid.toAddZeroClass.{u_3} A (AddCommMonoid.toAddMonoid.{u_3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39))) (DistribMulAction.toDistribSMul.{u_2, u_3} R' A (MonoidWithZero.toMonoid.{u_2} R' (Semiring.toMonoidWithZero.{u_2} R' inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.46)) (AddCommMonoid.toAddMonoid.{u_3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39)) (Module.toDistribMulAction.{u_2, u_3} R' A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.46 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39) inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.53))))] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.62 : SetLike.{u_1, u_3} S A] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.70 : SMulMemClass.{u_1, u_4, u_3} S R A (SMulZeroClass.toSMul.{u_4, u_3} R A (AddZero.toZero.{u_3} A (AddZeroClass.toAddZero.{u_3} A (AddMonoid.toAddZeroClass.{u_3} A (AddCommMonoid.toAddMonoid.{u_3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39))))) (DistribSMul.toSMulZeroClass.{u_4, u_3} R A (AddMonoid.toAddZeroClass.{u_3} A (AddCommMonoid.toAddMonoid.{u_3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39))) (DistribMulAction.toDistribSMul.{u_4, u_3} R A (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.36)) (AddCommMonoid.toAddMonoid.{u_3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39)) (Module.toDistribMulAction.{u_4, u_3} R A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.36 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39) inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.42)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.62], SMulMemClass.{u_1, u_2, u_3} S R' A (SMulZeroClass.toSMul.{u_2, u_3} R' A (AddZero.toZero.{u_3} A (AddZeroClass.toAddZero.{u_3} A (AddMonoid.toAddZeroClass.{u_3} A (AddCommMonoid.toAddMonoid.{u_3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39))))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R' A (AddMonoid.toAddZeroClass.{u_3} A (AddCommMonoid.toAddMonoid.{u_3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39))) (DistribMulAction.toDistribSMul.{u_2, u_3} R' A (MonoidWithZero.toMonoid.{u_2} R' (Semiring.toMonoidWithZero.{u_2} R' inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.46)) (AddCommMonoid.toAddMonoid.{u_3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39)) (Module.toDistribMulAction.{u_2, u_3} R' A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.46 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} A inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.39) inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.53)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.2915323218._hygCtx._hyg.62","typeFull":"∀ (S : Type u_1) (R' : Type u_2) (R : Type u_4) (A : Type u_3) [inst : Semiring R]\n  [inst_1 : NonUnitalNonAssocSemiring A] [inst_2 : Module R A] [inst_3 : Semiring R'] [inst_4 : SMul R' R]\n  [inst_5 : Module R' A] [IsScalarTower R' R A] [inst_7 : SetLike S A] [SMulMemClass S R A], SMulMemClass S R' A","typeReadable":"∀ (S : Type u_1) (R' : Type u_2) (R : Type u_4) (A : Type u_3) [inst : Semiring R]\n  [inst_1 : NonUnitalNonAssocSemiring A] [inst_2 : Module R A] [inst_3 : Semiring R'] [inst_4 : SMul R' R]\n  [inst_5 : Module R' A] [IsScalarTower R' R A] [inst_7 : SetLike S A] [SMulMemClass S R A], SMulMemClass S R' A","typeReferences":[["SMulMemClass"],["Module"],["IsScalarTower"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["SMul"],["MonoidWithZero","toMonoid"],["SetLike"],["AddZero","toZero"],["NonUnitalNonAssocSemiring"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["SMulMemClass","ofIsScalarTower"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","ofLinearComb","_proof_2"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.13 : Module.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10] (C : Set.{u_1} M), (Set.Nonempty.{u_1} M C) -> (forall (x : M), (Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) C x) -> (forall (y : M), (Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) C y) -> (forall (a : R) (b : R), Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) C (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddCommMagma.toAdd.{u_1} M (AddCommSemigroup.toAddCommMagma.{u_1} M (AddCommMonoid.toAddCommSemigroup.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10)))) (HSMul.hSMul.{u_2, u_1, u_1} R M M (instHSMul.{u_2, u_1} R M (SMulZeroClass.toSMul.{u_2, u_1} R 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.Submodule.Defs.2678515438._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.13))))) a x) (HSMul.hSMul.{u_2, u_1, u_1} R M M (instHSMul.{u_2, u_1} R M (SMulZeroClass.toSMul.{u_2, u_1} R 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.Submodule.Defs.2678515438._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2678515438._hygCtx._hyg.13))))) b y))))) -> (Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) C (OfNat.ofNat.{u_1} M 0 (Zero.toOfNat0.{u_1} 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.Submodule.Defs.2678515438._hygCtx._hyg.10)))))))","typeFull":"∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (C : Set M),\n  C.Nonempty → (∀ x ∈ C, ∀ y ∈ C, ∀ (a b : R), a • x + b • y ∈ C) → 0 ∈ C","typeReadable":"∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (C : Set M),\n  C.Nonempty → (∀ x ∈ C, ∀ y ∈ C, ∀ (a b : R), a • x + b • y ∈ C) → 0 ∈ C","typeReferences":[["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["Zero","toOfNat0"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["DistribSMul","toSMulZeroClass"],["Set","Nonempty"],["instHAdd"],["Set"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Set","instMembership"],["HAdd","hAdd"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["HSMul","hSMul"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Eq","trans"],["Eq","mp"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["congrArg"],["Semiring","toNonAssocSemiring"],["congr"],["MonoidWithZero","toMonoid"],["instHSMul"],["Zero","toOfNat0"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["MulActionWithZero","toSMulWithZero"],["DistribSMul","toSMulZeroClass"],["instHAdd"],["Set"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["Module","toMulActionWithZero"],["add_zero"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Set","instMembership"],["HAdd","hAdd"],["Exists","casesOn"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["HSMul","hSMul"],["zero_smul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","smulMemClass"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1397418928._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1397418928._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1397418928._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1397418928._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1397418928._hygCtx._hyg.10], SMulMemClass.{v, u, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1397418928._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1397418928._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1397418928._hygCtx._hyg.13) R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1397418928._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1397418928._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1397418928._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1397418928._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1397418928._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1397418928._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1397418928._hygCtx._hyg.13)))) (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1397418928._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1397418928._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1397418928._hygCtx._hyg.13)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n  SMulMemClass (Submodule R M) R M","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n  SMulMemClass (Submodule R M) R M","typeReferences":[["SMulMemClass"],["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submodule"],["Module","toDistribMulAction"],["AddCommMonoid"],["Submodule","setLike"],["MonoidWithZero","toMonoid"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["Submodule","toSubMulAction"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submodule"],["SMulMemClass","mk"],["Module","toDistribMulAction"],["SubMulAction","smul_mem'"],["Submodule","setLike"],["MonoidWithZero","toMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","coe_toAddSubmonoid"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3511054789._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3511054789._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3511054789._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3511054789._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3511054789._hygCtx._hyg.10] (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3511054789._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3511054789._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3511054789._hygCtx._hyg.13), Eq.{succ v} (Set.{v} M) (SetLike.coe.{v, v} (AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3511054789._hygCtx._hyg.10))) M (AddSubmonoid.instSetLike.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3511054789._hygCtx._hyg.10))) (Submodule.toAddSubmonoid.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3511054789._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3511054789._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3511054789._hygCtx._hyg.13 p)) (SetLike.coe.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3511054789._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3511054789._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3511054789._hygCtx._hyg.13) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3511054789._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3511054789._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3511054789._hygCtx._hyg.13) p)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (p : Submodule R M),\n  ↑p.toAddSubmonoid = ↑p","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (p : Submodule R M),\n  ↑p.toAddSubmonoid = ↑p","typeReferences":[["Submodule","toAddSubmonoid"],["Module"],["Set"],["AddSubmonoid","instSetLike"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["SetLike","coe"],["AddSubmonoid"],["Submodule","setLike"],["Eq"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["rfl"],["Submodule","toAddSubmonoid"],["SetLike","coe"],["AddSubmonoid"],["Set"],["AddSubmonoid","instSetLike"],["AddCommMonoid","toAddMonoid"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"definition","name":["Submodule","_sizeOf_inst"],"typeFallback":"forall (R : Type.{u}) (M : Type.{v}) {inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 : Semiring.{u} R} {inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 : AddCommMonoid.{v} M} {inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.34 : SizeOf.{succ u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.35 : SizeOf.{succ v} M], SizeOf.{succ v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20)","typeFull":"(R : Type u) →\n  (M : Type v) →\n    {inst : Semiring R} →\n      {inst_1 : AddCommMonoid M} → {inst_2 : Module R M} → [SizeOf R] → [SizeOf M] → SizeOf (Submodule R M)","typeReadable":"(R : Type u) →\n  (M : Type v) →\n    {inst : Semiring R} →\n      {inst_1 : AddCommMonoid M} → {inst_2 : Module R M} → [SizeOf R] → [SizeOf M] → SizeOf (Submodule R M)","typeReferences":[["AddCommMonoid"],["Module"],["SizeOf"],["Submodule"],["Semiring"]],"valueReferences":[["Submodule","_sizeOf_1"],["SizeOf","mk"],["Submodule"]]},{"isProp":false,"kind":"definition","name":["SMulMemClass","toModule"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10] {A : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 : SetLike.{u_1, v} A M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 : AddSubmonoidClass.{u_1, v} A M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.26 : SMulMemClass.{u_1, u, v} A R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.13)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18] (S' : A), Module.{u, v} R (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, u_1} M A (SetLike.instMembership.{u_1, v} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 (AddSubmonoidClass.toAddCommMonoid.{u_1, v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 S')","typeFull":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] →\n      [inst_1 : AddCommMonoid M] →\n        [inst_2 : Module R M] →\n          {A : Type u_1} →\n            [inst_3 : SetLike A M] → [inst_4 : AddSubmonoidClass A M] → [SMulMemClass A R M] → (S' : A) → Module R ↥S'","typeReadable":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] →\n      [inst_1 : AddCommMonoid M] →\n        [inst_2 : Module R M] →\n          {A : Type u_1} →\n            [inst_3 : SetLike A M] → [inst_4 : AddSubmonoidClass A M] → [SMulMemClass A R M] → (S' : A) → Module R ↥S'","typeReferences":[["SMulMemClass"],["SetLike","instMembership"],["Subtype"],["Module"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddSubmonoidClass","toAddCommMonoid"],["Module","toDistribMulAction"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["SetLike"],["AddSubmonoidClass"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["Subtype"],["SetLike","instMembership"],["SMulMemClass","toModule","_proof_4"],["Membership","mem"],["DistribMulAction","mk"],["Semiring","toMonoidWithZero"],["SMulMemClass","toModule","_proof_2"],["AddCommMonoid","toAddMonoid"],["AddSubmonoidClass","toAddCommMonoid"],["SubMulAction","SMulMemClass","toMulAction"],["Module","mk"],["Module","toDistribMulAction"],["SMulMemClass","toModule","_proof_1"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["SMulMemClass","toModule","_proof_3"]]},{"isProp":false,"kind":"inductive","name":["Submodule"],"typeFallback":"forall (R : Type.{u}) (M : Type.{v}) [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17], Type.{v}","typeFull":"(R : Type u) → (M : Type v) → [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [Module R M] → Type v","typeReadable":"(R : Type u) → (M : Type v) → [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [Module R M] → Type v","typeReferences":[["AddCommMonoid"],["Module"],["Semiring"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["Submodule","isScalarTower'"],"typeFallback":"forall {S : Type.{u'}} {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.10} (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.10 module_M) {S' : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.26 : SMul.{u', u} S R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.30 : SMul.{u', v} S M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.34 : SMul.{u_1, u} S' R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.38 : SMul.{u_1, v} S' M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.42 : SMul.{u', u_1} S S'] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.46 : IsScalarTower.{u_1, u, v} S' R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.34 (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.10 module_M)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.38] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.51 : IsScalarTower.{u', u_1, v} S S' M inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.42 inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.38 inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.30] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.56 : IsScalarTower.{u', u, v} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.26 (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.10 module_M)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.30], IsScalarTower.{u', u_1, v} S S' (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.10 module_M)) p x)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.42 (Submodule.smul.{u_1, u, v} S' R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.38 inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.46) (Submodule.smul.{u', u, v} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.Submodule.Defs.243307169._hygCtx._hyg.56)","typeFull":"∀ {S : Type u'} {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) {S' : Type u_1} [inst_2 : SMul S R] [inst_3 : SMul S M] [inst_4 : SMul S' R] [inst_5 : SMul S' M]\n  [inst_6 : SMul S S'] [inst_7 : IsScalarTower S' R M] [IsScalarTower S S' M] [inst_9 : IsScalarTower S R M],\n  IsScalarTower S S' ↥p","typeReadable":"∀ {S : Type u'} {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) {S' : Type u_1} [inst_2 : SMul S R] [inst_3 : SMul S M] [inst_4 : SMul S' R] [inst_5 : SMul S' M]\n  [inst_6 : SMul S S'] [inst_7 : IsScalarTower S' R M] [IsScalarTower S S' M] [inst_9 : IsScalarTower S R M],\n  IsScalarTower S S' ↥p","typeReferences":[["SetLike","instMembership"],["Subtype"],["Module"],["IsScalarTower"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submodule"],["Module","toDistribMulAction"],["AddCommMonoid"],["SMul"],["Submodule","setLike"],["MonoidWithZero","toMonoid"],["Submodule","smul"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["Submodule","toSubMulAction"],["Module","toDistribMulAction"],["MonoidWithZero","toMonoid"],["SubMulAction","isScalarTower'"],["DistribMulAction","toMulAction"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"]]},{"isProp":true,"kind":"theorem","name":["Submodule","mem_toAddSubmonoid"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.10] (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.13) (x : M), Iff (Membership.mem.{v, v} M (AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.10))) (SetLike.instMembership.{v, v} (AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.10))) M (AddSubmonoid.instSetLike.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.10)))) (Submodule.toAddSubmonoid.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.13 p) x) (Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.13) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.13) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1361378310._hygCtx._hyg.13)) p x)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (p : Submodule R M)\n  (x : M), x ∈ p.toAddSubmonoid ↔ x ∈ p","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (p : Submodule R M)\n  (x : M), x ∈ p.toAddSubmonoid ↔ x ∈ p","typeReferences":[["SetLike","instMembership"],["Submodule","toAddSubmonoid"],["Module"],["Membership","mem"],["AddSubmonoid","instSetLike"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["AddSubmonoid"],["Submodule","setLike"],["Iff"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Submodule","toAddSubmonoid"],["SetLike","instMembership"],["AddSubmonoid"],["AddSubmonoid","instSetLike"],["Membership","mem"],["AddCommMonoid","toAddMonoid"],["Iff","rfl"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","zero_mem"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3005011325._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3005011325._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3005011325._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3005011325._hygCtx._hyg.10} (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3005011325._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3005011325._hygCtx._hyg.10 module_M), Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3005011325._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3005011325._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3005011325._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3005011325._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3005011325._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3005011325._hygCtx._hyg.10 module_M)) p (OfNat.ofNat.{v} M 0 (Zero.toOfNat0.{v} M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3005011325._hygCtx._hyg.10))))))","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M),\n  0 ∈ p","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M),\n  0 ∈ p","typeReferences":[["SetLike","instMembership"],["Module"],["Membership","mem"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Submodule"],["AddCommMonoid"],["Submodule","setLike"],["Zero","toOfNat0"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["ZeroMemClass","zero_mem"],["AddSubmonoidClass","toZeroMemClass"],["Submodule","setLike"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toZero"],["Submodule","addSubmonoidClass"],["AddMonoid","toAddZeroClass"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["SMulMemClass","toModule","_proof_2"],"typeFallback":"forall {R : Type.{u_3}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 : Semiring.{u_3} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.13 : Module.{u_3, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 : SetLike.{u_2, u_1} A M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 : AddSubmonoidClass.{u_2, u_1} A M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.26 : SMulMemClass.{u_2, u_3, u_1} A R M (SMulZeroClass.toSMul.{u_3, u_1} R 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.Submodule.Defs.1426575160._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_3, u_1} R M (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.13)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18] (S' : A) (a : R) (x : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (y : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)), Eq.{succ u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (HSMul.hSMul.{u_3, u_1, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (instHSMul.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (SemigroupAction.toSMul.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (Monoid.toSemigroup.{u_3} R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (SubMulAction.SMulMemClass.toMulAction.{u_3, u_1, u_2} R M (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (DistribMulAction.toMulAction.{u_3, u_1} R M (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.13)) A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.26 S')))) a (HAdd.hAdd.{u_1, u_1, u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (instHAdd.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddSemigroup.toAdd.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddMonoid.toAddSemigroup.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddSubmonoidClass.toAddCommMonoid.{u_2, u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 S'))))) x y)) (HAdd.hAdd.{u_1, u_1, u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (instHAdd.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddSemigroup.toAdd.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddMonoid.toAddSemigroup.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (AddSubmonoidClass.toAddCommMonoid.{u_2, u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.22 S'))))) (HSMul.hSMul.{u_3, u_1, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (instHSMul.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (SemigroupAction.toSMul.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (Monoid.toSemigroup.{u_3} R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (SubMulAction.SMulMemClass.toMulAction.{u_3, u_1, u_2} R M (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (DistribMulAction.toMulAction.{u_3, u_1} R M (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.13)) A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.26 S')))) a x) (HSMul.hSMul.{u_3, u_1, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (instHSMul.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (SemigroupAction.toSMul.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (Monoid.toSemigroup.{u_3} R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_3, u_1} R (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M A (SetLike.instMembership.{u_2, u_1} A M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18) S' x)) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (SubMulAction.SMulMemClass.toMulAction.{u_3, u_1, u_2} R M (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (DistribMulAction.toMulAction.{u_3, u_1} R M (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.13)) A inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1426575160._hygCtx._hyg.26 S')))) a y))","typeFull":"∀ {R : Type u_3} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {A : Type u_2}\n  [inst_3 : SetLike A M] [inst_4 : AddSubmonoidClass A M] [inst_5 : SMulMemClass A R M] (S' : A) (a : R) (x y : ↥S'),\n  a • (x + y) = a • x + a • y","typeReadable":"∀ {R : Type u_3} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {A : Type u_2}\n  [inst_3 : SetLike A M] [inst_4 : AddSubmonoidClass A M] [inst_5 : SMulMemClass A R M] (S' : A) (a : R) (x y : ↥S'),\n  a • (x + y) = a • x + a • y","typeReferences":[["Subtype"],["Module"],["Membership","mem"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["SubMulAction","SMulMemClass","toMulAction"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["SetLike"],["Monoid","toSemigroup"],["Eq"],["AddSubmonoidClass"],["AddSemigroup","toAdd"],["DistribSMul","toSMulZeroClass"],["SMulMemClass"],["SetLike","instMembership"],["instHAdd"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["AddSubmonoidClass","toAddCommMonoid"],["HAdd","hAdd"],["Module","toDistribMulAction"],["AddMonoid","toAddSemigroup"],["HSMul","hSMul"],["AddZero","toZero"],["MulAction","toSemigroupAction"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Subtype"],["SetLike","instMembership"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["SetLike","val_smul"],["AddSubmonoidClass","toAddCommMonoid"],["SubMulAction","SMulMemClass","toMulAction"],["Module","toDistribMulAction"],["Subtype","coe_injective"],["AddSubmonoidClass","subtype"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Monoid","toSemigroup"],["AddZero","toZero"],["MulAction","toSemigroupAction"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"],["Function","Injective","distribMulAction","_proof_3"]]},{"isProp":false,"kind":"definition","name":["Submodule","ctorIdx"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} {inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 : Semiring.{u} R} {inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 : AddCommMonoid.{v} M} {inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17}, (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20) -> Nat","typeFull":"{R : Type u} →\n  {M : Type v} → {inst : Semiring R} → {inst_1 : AddCommMonoid M} → {inst_2 : Module R M} → Submodule R M → ℕ","typeReadable":"{R : Type u} →\n  {M : Type v} → {inst : Semiring R} → {inst_1 : AddCommMonoid M} → {inst_2 : Module R M} → Submodule R M → ℕ","typeReferences":[["Nat"],["AddCommMonoid"],["Module"],["Submodule"],["Semiring"]],"valueReferences":[]},{"isProp":false,"kind":"definition","name":["SubmoduleClass","module"],"typeFallback":"forall {S : Type.{u'}} {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340097._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340097._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340097._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340097._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340097._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340097._hygCtx._hyg.17 : SetLike.{u', v} S M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340097._hygCtx._hyg.21 : AddSubmonoidClass.{u', v} S M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340097._hygCtx._hyg.10)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340097._hygCtx._hyg.17] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340097._hygCtx._hyg.25 : SMulMemClass.{u', u, v} S R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340097._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340097._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340097._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340097._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340097._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340097._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340097._hygCtx._hyg.13)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340097._hygCtx._hyg.17] (s : S), Module.{u, v} R (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, u'} M S (SetLike.instMembership.{u', v} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340097._hygCtx._hyg.17) s x)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340097._hygCtx._hyg.7 (AddSubmonoidClass.toAddCommMonoid.{u', v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340097._hygCtx._hyg.10 S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340097._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1565340097._hygCtx._hyg.21 s)","typeFull":"{S : Type u'} →\n  {R : Type u} →\n    {M : Type v} →\n      [inst : Semiring R] →\n        [inst_1 : AddCommMonoid M] →\n          [inst_2 : Module R M] →\n            [inst_3 : SetLike S M] → [inst_4 : AddSubmonoidClass S M] → [SMulMemClass S R M] → (s : S) → Module R ↥s","typeReadable":"{S : Type u'} →\n  {R : Type u} →\n    {M : Type v} →\n      [inst : Semiring R] →\n        [inst_1 : AddCommMonoid M] →\n          [inst_2 : Module R M] →\n            [inst_3 : SetLike S M] → [inst_4 : AddSubmonoidClass S M] → [SMulMemClass S R M] → (s : S) → Module R ↥s","typeReferences":[["SMulMemClass"],["SetLike","instMembership"],["Subtype"],["Module"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddSubmonoidClass","toAddCommMonoid"],["Module","toDistribMulAction"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["SetLike"],["AddSubmonoidClass"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Submodule","module","_proof_1"],["SMulZeroClass","toSMul"],["SubmoduleClass","module'"],["AddZeroClass","toAddZero"],["AddMonoidWithOne","toAddMonoid"],["instDistribSMul"],["Semiring","toNonAssocSemiring"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["AddZero","toZero"],["NonAssocSemiring","toAddCommMonoidWithOne"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","setLike","_proof_1"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.7 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.13 : Module.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.10] (p : Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.13) (q : Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.13), (Eq.{succ u_1} (Set.{u_1} M) ((fun (s : Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.13) => AddSubsemigroup.carrier.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.10)))) (AddSubmonoid.toAddSubsemigroup.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.10)) (Submodule.toAddSubmonoid.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.13 s))) p) ((fun (s : Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.13) => AddSubsemigroup.carrier.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.10)))) (AddSubmonoid.toAddSubsemigroup.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.10)) (Submodule.toAddSubmonoid.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.13 s))) q)) -> (Eq.{succ u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2936539461._hygCtx._hyg.13) p q)","typeFull":"∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  (p q : Submodule R M), (fun s => s.carrier) p = (fun s => s.carrier) q → p = q","typeReadable":"∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  (p q : Submodule R M), (fun s => s.carrier) p = (fun s => s.carrier) q → p = q","typeReferences":[["Submodule","toAddSubmonoid"],["Module"],["Set"],["AddCommMonoid","toAddMonoid"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["Submodule"],["AddSubsemigroup","carrier"],["AddCommMonoid"],["AddSubmonoid","toAddSubsemigroup"],["Eq"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Submodule","toAddSubmonoid"],["Membership","mem"],["Submodule","casesOn"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddSubsemigroup","carrier"],["SetLike","coe_injective'"],["MonoidWithZero","toMonoid"],["Eq","symm"],["instHSMul"],["Eq","ndrec"],["Eq"],["Eq","rec"],["DistribSMul","toSMulZeroClass"],["Set"],["AddSubmonoid","instSetLike"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["Set","instMembership"],["Module","toDistribMulAction"],["Submodule","mk"],["AddSubmonoid"],["Eq","refl"],["HSMul","hSMul"],["AddSubmonoid","toAddSubsemigroup"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","ext_iff"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.10] {p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.13} {q : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.13}, Iff (Eq.{succ v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.13) p q) (forall (x : M), Iff (Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.13) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.13) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.13)) p x) (Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.13) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.13) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.722126854._hygCtx._hyg.13)) q x))","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {p q : Submodule R M},\n  p = q ↔ ∀ (x : M), x ∈ p ↔ x ∈ q","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {p q : Submodule R M},\n  p = q ↔ ∀ (x : M), x ∈ p ↔ x ∈ q","typeReferences":[["AddCommMonoid"],["SetLike","instMembership"],["Submodule","setLike"],["Module"],["Iff"],["Membership","mem"],["Eq"],["Submodule"],["Semiring"]],"valueReferences":[["Submodule","ext"],["SetLike","instMembership"],["HEq","refl"],["Membership","mem"],["Iff","rfl"],["Eq","casesOn"],["Iff","intro"],["Submodule"],["Submodule","setLike"],["Iff"],["Eq","refl"],["HEq"],["Eq","symm"],["Eq"],["Eq","ndrec"]]},{"isProp":false,"kind":"definition","name":["Submodule","add"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.10} (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.10 module_M), Add.{v} (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2314059840._hygCtx._hyg.10 module_M)) p x))","typeFull":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] → [inst_1 : AddCommMonoid M] → {module_M : Module R M} → (p : Submodule R M) → Add ↥p","typeReadable":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] → [inst_1 : AddCommMonoid M] → {module_M : Module R M} → (p : Submodule R M) → Add ↥p","typeReferences":[["AddCommMonoid"],["SetLike","instMembership"],["Subtype"],["Submodule","setLike"],["Add"],["Module"],["Membership","mem"],["Submodule"],["Semiring"]],"valueReferences":[["Subtype"],["SetLike","instMembership"],["instHAdd"],["Membership","mem"],["Subtype","val"],["Submodule"],["HAdd","hAdd"],["AddCommMonoid","toAddCommSemigroup"],["Submodule","add","_proof_1"],["Submodule","setLike"],["Add","mk"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Subtype","mk"]]},{"isProp":true,"kind":"theorem","name":["Submodule","toAddSubmonoid_inj"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.10] {p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.13} {q : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.13}, Iff (Eq.{succ v} (AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.10))) (Submodule.toAddSubmonoid.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.13 p) (Submodule.toAddSubmonoid.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.13 q)) (Eq.{succ v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2140789293._hygCtx._hyg.13) p q)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {p q : Submodule R M},\n  p.toAddSubmonoid = q.toAddSubmonoid ↔ p = q","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {p q : Submodule R M},\n  p.toAddSubmonoid = q.toAddSubmonoid ↔ p = q","typeReferences":[["AddCommMonoid"],["Submodule","toAddSubmonoid"],["AddSubmonoid"],["Module"],["Iff"],["AddCommMonoid","toAddMonoid"],["Eq"],["AddMonoid","toAddZeroClass"],["Submodule"],["Semiring"]],"valueReferences":[["Function","Injective","eq_iff"],["Submodule","toAddSubmonoid_injective"],["Submodule","toAddSubmonoid"],["AddSubmonoid"],["AddCommMonoid","toAddMonoid"],["AddMonoid","toAddZeroClass"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","smul_mem_iff'","_simp_1"],"typeFallback":"forall {G : Type.{u''}} {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.10} (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.10 module_M) {x : M} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.25 : Group.{u''} G] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.28 : MulAction.{u'', v} G M (DivInvMonoid.toMonoid.{u''} G (Group.toDivInvMonoid.{u''} G inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.25))] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.32 : SMul.{u'', u} G R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.36 : IsScalarTower.{u'', u, v} G R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.32 (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.10 module_M)))) (SemigroupAction.toSMul.{u'', v} G M (Monoid.toSemigroup.{u''} G (DivInvMonoid.toMonoid.{u''} G (Group.toDivInvMonoid.{u''} G inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.25))) (MulAction.toSemigroupAction.{u'', v} G M (DivInvMonoid.toMonoid.{u''} G (Group.toDivInvMonoid.{u''} G inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.25)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.28))] (g : G), Eq.{1} Prop (Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.10 module_M)) p (HSMul.hSMul.{u'', v, v} G M M (instHSMul.{u'', v} G M (SemigroupAction.toSMul.{u'', v} G M (Monoid.toSemigroup.{u''} G (DivInvMonoid.toMonoid.{u''} G (Group.toDivInvMonoid.{u''} G inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.25))) (MulAction.toSemigroupAction.{u'', v} G M (DivInvMonoid.toMonoid.{u''} G (Group.toDivInvMonoid.{u''} G inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.25)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.28))) g x)) (Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3455288177._hygCtx._hyg.10 module_M)) p x)","typeFull":"∀ {G : Type u''} {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) {x : M} [inst_2 : Group G] [inst_3 : MulAction G M] [inst_4 : SMul G R] [IsScalarTower G R M]\n  (g : G), (g • x ∈ p) = (x ∈ p)","typeReadable":"∀ {G : Type u''} {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) {x : M} [inst_2 : Group G] [inst_3 : MulAction G M] [inst_4 : SMul G R] [IsScalarTower G R M]\n  (g : G), (g • x ∈ p) = (x ∈ p)","typeReferences":[["IsScalarTower"],["Module"],["Membership","mem"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["SMul"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["Monoid","toSemigroup"],["Eq"],["Group","toDivInvMonoid"],["DistribSMul","toSMulZeroClass"],["Group"],["SetLike","instMembership"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["DivInvMonoid","toMonoid"],["MulAction"],["Submodule","setLike"],["HSMul","hSMul"],["AddZero","toZero"],["MulAction","toSemigroupAction"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Submodule","smul_mem_iff'"],["SetLike","instMembership"],["Membership","mem"],["SemigroupAction","toSMul"],["Submodule"],["DivInvMonoid","toMonoid"],["Submodule","setLike"],["HSMul","hSMul"],["instHSMul"],["Monoid","toSemigroup"],["Group","toDivInvMonoid"],["MulAction","toSemigroupAction"],["propext"]]},{"isProp":false,"kind":"definition","name":["Submodule","mk","_flat_ctor"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17] (carrier : Set.{v} M), (forall {a : M} {b : M}, (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) carrier a) -> (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) carrier b) -> (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) carrier (HAdd.hAdd.{v, v, v} M M M (instHAdd.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17))))) a b))) -> (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) carrier (OfNat.ofNat.{v} M 0 (Zero.toOfNat0.{v} M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17))))))) -> (forall (c : R) {x : M}, (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) carrier x) -> (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) carrier (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20))))) c x))) -> (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20)","typeFull":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] →\n      [inst_1 : AddCommMonoid M] →\n        [inst_2 : Module R M] →\n          (carrier : Set M) →\n            (∀ {a b : M}, a ∈ carrier → b ∈ carrier → a + b ∈ carrier) →\n              0 ∈ carrier → (∀ (c : R) {x : M}, x ∈ carrier → c • x ∈ carrier) → Submodule R M","typeReadable":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] →\n      [inst_1 : AddCommMonoid M] →\n        [inst_2 : Module R M] →\n          (carrier : Set M) →\n            (∀ {a b : M}, a ∈ carrier → b ∈ carrier → a + b ∈ carrier) →\n              0 ∈ carrier → (∀ (c : R) {x : M}, x ∈ carrier → c • x ∈ carrier) → Submodule R M","typeReferences":[["Module"],["Set"],["instHAdd"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Submodule"],["Set","instMembership"],["HAdd","hAdd"],["Module","toDistribMulAction"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["instHSMul"],["Zero","toOfNat0"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["AddSubsemigroup","mk"],["Submodule","mk"],["AddSubmonoid","mk"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","toAddSubgroup_inj"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3262223435._hygCtx._hyg.7 : Ring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3262223435._hygCtx._hyg.10 : AddCommGroup.{v} M] {module_M : Module.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3262223435._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3262223435._hygCtx._hyg.10)} (p : Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3262223435._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3262223435._hygCtx._hyg.10) module_M) (p' : Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3262223435._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3262223435._hygCtx._hyg.10) module_M), Iff (Eq.{succ v} (AddSubgroup.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3262223435._hygCtx._hyg.10)) (Submodule.toAddSubgroup.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3262223435._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3262223435._hygCtx._hyg.10 module_M p) (Submodule.toAddSubgroup.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3262223435._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3262223435._hygCtx._hyg.10 module_M p')) (Eq.{succ v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3262223435._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3262223435._hygCtx._hyg.10) module_M) p p')","typeFull":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M} (p p' : Submodule R M),\n  p.toAddSubgroup = p'.toAddSubgroup ↔ p = p'","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M} (p p' : Submodule R M),\n  p.toAddSubgroup = p'.toAddSubgroup ↔ p = p'","typeReferences":[["AddSubgroup"],["Module"],["AddCommGroup","toAddGroup"],["Iff"],["AddCommGroup","toAddCommMonoid"],["AddCommGroup"],["Submodule","toAddSubgroup"],["Eq"],["Submodule"],["Ring","toSemiring"],["Ring"]],"valueReferences":[["AddSubgroup"],["Function","Injective","eq_iff"],["Submodule","toAddSubgroup_injective"],["AddCommGroup","toAddGroup"],["AddCommGroup","toAddCommMonoid"],["Submodule","toAddSubgroup"],["Ring","toSemiring"],["Submodule"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","Module","Submodule","Defs",0,"Submodule","toAddSubgroup_injective","match_1_1"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.7 : Ring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.10 : AddCommGroup.{u_2} M] {module_M : Module.{u_1, u_2} R M (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.10)} (motive : forall (x._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx.35.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.63 : Submodule.{u_1, u_2} R M (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.10) module_M) (x._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx.36.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.66 : Submodule.{u_1, u_2} R M (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.10) module_M), (Eq.{succ u_2} (AddSubgroup.{u_2} M (AddCommGroup.toAddGroup.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.10)) (Submodule.toAddSubgroup.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.10 module_M x._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx.35.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.63) (Submodule.toAddSubgroup.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.10 module_M x._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx.36.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.66)) -> Prop) (x._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx.35.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.63 : Submodule.{u_1, u_2} R M (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.10) module_M) (x._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx.36.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.66 : Submodule.{u_1, u_2} R M (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.10) module_M) (x._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx.37.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.69 : Eq.{succ u_2} (AddSubgroup.{u_2} M (AddCommGroup.toAddGroup.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.10)) (Submodule.toAddSubgroup.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.10 module_M x._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx.35.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.63) (Submodule.toAddSubgroup.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.10 module_M x._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx.36.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.66)), (forall (x._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.79 : Submodule.{u_1, u_2} R M (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.10) module_M) (x._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.78 : Submodule.{u_1, u_2} R M (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.10) module_M) (h : Eq.{succ u_2} (AddSubgroup.{u_2} M (AddCommGroup.toAddGroup.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.10)) (Submodule.toAddSubgroup.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.10 module_M x._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.79) (Submodule.toAddSubgroup.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.10 module_M x._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.78)), motive x._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.79 x._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.78 h) -> (motive x._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx.35.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.63 x._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx.36.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.66 x._@.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx.37.Mathlib.Algebra.Module.Submodule.Defs.3633612391._hygCtx._hyg.69)","typeFull":"∀ {R : Type u_1} {M : Type u_2} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M}\n  (motive : (x x_1 : Submodule R M) → x.toAddSubgroup = x_1.toAddSubgroup → Prop) (x x_1 : Submodule R M)\n  (x_2 : x.toAddSubgroup = x_1.toAddSubgroup),\n  (∀ (x x_3 : Submodule R M) (h : x.toAddSubgroup = x_3.toAddSubgroup), motive x x_3 h) → motive x x_1 x_2","typeReadable":"∀ {R : Type u_1} {M : Type u_2} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M}\n  (motive : (x x_1 : Submodule R M) → x.toAddSubgroup = x_1.toAddSubgroup → Prop) (x x_1 : Submodule R M)\n  (x_2 : x.toAddSubgroup = x_1.toAddSubgroup),\n  (∀ (x x_3 : Submodule R M) (h : x.toAddSubgroup = x_3.toAddSubgroup), motive x x_3 h) → motive x x_1 x_2","typeReferences":[["AddSubgroup"],["Module"],["AddCommGroup","toAddGroup"],["AddCommGroup","toAddCommMonoid"],["AddCommGroup"],["Submodule","toAddSubgroup"],["Eq"],["Submodule"],["Ring","toSemiring"],["Ring"]],"valueReferences":[]},{"isProp":true,"kind":"theorem","name":["Submodule","sub_mem_iff_left"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3468852441._hygCtx._hyg.7 : Ring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.3468852441._hygCtx._hyg.10 : AddCommGroup.{v} M] {module_M : Module.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3468852441._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3468852441._hygCtx._hyg.10)} (p : Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3468852441._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3468852441._hygCtx._hyg.10) module_M) {x : M} {y : M}, (Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3468852441._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3468852441._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3468852441._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3468852441._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3468852441._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3468852441._hygCtx._hyg.10) module_M)) p y) -> (Iff (Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3468852441._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3468852441._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3468852441._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3468852441._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3468852441._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3468852441._hygCtx._hyg.10) module_M)) p (HSub.hSub.{v, v, v} M M M (instHSub.{v} M (SubNegMonoid.toSub.{v} M (AddGroup.toSubNegMonoid.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3468852441._hygCtx._hyg.10)))) x y)) (Membership.mem.{v, v} M (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3468852441._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3468852441._hygCtx._hyg.10) module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3468852441._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3468852441._hygCtx._hyg.10) module_M) M (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.3468852441._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.3468852441._hygCtx._hyg.10) module_M)) p x))","typeFull":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M} (p : Submodule R M)\n  {x y : M}, y ∈ p → (x - y ∈ p ↔ x ∈ p)","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M} (p : Submodule R M)\n  {x y : M}, y ∈ p → (x - y ∈ p ↔ x ∈ p)","typeReferences":[["SetLike","instMembership"],["Module"],["AddCommGroup","toAddGroup"],["Membership","mem"],["AddCommGroup"],["Ring","toSemiring"],["Submodule"],["Submodule","setLike"],["Iff"],["SubNegMonoid","toSub"],["AddCommGroup","toAddCommMonoid"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["instHSub"],["Ring"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Submodule","neg_mem"],["AddCommGroup","toAddGroup"],["Membership","mem"],["SubtractionCommMonoid","toSubtractionMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["Submodule"],["congrArg"],["sub_eq_add_neg"],["SubNegMonoid","toSub"],["HSub","hSub"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["AddGroup","toSubNegMonoid"],["Eq"],["propext"],["SetLike","instMembership"],["instHAdd"],["Neg","neg"],["SubNegMonoid","toNeg"],["Iff","rfl"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["HAdd","hAdd"],["NegZeroClass","toNeg"],["AddCommGroup","toDivisionAddCommMonoid"],["AddCommMonoid","toAddCommSemigroup"],["SubNegMonoid","toAddMonoid"],["Submodule","setLike"],["Iff"],["Submodule","add_mem_iff_left"],["AddCommGroup","toAddCommMonoid"],["id"],["Eq","mpr"],["instHSub"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["SubmoduleClass","module'","_proof_2"],"typeFallback":"forall {S : Type.{u_3}} {R : Type.{u_4}} {M : Type.{u_1}} {T : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 : Semiring.{u_4} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 : Semiring.{u_3} S] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17 : Module.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.21 : SMul.{u_3, u_4} S R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.25 : Module.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.29 : IsScalarTower.{u_3, u_4, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.21 (SMulZeroClass.toSMul.{u_4, u_1} R 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_4, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_4, u_1} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17)))) (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.25))))] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34 : SetLike.{u_2, u_1} T M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.42 : SMulMemClass.{u_2, u_4, u_1} T R M (SMulZeroClass.toSMul.{u_4, u_1} R 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_4, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_4, u_1} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34] (t : T) (x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.69 : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)), Eq.{succ u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (SetLike.smul'.{u_2, u_3, u_4, u_1} T S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.21 (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.25)))) (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (DistribMulAction.toMulAction.{u_4, u_1} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.42 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.29 t)) (OfNat.ofNat.{u_3} S 1 (One.toOfNat1.{u_3} S (MulOne.toOne.{u_3} S (MulOneClass.toMulOne.{u_3} S (Monoid.toMulOneClass.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14))))))) x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.69) x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.69","typeFull":"∀ {S : Type u_3} {R : Type u_4} {M : Type u_1} {T : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : Semiring S] [inst_3 : Module R M] [inst_4 : SMul S R] [inst_5 : Module S M] [inst_6 : IsScalarTower S R M]\n  [inst_7 : SetLike T M] [inst_8 : SMulMemClass T R M] (t : T) (x : ↥t), 1 • x = x","typeReadable":"∀ {S : Type u_3} {R : Type u_4} {M : Type u_1} {T : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : Semiring S] [inst_3 : Module R M] [inst_4 : SMul S R] [inst_5 : Module S M] [inst_6 : IsScalarTower S R M]\n  [inst_7 : SetLike T M] [inst_8 : SMulMemClass T R M] (t : T) (x : ↥t), 1 • x = x","typeReferences":[["MulOneClass","toMulOne"],["Subtype"],["SetLike","smul'"],["IsScalarTower"],["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["SMul"],["AddCommMonoid"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["SetLike"],["Eq"],["DistribSMul","toSMulZeroClass"],["SMulMemClass"],["SetLike","instMembership"],["MulOne","toOne"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Module","toDistribMulAction"],["One","toOfNat1"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["MulOneClass","toMulOne"],["Subtype"],["Eq","trans"],["SetLike","smul'"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Subtype","val"],["congrArg"],["MonoidWithZero","toMonoid"],["Monoid","toMulOneClass"],["DistribMulAction","toMulAction"],["instHSMul"],["congrFun'"],["Eq"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["True"],["MulOne","toOne"],["DistribMulAction","toDistribSMul"],["Subtype","ext"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["eq_self"],["Module","toDistribMulAction"],["One","toOfNat1"],["of_eq_true"],["one_smul"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","zero","_proof_1"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.24028081._hygCtx._hyg.7 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.24028081._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] {module_M : Module.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.24028081._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.24028081._hygCtx._hyg.10} (p : Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.24028081._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.24028081._hygCtx._hyg.10 module_M), Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.24028081._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.24028081._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.24028081._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.24028081._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.24028081._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.24028081._hygCtx._hyg.10 module_M)) p (OfNat.ofNat.{u_1} M 0 (Zero.toOfNat0.{u_1} 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.Submodule.Defs.24028081._hygCtx._hyg.10))))))","typeFull":"∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M), 0 ∈ p","typeReadable":"∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M), 0 ∈ p","typeReferences":[["SetLike","instMembership"],["Module"],["Membership","mem"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Submodule"],["AddCommMonoid"],["Submodule","setLike"],["Zero","toOfNat0"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["ZeroMemClass","zero_mem"],["AddSubmonoidClass","toZeroMemClass"],["Submodule","setLike"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toZero"],["Submodule","addSubmonoidClass"],["AddMonoid","toAddZeroClass"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","smul_mem_iff''"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.10} (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.10 module_M) {r : R} {x : M} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.25 : Invertible.{u} R (Distrib.toMul.{u} R (NonUnitalNonAssocSemiring.toDistrib.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.7)))) (AddMonoidWithOne.toOne.{u} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u} R (NonAssocSemiring.toAddCommMonoidWithOne.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.7)))) r], Iff (Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.10 module_M)) p (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.10 module_M))))) r x)) (Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2296171821._hygCtx._hyg.10 module_M)) p x)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M)\n  {r : R} {x : M} [Invertible r], r • x ∈ p ↔ x ∈ p","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M)\n  {r : R} {x : M} [Invertible r], r • x ∈ p ↔ x ∈ p","typeReferences":[["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["NonAssocSemiring","toAddCommMonoidWithOne"],["Invertible"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["NonUnitalNonAssocSemiring","toDistrib"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Distrib","toMul"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["Submodule","setLike"],["Iff"],["AddMonoidWithOne","toOne"],["HSMul","hSMul"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["MulOneClass","toMulOne"],["invOf_smul_smul"],["Invertible","invOf"],["Membership","mem"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["congrArg"],["Submodule"],["Iff","intro"],["MulOne","toMul"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["Eq","symm"],["DistribMulAction","toMulAction"],["instHSMul"],["Monoid","toSemigroup"],["Eq"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["MulOne","toOne"],["Submodule","smul_mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["Submodule","setLike"],["HSMul","hSMul"],["id"],["Eq","mpr"],["AddZero","toZero"],["MulAction","toSemigroupAction"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","toSubMulAction_inj"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2409323571._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2409323571._hygCtx._hyg.10 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2409323571._hygCtx._hyg.13 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2409323571._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2409323571._hygCtx._hyg.10] {p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2409323571._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2409323571._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2409323571._hygCtx._hyg.13} {q : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2409323571._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2409323571._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2409323571._hygCtx._hyg.13}, Iff (Eq.{succ v} (SubMulAction.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2409323571._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2409323571._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.2409323571._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2409323571._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2409323571._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2409323571._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2409323571._hygCtx._hyg.13))))) (Submodule.toSubMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2409323571._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2409323571._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2409323571._hygCtx._hyg.13 p) (Submodule.toSubMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2409323571._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2409323571._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2409323571._hygCtx._hyg.13 q)) (Eq.{succ v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2409323571._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2409323571._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2409323571._hygCtx._hyg.13) p q)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {p q : Submodule R M},\n  p.toSubMulAction = q.toSubMulAction ↔ p = q","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {p q : Submodule R M},\n  p.toSubMulAction = q.toSubMulAction ↔ p = q","typeReferences":[["Submodule","toSubMulAction"],["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["SubMulAction"],["AddZeroClass","toAddZero"],["Submodule"],["Module","toDistribMulAction"],["AddCommMonoid"],["Iff"],["MonoidWithZero","toMonoid"],["Eq"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["Submodule","toSubMulAction"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["SubMulAction"],["AddZeroClass","toAddZero"],["Submodule"],["Function","Injective","eq_iff"],["Module","toDistribMulAction"],["MonoidWithZero","toMonoid"],["Submodule","toSubMulAction_injective"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","copy","_proof_1"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.13 : Module.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10] (p : Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.13) (s : Set.{u_1} M), (Eq.{succ u_1} (Set.{u_1} M) s (SetLike.coe.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.13) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2619604494._hygCtx._hyg.13) p)) -> (Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) s (OfNat.ofNat.{u_1} M 0 (Zero.toOfNat0.{u_1} 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.Submodule.Defs.2619604494._hygCtx._hyg.10)))))))","typeFull":"∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (p : Submodule R M)\n  (s : Set M), s = ↑p → 0 ∈ s","typeReadable":"∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (p : Submodule R M)\n  (s : Set M), s = ↑p → 0 ∈ s","typeReferences":[["Module"],["Set"],["Membership","mem"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Submodule"],["Set","instMembership"],["AddCommMonoid"],["SetLike","coe"],["Submodule","setLike"],["Zero","toOfNat0"],["Eq"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["SetLike","instMembership"],["True"],["Eq","trans"],["Set"],["Membership","mem"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["congrArg"],["Set","instMembership"],["Submodule"],["ZeroMemClass","zero_mem","_simp_1"],["SetLike","coe"],["SetLike","mem_coe","_simp_1"],["of_eq_true"],["AddSubmonoidClass","toZeroMemClass"],["Submodule","setLike"],["congrFun'"],["Zero","toOfNat0"],["AddZero","toZero"],["Submodule","addSubmonoidClass"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","coe_zero"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2996138777._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.2996138777._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2996138777._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2996138777._hygCtx._hyg.10} {p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2996138777._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2996138777._hygCtx._hyg.10 module_M}, Eq.{succ v} M (Subtype.val.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2996138777._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2996138777._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2996138777._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2996138777._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2996138777._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2996138777._hygCtx._hyg.10 module_M)) p x) (OfNat.ofNat.{v} (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2996138777._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2996138777._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2996138777._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2996138777._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2996138777._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2996138777._hygCtx._hyg.10 module_M)) p x)) 0 (Zero.toOfNat0.{v} (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2996138777._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2996138777._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2996138777._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2996138777._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2996138777._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2996138777._hygCtx._hyg.10 module_M)) p x)) (Submodule.zero.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2996138777._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.2996138777._hygCtx._hyg.10 module_M p)))) (OfNat.ofNat.{v} M 0 (Zero.toOfNat0.{v} M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.2996138777._hygCtx._hyg.10))))))","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} {p : Submodule R M},\n  ↑0 = 0","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} {p : Submodule R M},\n  ↑0 = 0","typeReferences":[["SetLike","instMembership"],["Subtype"],["Module"],["Membership","mem"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Subtype","val"],["OfNat","ofNat"],["Submodule"],["AddCommMonoid"],["Submodule","setLike"],["Zero","toOfNat0"],["Submodule","zero"],["Eq"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["rfl"],["Subtype"],["SetLike","instMembership"],["Submodule","setLike"],["Membership","mem"],["Zero","toOfNat0"],["Submodule","zero"],["Subtype","val"],["OfNat","ofNat"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","addSubgroupClass"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1531369289._hygCtx._hyg.7 : Ring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1531369289._hygCtx._hyg.10 : AddCommGroup.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1531369289._hygCtx._hyg.25 : Module.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1531369289._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1531369289._hygCtx._hyg.10)], AddSubgroupClass.{v, v} (Submodule.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1531369289._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1531369289._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1531369289._hygCtx._hyg.25) M (AddGroup.toSubNegMonoid.{v} M (AddCommGroup.toAddGroup.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1531369289._hygCtx._hyg.10)) (Submodule.setLike.{u, v} R M (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1531369289._hygCtx._hyg.7) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1531369289._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1531369289._hygCtx._hyg.25)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M],\n  AddSubgroupClass (Submodule R M) M","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M],\n  AddSubgroupClass (Submodule R M) M","typeReferences":[["Submodule","setLike"],["AddSubgroupClass"],["Module"],["AddCommGroup","toAddGroup"],["AddCommGroup","toAddCommMonoid"],["AddCommGroup"],["AddGroup","toSubNegMonoid"],["Submodule"],["Ring","toSemiring"],["Ring"]],"valueReferences":[["Submodule","toSubMulAction"],["Submodule","setLike"],["AddCommGroup","toAddGroup"],["SubNegMonoid","toNeg"],["AddCommGroup","toAddCommMonoid"],["AddGroup","toSubNegMonoid"],["NegMemClass","mk"],["SubMulAction","neg_mem"],["AddSubgroupClass","mk"],["Submodule","addSubmonoidClass"],["Submodule"],["Ring","toSemiring"]]},{"isProp":true,"kind":"theorem","name":["Submodule","module'","_proof_10"],"typeFallback":"forall {S : Type.{u_3}} {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] {module_M : Module.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10} (p : Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 : Semiring.{u_3} S] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 : SMul.{u_3, u_2} S R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 : Module.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36 : IsScalarTower.{u_3, u_2, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (SMulZeroClass.toSMul.{u_2, u_1} R 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)))) (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32))))] (x : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)), Eq.{succ u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (SemigroupAction.toSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))) (MulAction.toSemigroupAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (DistribMulAction.toMulAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Submodule.addCommMonoid.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p)) (DistribMulAction.mk.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Submodule.addCommMonoid.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p)) (MulAction.mk.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (SemigroupAction.mk.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))) (Submodule.smul.{u_3, u_2, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36) (Submodule.module'._proof_5.{u_1, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36)) (Submodule.module'._proof_6.{u_1, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36)) (Submodule.module'._proof_7.{u_1, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36) (Submodule.module'._proof_8.{u_1, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36)))))) (OfNat.ofNat.{u_3} S 0 (Zero.toOfNat0.{u_3} S (MulZeroClass.toZero.{u_3} S (NonUnitalNonAssocSemiring.toMulZeroClass.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)))))) x) (OfNat.ofNat.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) 0 (Zero.toOfNat0.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddZero.toZero.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddZeroClass.toAddZero.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddMonoid.toAddZeroClass.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Submodule.addCommMonoid.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p)))))))","typeFull":"∀ {S : Type u_3} {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) [inst_2 : Semiring S] [inst_3 : SMul S R] [inst_4 : Module S M] [inst_5 : IsScalarTower S R M]\n  (x : ↥p), 0 • x = 0","typeReadable":"∀ {S : Type u_3} {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) [inst_2 : Semiring S] [inst_3 : SMul S R] [inst_4 : Module S M] [inst_5 : IsScalarTower S R M]\n  (x : ↥p), 0 • x = 0","typeReferences":[["Submodule","module'","_proof_6"],["Subtype"],["SemigroupAction","mk"],["IsScalarTower"],["Module"],["Membership","mem"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["SMul"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["Zero","toOfNat0"],["Monoid","toSemigroup"],["Eq"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","mk"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Submodule","module'","_proof_7"],["Module","toDistribMulAction"],["MulZeroClass","toZero"],["Submodule","setLike"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["HSMul","hSMul"],["Submodule","smul"],["Submodule","addCommMonoid"],["Submodule","module'","_proof_8"],["MulAction","toSemigroupAction"],["AddZero","toZero"],["Submodule","module'","_proof_5"],["MulAction","mk"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Submodule","module'","_proof_4"]]},{"isProp":false,"kind":"definition","name":["Submodule","casesOn"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17] {motive : (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20) -> Sort.{u_1}} (t : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20), (forall (toAddSubmonoid : AddSubmonoid.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17))) (smul_mem' : forall (c : R) {x : M}, (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) toAddSubmonoid)) x) -> (Membership.mem.{v, v} M (Set.{v} M) (Set.instMembership.{v} M) (AddSubsemigroup.carrier.{v} M (AddZero.toAdd.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (AddSubmonoid.toAddSubsemigroup.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) toAddSubmonoid)) (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20))))) c x))), motive (Submodule.mk.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4285676355._hygCtx._hyg.20 toAddSubmonoid smul_mem')) -> (motive t)","typeFull":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] →\n      [inst_1 : AddCommMonoid M] →\n        [inst_2 : Module R M] →\n          {motive : Submodule R M → Sort u_1} →\n            (t : Submodule R M) →\n              ((toAddSubmonoid : AddSubmonoid M) →\n                  (smul_mem' : ∀ (c : R) {x : M}, x ∈ toAddSubmonoid.carrier → c • x ∈ toAddSubmonoid.carrier) →\n                    motive { toAddSubmonoid := toAddSubmonoid, smul_mem' := smul_mem' }) →\n                motive t","typeReadable":"{R : Type u} →\n  {M : Type v} →\n    [inst : Semiring R] →\n      [inst_1 : AddCommMonoid M] →\n        [inst_2 : Module R M] →\n          {motive : Submodule R M → Sort u_1} →\n            (t : Submodule R M) →\n              ((toAddSubmonoid : AddSubmonoid M) →\n                  (smul_mem' : ∀ (c : R) {x : M}, x ∈ toAddSubmonoid.carrier → c • x ∈ toAddSubmonoid.carrier) →\n                    motive { toAddSubmonoid := toAddSubmonoid, smul_mem' := smul_mem' }) →\n                motive t","typeReferences":[["Module"],["Set"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["Submodule"],["Set","instMembership"],["AddSubsemigroup","carrier"],["Module","toDistribMulAction"],["AddCommMonoid"],["Submodule","mk"],["AddSubmonoid"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["instHSMul"],["AddSubmonoid","toAddSubsemigroup"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Submodule","rec"]]},{"isProp":true,"kind":"theorem","name":["Submodule","coe_eq_zero","_simp_1"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10} {p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M} {x : Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M)) p x)}, Eq.{1} Prop (Eq.{succ v} M (Subtype.val.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M)) p x) x) (OfNat.ofNat.{v} M 0 (Zero.toOfNat0.{v} M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10))))))) (Eq.{succ v} (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M)) p x)) x (OfNat.ofNat.{v} (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M)) p x)) 0 (Zero.toOfNat0.{v} (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M)) p x)) (Submodule.zero.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4202572969._hygCtx._hyg.10 module_M p))))","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} {p : Submodule R M}\n  {x : ↥p}, (↑x = 0) = (x = 0)","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} {p : Submodule R M}\n  {x : ↥p}, (↑x = 0) = (x = 0)","typeReferences":[["Subtype"],["SetLike","instMembership"],["Module"],["Membership","mem"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Subtype","val"],["OfNat","ofNat"],["Submodule"],["AddCommMonoid"],["Submodule","setLike"],["Zero","toOfNat0"],["Submodule","zero"],["Eq"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Subtype"],["SetLike","instMembership"],["Submodule","coe_eq_zero"],["Membership","mem"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Subtype","val"],["OfNat","ofNat"],["Submodule"],["Submodule","setLike"],["Zero","toOfNat0"],["Submodule","zero"],["Eq"],["AddZero","toZero"],["propext"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["SubmoduleClass","module'","_proof_4"],"typeFallback":"forall {S : Type.{u_3}} {R : Type.{u_4}} {M : Type.{u_1}} {T : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 : Semiring.{u_4} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 : Semiring.{u_3} S] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17 : Module.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.21 : SMul.{u_3, u_4} S R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.25 : Module.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.29 : IsScalarTower.{u_3, u_4, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.21 (SMulZeroClass.toSMul.{u_4, u_1} R 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_4, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_4, u_1} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17)))) (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.25))))] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34 : SetLike.{u_2, u_1} T M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.38 : AddSubmonoidClass.{u_2, u_1} T M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.42 : SMulMemClass.{u_2, u_4, u_1} T R M (SMulZeroClass.toSMul.{u_4, u_1} R 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_4, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_4, u_1} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34] (t : T) (x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.79 : S) (x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.81 : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.83 : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)), Eq.{succ u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (SetLike.smul'.{u_2, u_3, u_4, u_1} T S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.21 (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.25)))) (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (DistribMulAction.toMulAction.{u_4, u_1} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.42 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.29 t)) x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.79 (HAdd.hAdd.{u_1, u_1, u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (instHAdd.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (AddSemigroup.toAdd.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (AddMonoid.toAddSemigroup.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (AddSubmonoidClass.toAddCommMonoid.{u_2, u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 T inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.38 t))))) x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.81 x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.83)) (HAdd.hAdd.{u_1, u_1, u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (instHAdd.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (AddSemigroup.toAdd.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (AddMonoid.toAddSemigroup.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (AddSubmonoidClass.toAddCommMonoid.{u_2, u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 T inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.38 t))))) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (SetLike.smul'.{u_2, u_3, u_4, u_1} T S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.21 (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.25)))) (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (DistribMulAction.toMulAction.{u_4, u_1} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.42 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.29 t)) x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.79 x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.81) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_2} M T (SetLike.instMembership.{u_2, u_1} T M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34) t x)) (SetLike.smul'.{u_2, u_3, u_4, u_1} T S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.21 (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757500._hygCtx._hyg.11)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.25)))) (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (DistribMulAction.toMulAction.{u_4, u_1} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11) (Module.toDistribMulAction.{u_4, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.17)) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.42 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.29 t)) x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.79 x._@.Mathlib.Algebra.Module.Submodule.Defs.1227757500._hygCtx._hyg.83))","typeFull":"∀ {S : Type u_3} {R : Type u_4} {M : Type u_1} {T : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : Semiring S] [inst_3 : Module R M] [inst_4 : SMul S R] [inst_5 : Module S M] [inst_6 : IsScalarTower S R M]\n  [inst_7 : SetLike T M] [inst_8 : AddSubmonoidClass T M] [inst_9 : SMulMemClass T R M] (t : T) (x : S) (x_1 x_2 : ↥t),\n  x • (x_1 + x_2) = x • x_1 + x • x_2","typeReadable":"∀ {S : Type u_3} {R : Type u_4} {M : Type u_1} {T : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : Semiring S] [inst_3 : Module R M] [inst_4 : SMul S R] [inst_5 : Module S M] [inst_6 : IsScalarTower S R M]\n  [inst_7 : SetLike T M] [inst_8 : AddSubmonoidClass T M] [inst_9 : SMulMemClass T R M] (t : T) (x : S) (x_1 x_2 : ↥t),\n  x • (x_1 + x_2) = x • x_1 + x • x_2","typeReferences":[["Subtype"],["SetLike","smul'"],["IsScalarTower"],["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["SMul"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["SetLike"],["Eq"],["AddSubmonoidClass"],["AddSemigroup","toAdd"],["DistribSMul","toSMulZeroClass"],["SMulMemClass"],["SetLike","instMembership"],["instHAdd"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["AddSubmonoidClass","toAddCommMonoid"],["HAdd","hAdd"],["Module","toDistribMulAction"],["AddMonoid","toAddSemigroup"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Subtype"],["Eq","trans"],["SetLike","smul'"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Subtype","val"],["congrArg"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["congrFun'"],["Eq"],["DistribSMul","toSMulZeroClass"],["AddSemigroup","toAdd"],["SetLike","instMembership"],["True"],["instHAdd"],["DistribMulAction","toDistribSMul"],["Subtype","ext"],["Semiring","toMonoidWithZero"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["AddSubmonoidClass","toAddCommMonoid"],["HAdd","hAdd"],["eq_self"],["Module","toDistribMulAction"],["of_eq_true"],["AddMonoid","toAddSemigroup"],["HSMul","hSMul"],["smul_add"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"definition","name":["Submodule","smul"],"typeFallback":"forall {S : Type.{u'}} {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.10} (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.10 module_M) [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.25 : SMul.{u', u} S R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.29 : SMul.{u', v} S M] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.33 : IsScalarTower.{u', u, v} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.25 (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.10 module_M)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.29], SMul.{u', v} S (Subtype.{succ v} M (fun (x : M) => Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.4232235684._hygCtx._hyg.10 module_M)) p x))","typeFull":"{S : Type u'} →\n  {R : Type u} →\n    {M : Type v} →\n      [inst : Semiring R] →\n        [inst_1 : AddCommMonoid M] →\n          {module_M : Module R M} →\n            (p : Submodule R M) → [inst_2 : SMul S R] → [inst_3 : SMul S M] → [IsScalarTower S R M] → SMul S ↥p","typeReadable":"{S : Type u'} →\n  {R : Type u} →\n    {M : Type v} →\n      [inst : Semiring R] →\n        [inst_1 : AddCommMonoid M] →\n          {module_M : Module R M} →\n            (p : Submodule R M) → [inst_2 : SMul S R] → [inst_3 : SMul S M] → [IsScalarTower S R M] → SMul S ↥p","typeReferences":[["SetLike","instMembership"],["Subtype"],["Module"],["IsScalarTower"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submodule"],["Module","toDistribMulAction"],["AddCommMonoid"],["SMul"],["Submodule","setLike"],["MonoidWithZero","toMonoid"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["SMul","mk"],["SetLike","instMembership"],["Subtype"],["Submodule","setLike"],["HSMul","hSMul"],["Membership","mem"],["instHSMul"],["Submodule","smul","_proof_1"],["Subtype","mk"],["Subtype","val"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","module'","_proof_8"],"typeFallback":"forall {S : Type.{u_3}} {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] {module_M : Module.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10} (p : Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 : Semiring.{u_3} S] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 : SMul.{u_3, u_2} S R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 : Module.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36 : IsScalarTower.{u_3, u_2, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (SMulZeroClass.toSMul.{u_2, u_1} R 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)))) (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32))))] (a : S) (x : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (y : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)), Eq.{succ u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (SemigroupAction.toSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))) (MulAction.toSemigroupAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (MulAction.mk.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (SemigroupAction.mk.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))) (Submodule.smul.{u_3, u_2, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36) (Submodule.module'._proof_5.{u_1, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36)) (Submodule.module'._proof_6.{u_1, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36))))) a (HAdd.hAdd.{u_1, u_1, u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (instHAdd.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddSemigroup.toAdd.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddMonoid.toAddSemigroup.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Submodule.addCommMonoid.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p))))) x y)) (HAdd.hAdd.{u_1, u_1, u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (instHAdd.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddSemigroup.toAdd.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddMonoid.toAddSemigroup.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (AddCommMonoid.toAddMonoid.{u_1} (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Submodule.addCommMonoid.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p))))) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (SemigroupAction.toSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))) (MulAction.toSemigroupAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (MulAction.mk.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (SemigroupAction.mk.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))) (Submodule.smul.{u_3, u_2, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36) (Submodule.module'._proof_5.{u_1, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36)) (Submodule.module'._proof_6.{u_1, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36))))) a x) (HSMul.hSMul.{u_3, u_1, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (instHSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (SemigroupAction.toSMul.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))) (MulAction.toSemigroupAction.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (MulAction.mk.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (SemigroupAction.mk.{u_3, u_1} S (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M)) p x)) (Monoid.toSemigroup.{u_3} S (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25))) (Submodule.smul.{u_3, u_2, u_1} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 (SMulZeroClass.toSMul.{u_3, u_1} S 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.Submodule.Defs.1227757499._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} S M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u_3, u_1} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10) (Module.toDistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32)))) inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36) (Submodule.module'._proof_5.{u_1, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36)) (Submodule.module'._proof_6.{u_1, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.10 module_M p inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.Submodule.Defs.1227757499._hygCtx._hyg.36))))) a y))","typeFull":"∀ {S : Type u_3} {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) [inst_2 : Semiring S] [inst_3 : SMul S R] [inst_4 : Module S M] [inst_5 : IsScalarTower S R M]\n  (a : S) (x y : ↥p), a • (x + y) = a • x + a • y","typeReadable":"∀ {S : Type u_3} {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M}\n  (p : Submodule R M) [inst_2 : Semiring S] [inst_3 : SMul S R] [inst_4 : Module S M] [inst_5 : IsScalarTower S R M]\n  (a : S) (x y : ↥p), a • (x + y) = a • x + a • y","typeReferences":[["Submodule","module'","_proof_6"],["Subtype"],["SemigroupAction","mk"],["IsScalarTower"],["Module"],["Membership","mem"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["SMul"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["Monoid","toSemigroup"],["Eq"],["AddSemigroup","toAdd"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["instHAdd"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["HAdd","hAdd"],["Module","toDistribMulAction"],["Submodule","setLike"],["AddMonoid","toAddSemigroup"],["HSMul","hSMul"],["Submodule","smul"],["Submodule","addCommMonoid"],["MulAction","toSemigroupAction"],["AddZero","toZero"],["Submodule","module'","_proof_5"],["MulAction","mk"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Submodule","module'","_proof_2"]]}]