diff --git "a/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.LinearMap.Defs.sym.json" "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.LinearMap.Defs.sym.json"
new file mode 100644--- /dev/null
+++ "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.LinearMap.Defs.sym.json"
@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["DistribMulActionHom","instLinearMapClassId"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_8}} {M₃ : Type.{u_11}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.670415694._hygCtx._hyg.15 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.670415694._hygCtx._hyg.21 : AddCommMonoid.{u_11} M₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.670415694._hygCtx._hyg.24 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.670415694._hygCtx._hyg.27 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.670415694._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.670415694._hygCtx._hyg.15] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.670415694._hygCtx._hyg.38 : Module.{u_1, u_11} R M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.670415694._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.670415694._hygCtx._hyg.21], LinearMapClass.{max u_11 u_8, u_1, u_8, u_11} 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(R : Type.{u_14}) {A : Type.{u_15}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.714807159._hygCtx._hyg.17 : Semiring.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.714807159._hygCtx._hyg.20 : NonUnitalNonAssocSemiring.{u_15} A] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.714807159._hygCtx._hyg.23 : Module.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.714807159._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.714807159._hygCtx._hyg.20)] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.714807159._hygCtx._hyg.27 : SMulCommClass.{u_14, u_15, u_15} R A A (SMulZeroClass.toSMul.{u_14, u_15} R A (AddZero.toZero.{u_15} A (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.714807159._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_14, u_15} R A 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[inst : Semiring R] [inst_1 : NonUnitalNonAssocSemiring A] [inst_2 : Module R A]\n  [inst_3 : SMulCommClass R A A] (a b : A), (LinearMap.mulLeft R a) b = a * 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(AddCommGroup.toAddCommMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2996033385._hygCtx._hyg.21) (AddCommGroup.toAddCommMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2996033385._hygCtx._hyg.24) module_M module_M₂ σ) f x) (DFunLike.coe.{max (succ u_10) (succ u_8), succ u_8, succ u_10} (LinearMap.{u_1, u_5, u_8, u_10} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2996033385._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2996033385._hygCtx._hyg.18 σ M M₂ (AddCommGroup.toAddCommMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2996033385._hygCtx._hyg.21) (AddCommGroup.toAddCommMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2996033385._hygCtx._hyg.24) module_M module_M₂) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) (LinearMap.instFunLike.{u_1, u_5, u_8, u_10} R S M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2996033385._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2996033385._hygCtx._hyg.18 (AddCommGroup.toAddCommMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2996033385._hygCtx._hyg.21) (AddCommGroup.toAddCommMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2996033385._hygCtx._hyg.24) module_M module_M₂ σ) f y))","typeFull":"∀ {R : Type u_1} {S : Type u_5} {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommGroup M] [inst_3 : AddCommGroup M₂] {module_M : Module R M} {module_M₂ : Module S M₂} {σ : R →+* S}\n  (f : M →ₛₗ[σ] M₂) (x y : M), f (x - y) = f x - f y","typeReadable":"∀ {R : Type u_1} {S : Type u_5} {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommGroup M] [inst_3 : AddCommGroup M₂] {module_M : Module R M} {module_M₂ : Module S M₂} {σ : R →+* S}\n  (f : M →ₛₗ[σ] M₂) (x y : M), f (x - y) = f x - f y","typeReferences":[["RingHom"],["LinearMap","instFunLike"],["Module"],["AddCommGroup","toAddGroup"],["AddCommGroup"],["LinearMap"],["DFunLike","coe"],["Semiring","toNonAssocSemiring"],["SubNegMonoid","toSub"],["AddCommGroup","toAddCommMonoid"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["Eq"],["instHSub"],["Semiring"]],"valueReferences":[["RingHom"],["LinearMap","instFunLike"],["AddCommGroup","toAddGroup"],["Semiring","toMonoidWithZero"],["RingHom","instFunLike"],["SubtractionCommMonoid","toSubtractionMonoid"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["DFunLike","coe"],["map_sub"],["LinearMap","semilinearMapClass"],["DistribMulActionSemiHomClass","toAddMonoidHomClass"],["Module","toDistribMulAction"],["Semiring","toNonAssocSemiring"],["AddCommGroup","toDivisionAddCommMonoid"],["MonoidWithZero","toMonoid"],["AddCommGroup","toAddCommMonoid"],["SemilinearMapClass","distribMulActionSemiHomClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","addCommMonoid","_proof_1"],"typeFallback":"forall {R₂ : Type.{u_1}} {M₂ : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4245081335._hygCtx._hyg.18 : Semiring.{u_1} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4245081335._hygCtx._hyg.27 : AddCommMonoid.{u_2} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4245081335._hygCtx._hyg.43 : Module.{u_1, u_2} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4245081335._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4245081335._hygCtx._hyg.27], SMulCommClass.{u_1, 0, u_2} R₂ Nat M₂ (SMulZeroClass.toSMul.{u_1, u_2} R₂ M₂ (AddZero.toZero.{u_2} M₂ (AddZeroClass.toAddZero.{u_2} M₂ (AddMonoid.toAddZeroClass.{u_2} M₂ (AddCommMonoid.toAddMonoid.{u_2} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4245081335._hygCtx._hyg.27)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R₂ M₂ (AddMonoid.toAddZeroClass.{u_2} M₂ (AddCommMonoid.toAddMonoid.{u_2} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4245081335._hygCtx._hyg.27)) (DistribMulAction.toDistribSMul.{u_1, u_2} R₂ M₂ (MonoidWithZero.toMonoid.{u_1} R₂ (Semiring.toMonoidWithZero.{u_1} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4245081335._hygCtx._hyg.18)) (AddCommMonoid.toAddMonoid.{u_2} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4245081335._hygCtx._hyg.27) (Module.toDistribMulAction.{u_1, u_2} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4245081335._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4245081335._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4245081335._hygCtx._hyg.43)))) (AddMonoid.toNSMul.{u_2} M₂ (AddCommMonoid.toAddMonoid.{u_2} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4245081335._hygCtx._hyg.27))","typeFull":"∀ {R₂ : Type u_1} {M₂ : Type u_2} [inst : Semiring R₂] [inst_1 : AddCommMonoid M₂] [inst_2 : Module R₂ M₂],\n  SMulCommClass R₂ ℕ M₂","typeReadable":"∀ {R₂ : Type u_1} {M₂ : Type u_2} [inst : Semiring R₂] [inst_1 : AddCommMonoid M₂] [inst_2 : Module R₂ M₂],\n  SMulCommClass R₂ ℕ M₂","typeReferences":[["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["SMulCommClass"],["Module","toDistribMulAction"],["AddCommMonoid"],["Nat"],["AddMonoid","toNSMul"],["MonoidWithZero","toMonoid"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["Module","toDistribMulAction"],["AddMonoid","nat_smulCommClass'"],["MonoidWithZero","toMonoid"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","mulRight_toAddMonoidHom"],"typeFallback":"forall (R : Type.{u_14}) {A : Type.{u_15}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 : Semiring.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20 : NonUnitalNonAssocSemiring.{u_15} A] 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(Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17))) (DistribMulActionSemiHomClass.toAddMonoidHomClass.{u_15, u_14, u_14, u_15, u_15} (LinearMap.{u_14, u_14, u_15, u_15} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23) R R (DFunLike.coe.{succ u_14, succ u_14, succ u_14} (RingHom.{u_14, u_14} R R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17) (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17)) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => R) (RingHom.instFunLike.{u_14, u_14} R R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17) (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17)) (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17))) A A (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17)) (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17)) 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u_15, u_15} R R A A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17))) (SemilinearMapClass.distribMulActionSemiHomClass.{u_14, u_14, u_15, u_15, u_15} R R A A (LinearMap.{u_14, u_14, u_15, u_15} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 (RingHom.id.{u_14} R 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17)) (LinearMap.instFunLike.{u_14, u_14, u_15, u_15} R R A A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17))) (LinearMap.semilinearMapClass.{u_14, u_14, u_15, u_15} R R 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{M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3441326568._hygCtx._hyg.21 : AddCommGroup.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3441326568._hygCtx._hyg.24 : AddCommGroup.{u_10} M₂] {R : Type.{u_14}} {S : Type.{u_15}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3441326568._hygCtx._hyg.46 : Monoid.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3441326568._hygCtx._hyg.49 : MulAction.{u_14, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3441326568._hygCtx._hyg.46] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3441326568._hygCtx._hyg.53 : MulAction.{u_14, u_10} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3441326568._hygCtx._hyg.46] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3441326568._hygCtx._hyg.57 : Semiring.{u_15} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3441326568._hygCtx._hyg.60 : Module.{u_15, u_8} S M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3441326568._hygCtx._hyg.57 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3441326568._hygCtx._hyg.46) S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3441326568._hygCtx._hyg.57 (Units.instSMul.{u_14, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3441326568._hygCtx._hyg.46 (SemigroupAction.toSMul.{u_14, u_8} R M (Monoid.toSemigroup.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3441326568._hygCtx._hyg.46) (MulAction.toSemigroupAction.{u_14, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3441326568._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3441326568._hygCtx._hyg.49))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3441326568._hygCtx._hyg.60 (Units.instSMul.{u_14, u_10} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3441326568._hygCtx._hyg.46 (SemigroupAction.toSMul.{u_14, u_10} R M₂ (Monoid.toSemigroup.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3441326568._hygCtx._hyg.46) (MulAction.toSemigroupAction.{u_14, u_10} R M₂ 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S","typeReferences":[["LinearMap","CompatibleSMul"],["MulAction"],["Module"],["AddCommGroup","toAddCommMonoid"],["Monoid"],["SemigroupAction","toSMul"],["AddCommGroup"],["Monoid","toSemigroup"],["Units","instSMul"],["MulAction","toSemigroupAction"],["Units"],["Semiring"]],"valueReferences":[["LinearMap","CompatibleSMul","mk"],["Units","val"],["AddCommGroup","toAddCommMonoid"],["SemigroupAction","toSMul"],["Monoid","toSemigroup"],["Units","instSMul"],["LinearMap","CompatibleSMul","map_smul"],["MulAction","toSemigroupAction"],["Units"]]},{"isProp":true,"kind":"theorem","name":["LinearMapClass","map_smul_of_tower"],"typeFallback":"forall {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3376419935._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3376419935._hygCtx._hyg.27 : AddCommMonoid.{u_10} M₂] {R : Type.{u_14}} {S : Type.{u_15}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3376419935._hygCtx._hyg.73 : Semiring.{u_15} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3376419935._hygCtx._hyg.76 : SMul.{u_14, u_8} R M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3376419935._hygCtx._hyg.80 : Module.{u_15, u_8} S M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3376419935._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3376419935._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3376419935._hygCtx._hyg.84 : SMul.{u_14, u_10} R M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3376419935._hygCtx._hyg.88 : Module.{u_15, u_10} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3376419935._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3376419935._hygCtx._hyg.27] {F : Type.{u_16}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3376419935._hygCtx._hyg.93 : LinearMap.CompatibleSMul.{u_8, u_10, u_14, u_15} M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3376419935._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3376419935._hygCtx._hyg.27 R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3376419935._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3376419935._hygCtx._hyg.76 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3376419935._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3376419935._hygCtx._hyg.84 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3376419935._hygCtx._hyg.88] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3376419935._hygCtx._hyg.99 : FunLike.{succ u_16, succ u_8, succ u_10} F M M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3376419935._hygCtx._hyg.104 : LinearMapClass.{u_16, u_15, u_8, u_10} F S M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3376419935._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3376419935._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3376419935._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3376419935._hygCtx._hyg.80 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u_14} {S : Type u_15}\n  [inst_2 : Semiring S] [inst_3 : SMul R M] [inst_4 : Module S M] [inst_5 : SMul R M₂] [inst_6 : Module S M₂]\n  {F : Type u_16} [LinearMap.CompatibleSMul M M₂ R S] [inst_8 : FunLike F M M₂] [LinearMapClass F S M M₂] (fₗ : F)\n  (c : R) (x : M), fₗ (c • x) = c • fₗ x","typeReadable":"∀ {M : Type u_8} {M₂ : Type u_10} [inst : AddCommMonoid M] [inst_1 : AddCommMonoid M₂] {R : Type u_14} {S : Type u_15}\n  [inst_2 : Semiring S] [inst_3 : SMul R M] [inst_4 : Module S M] [inst_5 : SMul R M₂] [inst_6 : Module S M₂]\n  {F : Type u_16} [LinearMap.CompatibleSMul M M₂ R S] [inst_8 : FunLike F M M₂] [LinearMapClass F S M M₂] (fₗ : F)\n  (c : R) (x : M), fₗ (c • x) = c • fₗ 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(R : Type.{u_1}) {S : Type.{u_5}} {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.18 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.27 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.33 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.37 : Module.{u_1, u_10} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.27] 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[inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.33 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.21] {σ : RingHom.{u_1, u_1} R R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.15)} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.55 : RingHomId.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.15 σ], LinearMap.{u_1, u_1, u_8, u_8} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.15 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M","typeReferences":[["RingHom"],["RingHomId"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["Module"],["LinearMap"],["Semiring"]],"valueReferences":[["AddCommMonoid","toAddCommSemigroup"],["LinearMap","mk"],["AddHom","mk"],["LinearMap","id'","_proof_1"],["LinearMap","id'","_proof_2"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","coe_smul"],"typeFallback":"forall {R : Type.{u_1}} {R₂ : Type.{u_3}} {S : Type.{u_5}} {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1393146857._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1393146857._hygCtx._hyg.18 : Semiring.{u_3} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1393146857._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1393146857._hygCtx._hyg.24 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1393146857._hygCtx._hyg.27 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1393146857._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1393146857._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1393146857._hygCtx._hyg.31 : Module.{u_3, u_10} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1393146857._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1393146857._hygCtx._hyg.24] {σ₁₂ : RingHom.{u_1, u_3} R R₂ (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1393146857._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1393146857._hygCtx._hyg.18)} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1393146857._hygCtx._hyg.40 : DistribSMul.{u_5, u_10} S M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1393146857._hygCtx._hyg.24))] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1393146857._hygCtx._hyg.44 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1393146857._hygCtx._hyg.31)))) (SMulZeroClass.toSMul.{u_5, u_10} S M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1393146857._hygCtx._hyg.24)))) (DistribSMul.toSMulZeroClass.{u_5, u_10} S M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1393146857._hygCtx._hyg.24)) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1393146857._hygCtx._hyg.40))] (a : S) (f : LinearMap.{u_1, u_3, u_8, u_10} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1393146857._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1393146857._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1393146857._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1393146857._hygCtx._hyg.24 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: Semiring R₂] [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M]\n  [inst_5 : Module R₂ M₂] {σ₁₂ : R →+* R₂} [inst_6 : DistribSMul S M₂] [inst_7 : SMulCommClass R₂ S M₂] (a : S)\n  (f : M →ₛₗ[σ₁₂] M₂), ⇑(a • f) = a • 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46}, (LinearMap.{u_14, u_15, u_16, u_17} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53) -> Nat","typeFull":"{R : Type u_14} →\n  {S : Type u_15} →\n    {inst : Semiring R} →\n      {inst_1 : Semiring S} →\n        {σ : R →+* S} →\n          {M : Type u_16} →\n            {M₂ : Type u_17} →\n              {inst_2 : AddCommMonoid M} →\n                {inst_3 : AddCommMonoid M₂} → {inst_4 : Module R M} → {inst_5 : Module S M₂} → (M →ₛₗ[σ] M₂) → ℕ","typeReadable":"{R : Type u_14} →\n  {S : Type u_15} →\n    {inst : Semiring R} →\n      {inst_1 : Semiring S} →\n        {σ : R →+* S} →\n          {M : Type u_16} →\n            {M₂ : Type u_17} →\n              {inst_2 : AddCommMonoid M} →\n                {inst_3 : AddCommMonoid M₂} → {inst_4 : Module R M} → {inst_5 : Module S M₂} → (M →ₛₗ[σ] M₂) → ℕ","typeReferences":[["RingHom"],["Nat"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["Module"],["LinearMap"],["Semiring"]],"valueReferences":[]},{"isProp":false,"kind":"definition","name":["LinearMap","instSMul"],"typeFallback":"forall {R : Type.{u_1}} {R₂ : Type.{u_3}} {S : Type.{u_5}} {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1097252691._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1097252691._hygCtx._hyg.18 : Semiring.{u_3} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1097252691._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] 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 [inst_5 : Module S M₃] {σ : R →+* S} [inst_6 : FunLike F M M₃] [SemilinearMapClass F σ M M₃],\n  DistribMulActionSemiHomClass F (⇑σ) M M₃","typeReadable":"∀ {R : Type u_1} {S : Type u_5} {M : Type u_8} {M₃ : Type u_11} (F : Type u_14) [inst : Semiring R]\n  [inst_1 : Semiring S] [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₃] [inst_4 : Module R M]\n  [inst_5 : Module S M₃] {σ : R →+* S} [inst_6 : FunLike F M M₃] [SemilinearMapClass F σ M M₃],\n  DistribMulActionSemiHomClass F (⇑σ) M 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M} →\n                    {inst_5 : Module S M₂} → [SizeOf R] → [SizeOf S] → [SizeOf M] → [SizeOf M₂] → SizeOf (M →ₛₗ[σ] M₂)","typeReferences":[["RingHom"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["Module"],["LinearMap"],["SizeOf"],["Semiring"]],"valueReferences":[["LinearMap"],["LinearMap","_sizeOf_1"],["SizeOf","mk"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","restrictScalars","_proof_2"],"typeFallback":"forall (R : Type.{u_4}) {S : Type.{u_3}} {M : Type.{u_2}} {M₂ : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15 : Semiring.{u_4} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 : Semiring.{u_3} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27 : AddCommMonoid.{u_1} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.33 : Module.{u_4, u_2} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.37 : Module.{u_4, u_1} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.71 : Module.{u_3, u_2} S M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.75 : Module.{u_3, u_1} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.79 : LinearMap.CompatibleSMul.{u_2, u_1, u_4, u_3} M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27 R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 (SMulZeroClass.toSMul.{u_4, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21)))) (DistribSMul.toSMulZeroClass.{u_4, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21)) (DistribMulAction.toDistribSMul.{u_4, u_2} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21) (Module.toDistribMulAction.{u_4, u_2} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.33)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.71 (SMulZeroClass.toSMul.{u_4, u_1} R M₂ (AddZero.toZero.{u_1} M₂ (AddZeroClass.toAddZero.{u_1} M₂ (AddMonoid.toAddZeroClass.{u_1} M₂ (AddCommMonoid.toAddMonoid.{u_1} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27)))) (DistribSMul.toSMulZeroClass.{u_4, u_1} R M₂ (AddMonoid.toAddZeroClass.{u_1} M₂ (AddCommMonoid.toAddMonoid.{u_1} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27)) (DistribMulAction.toDistribSMul.{u_4, u_1} R M₂ (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15)) (AddCommMonoid.toAddMonoid.{u_1} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27) (Module.toDistribMulAction.{u_4, u_1} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.37)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.75] (fₗ : LinearMap.{u_3, u_3, u_2, u_1} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 (RingHom.id.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.71 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.75) (c : R) (x : M), Eq.{succ u_1} M₂ (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_2, succ u_1} (LinearMap.{u_3, u_3, u_2, u_1} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 (RingHom.id.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.71 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.75) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) (LinearMap.instFunLike.{u_3, u_3, u_2, u_1} S S M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.71 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.75 (RingHom.id.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18))) fₗ (HSMul.hSMul.{u_4, u_2, u_2} R M M (instHSMul.{u_4, u_2} R M (SMulZeroClass.toSMul.{u_4, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21)))) (DistribSMul.toSMulZeroClass.{u_4, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21)) (DistribMulAction.toDistribSMul.{u_4, u_2} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21) (Module.toDistribMulAction.{u_4, u_2} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.33))))) c x)) (HSMul.hSMul.{u_4, u_1, u_1} R M₂ M₂ (instHSMul.{u_4, u_1} R M₂ (SMulZeroClass.toSMul.{u_4, u_1} R M₂ (AddZero.toZero.{u_1} M₂ (AddZeroClass.toAddZero.{u_1} M₂ (AddMonoid.toAddZeroClass.{u_1} M₂ (AddCommMonoid.toAddMonoid.{u_1} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27)))) (DistribSMul.toSMulZeroClass.{u_4, u_1} R M₂ (AddMonoid.toAddZeroClass.{u_1} M₂ (AddCommMonoid.toAddMonoid.{u_1} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27)) (DistribMulAction.toDistribSMul.{u_4, u_1} R M₂ (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15)) (AddCommMonoid.toAddMonoid.{u_1} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27) (Module.toDistribMulAction.{u_4, u_1} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.37))))) c (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_2, succ u_1} (LinearMap.{u_3, u_3, u_2, u_1} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 (RingHom.id.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.71 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.75) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) (LinearMap.instFunLike.{u_3, u_3, u_2, u_1} S S M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.71 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.75 (RingHom.id.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18))) fₗ x))","typeFull":"∀ (R : Type u_4) {S : Type u_3} {M : Type u_2} {M₂ : Type u_1} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M] [inst_5 : Module R M₂]\n  [inst_6 : Module S M] [inst_7 : Module S M₂] [LinearMap.CompatibleSMul M M₂ R S] (fₗ : M →ₗ[S] M₂) (c : R) (x : M),\n  fₗ (c • x) = c • fₗ x","typeReadable":"∀ (R : Type u_4) {S : Type u_3} {M : Type u_2} {M₂ : Type u_1} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M] [inst_5 : Module R M₂]\n  [inst_6 : Module S M] [inst_7 : Module S M₂] [LinearMap.CompatibleSMul M M₂ R S] (fₗ : M →ₗ[S] M₂) (c : R) (x : M),\n  fₗ (c • x) = c • fₗ x","typeReferences":[["LinearMap","CompatibleSMul"],["LinearMap","instFunLike"],["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["HSMul","hSMul"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["Module","toDistribMulAction"],["MonoidWithZero","toMonoid"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["LinearMap","map_smul_of_tower"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","instAdd","_proof_2"],"typeFallback":"forall {R₁ : Type.{u_3}} {R₂ : Type.{u_4}} {M : Type.{u_2}} {M₂ : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.15 : Semiring.{u_3} R₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.18 : Semiring.{u_4} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.24 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.27 : AddCommMonoid.{u_1} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.39 : Module.{u_3, u_2} R₁ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.43 : Module.{u_4, u_1} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.27] {σ₁₂ : RingHom.{u_3, u_4} R₁ R₂ (Semiring.toNonAssocSemiring.{u_3} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_4} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.18)} (f : LinearMap.{u_3, u_4, u_2, u_1} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.43) (g : LinearMap.{u_3, u_4, u_2, u_1} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.43) (m : R₁) (x : M), Eq.{succ u_1} M₂ (HAdd.hAdd.{max u_1 u_2, max u_1 u_2, max u_1 u_2} (M -> M₂) (M -> M₂) (M -> M₂) (instHAdd.{max u_1 u_2} (M -> M₂) (Pi.instAdd.{u_2, u_1} M (fun (a : M) => M₂) (fun (i : M) => AddCommMagma.toAdd.{u_1} M₂ (AddCommSemigroup.toAddCommMagma.{u_1} M₂ (AddCommMonoid.toAddCommSemigroup.{u_1} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.27))))) (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_2, succ u_1} (LinearMap.{u_3, u_4, u_2, u_1} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.43) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) (LinearMap.instFunLike.{u_3, u_4, u_2, u_1} R₁ R₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.43 σ₁₂) f) (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_2, succ u_1} (LinearMap.{u_3, u_4, u_2, u_1} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1387507625._hygCtx._hyg.15 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[inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1097252691._hygCtx._hyg.15 : Semiring.{u_4} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1097252691._hygCtx._hyg.18 : Semiring.{u_5} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1097252691._hygCtx._hyg.21 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1097252691._hygCtx._hyg.24 : AddCommMonoid.{u_1} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1097252691._hygCtx._hyg.27 : Module.{u_4, u_3} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1097252691._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1097252691._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1097252691._hygCtx._hyg.31 : Module.{u_5, u_1} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1097252691._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1097252691._hygCtx._hyg.24] {σ₁₂ : RingHom.{u_4, u_5} R R₂ (Semiring.toNonAssocSemiring.{u_4} R 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R).toFun x","typeReadable":"∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (m : R) (x : M),\n  (DistribMulActionHom.id R).toFun (m • x) = (MonoidHom.id R) m • (DistribMulActionHom.id R).toFun 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.43 σ₁₂))))))) a)","typeFull":"∀ {R₁ : Type u_3} {R₂ : Type u_4} {M : Type u_2} {M₂ : Type u_1} [inst : Semiring R₁] [inst_1 : Semiring R₂]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R₁ M] [inst_5 : Module R₂ M₂]\n  {σ₁₂ : R₁ →+* R₂} (a : M), 0 a = 0 a","typeReadable":"∀ {R₁ : Type u_3} {R₂ : Type u_4} {M : Type u_2} {M₂ : Type u_1} [inst : Semiring R₁] [inst_1 : Semiring R₂]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R₁ M] [inst_5 : Module R₂ M₂]\n  {σ₁₂ : R₁ →+* R₂} (a : M), 0 a = 0 a","typeReferences":[["RingHom"],["LinearMap","instFunLike"],["Module"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["OfNat","ofNat"],["LinearMap","addCommMonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["Zero","toOfNat0"],["Eq"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["LinearMap","addCommMonoid"],["rfl"],["LinearMap","instFunLike"],["Zero","toOfNat0"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["LinearMap"],["AddZero","toZero"],["DFunLike","coe"],["OfNat","ofNat"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"definition","name":["term_→ₗ[_]_"],"typeFallback":"Lean.TrailingParserDescr","typeFull":"Lean.TrailingParserDescr","typeReadable":"Lean.TrailingParserDescr","typeReferences":[["Lean","TrailingParserDescr"]],"valueReferences":[["Nat"],["Lean","ParserDescr","cat"],["Lean","Name","mkStr1"],["instOfNatNat"],["Lean","ParserDescr","trailingNode"],["Lean","ParserDescr","symbol"],["OfNat","ofNat"],["Lean","ParserDescr","binary"]]},{"isProp":false,"kind":"definition","name":["LinearMap","_sizeOf_1"],"typeFallback":"forall {R : Type.{u_14}} {S : Type.{u_15}} {inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 : Semiring.{u_14} R} {inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 : Semiring.{u_15} S} {σ : RingHom.{u_14, u_15} R S (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30) (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33)} {M : Type.{u_16}} {M₂ : Type.{u_17}} {inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 : AddCommMonoid.{u_16} M} {inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 : AddCommMonoid.{u_17} M₂} {inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 : Module.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43} {inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53 : Module.{u_15, u_17} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.68 : SizeOf.{succ u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.69 : SizeOf.{succ u_15} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.70 : SizeOf.{succ u_16} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.71 : SizeOf.{succ u_17} M₂], (LinearMap.{u_14, u_15, u_16, u_17} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53) -> Nat","typeFull":"{R : Type u_14} →\n  {S : Type u_15} →\n    {inst : Semiring R} →\n      {inst_1 : Semiring S} →\n        {σ : R →+* S} →\n          {M : Type u_16} →\n            {M₂ : Type u_17} →\n              {inst_2 : AddCommMonoid M} →\n                {inst_3 : AddCommMonoid M₂} →\n                  {inst_4 : Module R M} →\n                    {inst_5 : Module S M₂} → [SizeOf R] → [SizeOf S] → [SizeOf M] → [SizeOf M₂] → (M →ₛₗ[σ] M₂) → ℕ","typeReadable":"{R : Type u_14} →\n  {S : Type u_15} →\n    {inst : Semiring R} →\n      {inst_1 : Semiring S} →\n        {σ : R →+* S} →\n          {M : Type u_16} →\n            {M₂ : Type u_17} →\n              {inst_2 : AddCommMonoid M} →\n                {inst_3 : AddCommMonoid M₂} →\n                  {inst_4 : Module R M} →\n                    {inst_5 : Module S M₂} → [SizeOf R] → [SizeOf S] → [SizeOf M] → [SizeOf M₂] → (M →ₛₗ[σ] M₂) → ℕ","typeReferences":[["RingHom"],["Nat"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["Module"],["LinearMap"],["SizeOf"],["Semiring"]],"valueReferences":[["instAddNat"],["instHAdd"],["LinearMap","rec"],["AddHom","_sizeOf_inst"],["OfNat","ofNat"],["HAdd","hAdd"],["Nat"],["SizeOf","sizeOf"],["AddCommMonoid","toAddCommSemigroup"],["instOfNatNat"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["AddHom"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","restrictScalars_zero"],"typeFallback":"forall (R : Type.{u_14}) (S : Type.{u_15}) (M : Type.{u_16}) (N : Type.{u_17}) [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.711350329._hygCtx._hyg.20 : Semiring.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.711350329._hygCtx._hyg.23 : Semiring.{u_15} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.711350329._hygCtx._hyg.26 : AddCommMonoid.{u_16} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.711350329._hygCtx._hyg.29 : AddCommMonoid.{u_17} N] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.711350329._hygCtx._hyg.32 : Module.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.711350329._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.711350329._hygCtx._hyg.26] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.711350329._hygCtx._hyg.36 : Module.{u_14, u_17} R N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.711350329._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.711350329._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.711350329._hygCtx._hyg.40 : Module.{u_15, u_16} S M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.711350329._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.711350329._hygCtx._hyg.26] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.711350329._hygCtx._hyg.44 : Module.{u_15, u_17} S N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.711350329._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.711350329._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.711350329._hygCtx._hyg.48 : LinearMap.CompatibleSMul.{u_16, u_17, u_14, u_15} M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.711350329._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.711350329._hygCtx._hyg.29 R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.711350329._hygCtx._hyg.23 (SMulZeroClass.toSMul.{u_14, u_16} R M (AddZero.toZero.{u_16} M (AddZeroClass.toAddZero.{u_16} M (AddMonoid.toAddZeroClass.{u_16} M (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.711350329._hygCtx._hyg.26)))) (DistribSMul.toSMulZeroClass.{u_14, u_16} R M (AddMonoid.toAddZeroClass.{u_16} M (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.711350329._hygCtx._hyg.26)) (DistribMulAction.toDistribSMul.{u_14, u_16} R M 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.43 : Module.{u_3, u_10} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.27] {σ₁₂ : RingHom.{u_2, u_3} R₁ R₂ (Semiring.toNonAssocSemiring.{u_2} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.18)} (f : LinearMap.{u_2, u_3, u_8, u_10} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.43), Eq.{max (succ u_8) (succ u_10)} (AddMonoidHom.{u_8, u_10} M M₂ (AddZeroClass.toAddZero.{u_8} M (AddMonoid.toAddZeroClass.{u_8} M (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.24))) (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.27)))) (DFunLike.coe.{succ (max u_10 u_8), succ (max u_10 u_8), succ (max u_10 u_8)} (AddMonoidHom.{max u_10 u_8, max u_10 u_8} (LinearMap.{u_2, u_3, u_8, u_10} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.24 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.43) (LinearMap.addCommMonoid.{u_2, u_3, u_8, u_10} R₁ R₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.43 σ₁₂)))) (AddZeroClass.toAddZero.{max u_10 u_8} (AddMonoidHom.{u_8, u_10} M M₂ (AddZeroClass.toAddZero.{u_8} M (AddMonoid.toAddZeroClass.{u_8} M (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.24))) (AddZeroClass.toAddZero.{u_10} M₂ 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[inst_1 : Semiring R₂]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R₁ M] [inst_5 : Module R₂ M₂]\n  {σ₁₂ : R₁ →+* R₂} (f : M →ₛₗ[σ₁₂] M₂), LinearMap.toAddMonoidHom' f = 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[inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 : Semiring.{u_15} S] {σ : RingHom.{u_14, u_15} R S (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30) (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33)} {M : Type.{u_16}} {M₂ : Type.{u_17}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 : AddCommMonoid.{u_16} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 : AddCommMonoid.{u_17} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 : Module.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53 : Module.{u_15, u_17} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46] {motive : (LinearMap.{u_14, u_15, u_16, u_17} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53) -> Sort.{u}} (t : LinearMap.{u_14, u_15, u_16, u_17} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53), (forall (toAddHom : AddHom.{u_16, u_17} M M₂ (AddCommMagma.toAdd.{u_16} M (AddCommSemigroup.toAddCommMagma.{u_16} M (AddCommMonoid.toAddCommSemigroup.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43))) (AddCommMagma.toAdd.{u_17} M₂ (AddCommSemigroup.toAddCommMagma.{u_17} M₂ (AddCommMonoid.toAddCommSemigroup.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46)))) (map_smul' : forall (m : R) (x : M), Eq.{succ u_17} M₂ (AddHom.toFun.{u_16, u_17} M M₂ (AddCommMagma.toAdd.{u_16} M (AddCommSemigroup.toAddCommMagma.{u_16} M (AddCommMonoid.toAddCommSemigroup.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43))) (AddCommMagma.toAdd.{u_17} M₂ (AddCommSemigroup.toAddCommMagma.{u_17} M₂ (AddCommMonoid.toAddCommSemigroup.{u_17} M₂ 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σ m) (AddHom.toFun.{u_16, u_17} M M₂ (AddCommMagma.toAdd.{u_16} M (AddCommSemigroup.toAddCommMagma.{u_16} M (AddCommMonoid.toAddCommSemigroup.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43))) (AddCommMagma.toAdd.{u_17} M₂ (AddCommSemigroup.toAddCommMagma.{u_17} M₂ (AddCommMonoid.toAddCommSemigroup.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46))) toAddHom x))), motive (LinearMap.mk.{u_14, u_15, u_16, u_17} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53 toAddHom map_smul')) -> (motive t)","typeFull":"{R : Type u_14} →\n  {S : Type u_15} →\n    [inst : Semiring R] →\n      [inst_1 : Semiring S] →\n        {σ : R →+* S} →\n          {M : Type u_16} →\n            {M₂ : Type u_17} →\n              [inst_2 : AddCommMonoid M] →\n                [inst_3 : AddCommMonoid M₂] →\n                  [inst_4 : Module R M] →\n                    [inst_5 : Module S M₂] →\n                      {motive : (M →ₛₗ[σ] M₂) → Sort u} →\n                        (t : M →ₛₗ[σ] M₂) →\n                          ((toAddHom : M →ₙ+ M₂) →\n                              (map_smul' : ∀ (m : R) (x : M), toAddHom.toFun (m • x) = σ m • toAddHom.toFun x) →\n                                motive { toAddHom := toAddHom, map_smul' := map_smul' }) →\n                            motive t","typeReadable":"{R : Type u_14} →\n  {S : Type u_15} →\n    [inst : Semiring R] →\n      [inst_1 : Semiring S] →\n        {σ : R →+* S} →\n          {M : Type u_16} →\n            {M₂ : Type u_17} →\n              [inst_2 : AddCommMonoid M] →\n                [inst_3 : AddCommMonoid M₂] →\n                  [inst_4 : Module R M] →\n                    [inst_5 : Module S M₂] →\n                      {motive : (M →ₛₗ[σ] M₂) → Sort u} →\n                        (t : M →ₛₗ[σ] M₂) →\n                          ((toAddHom : M →ₙ+ M₂) →\n                              (map_smul' : ∀ (m : R) (x : M), toAddHom.toFun (m • x) = σ m • toAddHom.toFun x) →\n                                motive { toAddHom := toAddHom, map_smul' := map_smul' }) →\n                            motive 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[inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3102259698._hygCtx._hyg.18 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3102259698._hygCtx._hyg.24 : AddCommMonoid.{u_9} M₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3102259698._hygCtx._hyg.27 : AddCommMonoid.{u_10} M₂] {modM₁ : Module.{u_1, u_9} R M₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3102259698._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3102259698._hygCtx._hyg.24} {modM₂ : Module.{u_5, u_10} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3102259698._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3102259698._hygCtx._hyg.27} {σ : RingHom.{u_1, u_5} R S (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3102259698._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3102259698._hygCtx._hyg.18)}, (LinearMap.{u_1, u_5, u_9, u_10} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3102259698._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3102259698._hygCtx._hyg.18 σ M₁ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3102259698._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3102259698._hygCtx._hyg.27 modM₁ modM₂) -> (AddMonoidHom.{u_9, u_10} M₁ M₂ (AddZeroClass.toAddZero.{u_9} M₁ (AddMonoid.toAddZeroClass.{u_9} M₁ (AddCommMonoid.toAddMonoid.{u_9} M₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3102259698._hygCtx._hyg.24))) (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3102259698._hygCtx._hyg.27))))","typeFull":"{R : Type u_1} →\n  {S : Type u_5} →\n    {M₁ : Type u_9} →\n      {M₂ : Type u_10} →\n        [inst : Semiring R] →\n          [inst_1 : Semiring S] →\n            [inst_2 : AddCommMonoid M₁] →\n              [inst_3 : AddCommMonoid M₂] →\n                {modM₁ : Module R M₁} → {modM₂ : Module S M₂} → {σ : R →+* S} → (M₁ →ₛₗ[σ] M₂) → M₁ →+ M₂","typeReadable":"{R : Type u_1} →\n  {S : Type u_5} →\n    {M₁ : Type u_9} →\n      {M₂ : Type u_10} →\n        [inst : Semiring R] →\n          [inst_1 : Semiring S] →\n            [inst_2 : AddCommMonoid M₁] →\n              [inst_3 : AddCommMonoid M₂] →\n                {modM₁ : Module R M₁} → {modM₂ : Module S M₂} → {σ : R →+* S} → (M₁ →ₛₗ[σ] M₂) → M₁ →+ M₂","typeReferences":[["RingHom"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["Module"],["AddMonoidHom"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["LinearMap"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["LinearMap","map_add"],["LinearMap","instFunLike"],["AddCommMonoid","toAddMonoid"],["AddMonoidHom","mk"],["LinearMap"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["AddZero","toZero"],["ZeroHom","mk"],["LinearMap","map_zero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","toAddMonoidHom_mulLeft"],"typeFallback":"forall (R : Type.{u_14}) {A : Type.{u_15}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17 : Semiring.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20 : NonUnitalNonAssocSemiring.{u_15} A] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23 : Module.{u_14, u_15} R A 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(RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23) (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20)))) (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23) R R (DFunLike.coe.{succ u_14, succ u_14, succ u_14} (RingHom.{u_14, u_14} R R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17) (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17)) R (fun 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20 a)","typeFull":"∀ (R : Type u_14) {A : Type u_15} [inst : Semiring R] [inst_1 : NonUnitalNonAssocSemiring A] [inst_2 : Module R A]\n  [inst_3 : SMulCommClass R A A] (a : A), ↑(LinearMap.mulLeft R a) = AddMonoidHom.mulLeft a","typeReadable":"∀ (R : Type u_14) {A : Type u_15} [inst : Semiring R] [inst_1 : NonUnitalNonAssocSemiring A] [inst_2 : Module R A]\n  [inst_3 : SMulCommClass R A A] (a : A), ↑(LinearMap.mulLeft R a) = AddMonoidHom.mulLeft 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{R₁ : Type.{u_1}} {R₂ : Type.{u_2}} {M : Type.{u_3}} {M₂ : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4245081335._hygCtx._hyg.15 : Semiring.{u_1} R₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4245081335._hygCtx._hyg.18 : Semiring.{u_2} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4245081335._hygCtx._hyg.24 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4245081335._hygCtx._hyg.27 : AddCommMonoid.{u_4} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4245081335._hygCtx._hyg.39 : Module.{u_1, u_3} R₁ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4245081335._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4245081335._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4245081335._hygCtx._hyg.43 : Module.{u_2, u_4} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4245081335._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4245081335._hygCtx._hyg.27] {σ₁₂ : RingHom.{u_1, u_2} R₁ R₂ 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x)","typeReferences":[["RingHom"],["LinearMap","instFunLike"],["Module"],["DistribMulAction","toDistribSMul"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["DFunLike","coe"],["LinearMap","addCommMonoid","_proof_1"],["Module","toDistribMulAction"],["AddCommMonoid"],["Nat"],["Nat","instMonoid"],["Semiring","toNonAssocSemiring"],["Nat","instSemiring"],["HSMul","hSMul"],["instHSMul"],["LinearMap","instSMul"],["AddCommMonoid","toNatModule"],["Eq"],["Semiring"]],"valueReferences":[["rfl"],["AddMonoid","nat_smulCommClass'"],["LinearMap","instFunLike"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["DFunLike","coe"],["Nat"],["Module","toDistribMulAction"],["Nat","instMonoid"],["Nat","instSemiring"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["instHSMul"],["LinearMap","instSMul"],["AddCommMonoid","toNatModule"]]},{"isProp":false,"kind":"definition","name":["IsLinearMap","recOn"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} {M₂ : Type.{w}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.31 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.34 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.37 : AddCommMonoid.{w} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.40 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.34] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.44 : Module.{u, w} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.37] {f : M -> M₂} {motive : (IsLinearMap.{u, v, w} R M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.44 f) -> Sort.{u_1}} (t : IsLinearMap.{u, v, w} R M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.44 f), (forall (map_add : forall (x : M) (y : M), Eq.{succ w} M₂ (f (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.LinearMap.Defs.470043658._hygCtx._hyg.34)))) x y)) (HAdd.hAdd.{w, w, w} M₂ M₂ M₂ (instHAdd.{w} M₂ (AddCommMagma.toAdd.{w} M₂ (AddCommSemigroup.toAddCommMagma.{w} M₂ (AddCommMonoid.toAddCommSemigroup.{w} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.37)))) (f x) (f y))) (map_smul : forall (c : R) (x : M), Eq.{succ w} M₂ (f (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.LinearMap.Defs.470043658._hygCtx._hyg.34)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.34)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.31)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.34) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.40))))) c x)) (HSMul.hSMul.{u, w, w} R M₂ M₂ (instHSMul.{u, w} R M₂ (SMulZeroClass.toSMul.{u, w} R M₂ (AddZero.toZero.{w} M₂ (AddZeroClass.toAddZero.{w} M₂ (AddMonoid.toAddZeroClass.{w} M₂ (AddCommMonoid.toAddMonoid.{w} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.37)))) (DistribSMul.toSMulZeroClass.{u, w} R M₂ (AddMonoid.toAddZeroClass.{w} M₂ (AddCommMonoid.toAddMonoid.{w} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.37)) (DistribMulAction.toDistribSMul.{u, w} R M₂ (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.31)) (AddCommMonoid.toAddMonoid.{w} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.37) 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.18)} (f : LinearMap.{u_1, u_5, u_8, u_11} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.41) (f' : M -> M₃), (Eq.{max (succ u_11) (succ u_8)} (M -> M₃) f' (DFunLike.coe.{max (succ u_11) (succ u_8), succ u_8, succ u_11} (LinearMap.{u_1, u_5, u_8, u_11} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.41) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₃) (LinearMap.instFunLike.{u_1, u_5, u_8, u_11} R S M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.41 σ) f)) -> (LinearMap.{u_1, u_5, u_8, u_11} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.15 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.41 σ) (SemilinearMapClass.toAddHomClass.{max u_11 u_8, u_1, u_5, u_8, u_11} (LinearMap.{u_1, u_5, u_8, u_11} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.41) R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.41 (LinearMap.instFunLike.{u_1, u_5, u_8, u_11} R S M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.41 σ) (LinearMap.semilinearMapClass.{u_1, u_5, u_8, u_11} R S M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.41 σ)) f) (HSMul.hSMul.{u_1, u_8, u_8} R M M (instHSMul.{u_1, u_8} R M (SMulZeroClass.toSMul.{u_1, u_8} R M (AddZero.toZero.{u_8} M (AddZeroClass.toAddZero.{u_8} M (AddMonoid.toAddZeroClass.{u_8} M (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.21)))) (DistribSMul.toSMulZeroClass.{u_1, u_8} R M (AddMonoid.toAddZeroClass.{u_8} M (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1564126289._hygCtx._hyg.21)) (DistribMulAction.toDistribSMul.{u_1, u_8} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2250429500._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2250429500._hygCtx._hyg.76 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2250429500._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2250429500._hygCtx._hyg.84 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2250429500._hygCtx._hyg.88","typeFull":"∀ {M : Type u_8} {M₂ : Type u_10} [inst : AddCommMonoid M] [inst_1 : AddCommMonoid M₂] {R : Type u_14} {S : Type u_15}\n  [inst_2 : Semiring S] [inst_3 : SMul R M] [inst_4 : Module S M] [inst_5 : SMul R M₂] [inst_6 : Module S M₂]\n  [inst_7 : SMul R S] [IsScalarTower R S M] [IsScalarTower R S M₂], LinearMap.CompatibleSMul M M₂ R S","typeReadable":"∀ {M : Type u_8} {M₂ : Type u_10} [inst : AddCommMonoid M] [inst_1 : AddCommMonoid M₂] {R : Type u_14} {S : Type u_15}\n  [inst_2 : Semiring S] [inst_3 : SMul R M] [inst_4 : Module S M] [inst_5 : SMul R M₂] [inst_6 : Module S M₂]\n  [inst_7 : SMul R S] [IsScalarTower R S M] [IsScalarTower R S M₂], LinearMap.CompatibleSMul M M₂ R 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{R : Type.{u_1}} {S : Type.{u_5}} {M : Type.{u_8}} {M₃ : Type.{u_11}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168034._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168034._hygCtx._hyg.18 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168034._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168034._hygCtx._hyg.30 : AddCommMonoid.{u_11} M₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168034._hygCtx._hyg.33 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168034._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168034._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168034._hygCtx._hyg.41 : Module.{u_5, u_11} S M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168034._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168034._hygCtx._hyg.30] {σ : RingHom.{u_1, u_5} R S (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168034._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168034._hygCtx._hyg.18)} (f : LinearMap.{u_1, u_5, u_8, u_11} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168034._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168034._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168034._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168034._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168034._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168034._hygCtx._hyg.41) {σ' : RingHom.{u_5, u_1} S R (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168034._hygCtx._hyg.18) (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168034._hygCtx._hyg.15)} 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_4} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.33) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.51 σ₁₂)))) x._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.122 x._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.120)) (DFunLike.coe.{succ (max u_4 u_3), succ u_3, succ u_4} (LinearMap.{u_1, u_2, u_3, u_4} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.18 σ₁₂ M N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_4} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.33) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.51) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => N₂) (LinearMap.instFunLike.{u_1, u_2, u_3, u_4} R₁ R₂ M N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_4} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.33) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.51 σ₁₂) (HSMul.hSMul.{0, max u_4 u_3, max u_4 u_3} Nat (LinearMap.{u_1, u_2, u_3, u_4} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.18 σ₁₂ M N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_4} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.33) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.51) (LinearMap.{u_1, u_2, u_3, u_4} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.18 σ₁₂ M N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_4} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.33) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.51) (instHSMul.{0, max u_4 u_3} Nat (LinearMap.{u_1, u_2, u_3, u_4} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.18 σ₁₂ M N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_4} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.33) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.51) (AddMonoid.toNSMul.{max u_4 u_3} (LinearMap.{u_1, u_2, u_3, u_4} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.18 σ₁₂ M N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_4} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.33) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.51) (AddCommMonoid.toAddMonoid.{max u_4 u_3} (LinearMap.{u_1, u_2, u_3, u_4} R₁ R₂ 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(Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.142))) fₗ (HSMul.hSMul.{u_14, u_8, u_8} R M M (instHSMul.{u_14, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.145) c x)) (HSMul.hSMul.{u_14, u_10, u_10} R M₂ M₂ (instHSMul.{u_14, u_10} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.153) c (DFunLike.coe.{max (succ u_10) (succ u_8), succ u_8, succ u_10} (LinearMap.{u_15, u_15, u_8, u_10} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.142 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.142 (RingHom.id.{u_15} S (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.142)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.90 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.96 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.149 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.157) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) (LinearMap.instFunLike.{u_15, u_15, u_8, u_10} S S M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.142 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.142 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.90 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.96 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.149 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.157 (RingHom.id.{u_15} S (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.142))) fₗ x))), motive (LinearMap.CompatibleSMul.mk.{u_8, u_10, u_14, u_15} M M₂ 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u} →\n                        (t : LinearMap.CompatibleSMul M M₂ R S) →\n                          ((map_smul : ∀ (fₗ : M →ₗ[S] M₂) (c : R) (x : M), fₗ (c • x) = c • fₗ x) → motive ⋯) →\n                            motive t","typeReadable":"{M : Type u_8} →\n  {M₂ : Type u_10} →\n    [inst : AddCommMonoid M] →\n      [inst_1 : AddCommMonoid M₂] →\n        {R : Type u_14} →\n          {S : Type u_15} →\n            [inst_2 : Semiring S] →\n              [inst_3 : SMul R M] →\n                [inst_4 : Module S M] →\n                  [inst_5 : SMul R M₂] →\n                    [inst_6 : Module S M₂] →\n                      {motive : LinearMap.CompatibleSMul M M₂ R S → Sort u} →\n                        (t : LinearMap.CompatibleSMul M M₂ R S) →\n                          ((map_smul : ∀ (fₗ : M →ₗ[S] M₂) (c : R) (x : M), fₗ (c • x) = c • fₗ x) → motive ⋯) →\n                            motive t","typeReferences":[["LinearMap","CompatibleSMul"],["LinearMap","instFunLike"],["Module"],["LinearMap"],["DFunLike","coe"],["AddCommMonoid"],["SMul"],["LinearMap","CompatibleSMul","mk"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["HSMul","hSMul"],["instHSMul"],["Eq"],["Semiring"]],"valueReferences":[["LinearMap","CompatibleSMul","rec"]]},{"isProp":true,"kind":"theorem","name":["IsLinearMap","map_neg"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.580316089._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.580316089._hygCtx._hyg.18 : AddCommGroup.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.580316089._hygCtx._hyg.21 : AddCommGroup.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.580316089._hygCtx._hyg.24 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.580316089._hygCtx._hyg.15 (AddCommGroup.toAddCommMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.580316089._hygCtx._hyg.18)] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.580316089._hygCtx._hyg.28 : Module.{u_1, u_10} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.580316089._hygCtx._hyg.15 (AddCommGroup.toAddCommMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.580316089._hygCtx._hyg.21)] {f : M -> M₂}, (IsLinearMap.{u_1, u_8, u_10} R M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.580316089._hygCtx._hyg.15 (AddCommGroup.toAddCommMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.580316089._hygCtx._hyg.18) (AddCommGroup.toAddCommMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.580316089._hygCtx._hyg.21) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.580316089._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.580316089._hygCtx._hyg.28 f) -> (forall (x : M), Eq.{succ u_10} M₂ (f (Neg.neg.{u_8} M (NegZeroClass.toNeg.{u_8} M (SubNegZeroMonoid.toNegZeroClass.{u_8} M (SubtractionMonoid.toSubNegZeroMonoid.{u_8} M (SubtractionCommMonoid.toSubtractionMonoid.{u_8} M (AddCommGroup.toDivisionAddCommMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.580316089._hygCtx._hyg.18))))) x)) (Neg.neg.{u_10} M₂ (NegZeroClass.toNeg.{u_10} M₂ (SubNegZeroMonoid.toNegZeroClass.{u_10} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u_10} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u_10} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.580316089._hygCtx._hyg.21))))) (f x)))","typeFull":"∀ {R : Type u_1} {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R] [inst_1 : AddCommGroup M]\n  [inst_2 : AddCommGroup M₂] [inst_3 : Module R M] [inst_4 : Module R M₂] {f : M → M₂},\n  IsLinearMap R f → ∀ (x : M), f (-x) = -f x","typeReadable":"∀ {R : Type u_1} {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R] [inst_1 : AddCommGroup M]\n  [inst_2 : AddCommGroup M₂] [inst_3 : Module R M] [inst_4 : Module R M₂] {f : M → M₂},\n  IsLinearMap R f → ∀ (x : M), f (-x) = -f x","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["NegZeroClass","toNeg"],["AddCommGroup","toDivisionAddCommMonoid"],["Neg","neg"],["Module"],["AddCommGroup","toAddCommMonoid"],["SubtractionCommMonoid","toSubtractionMonoid"],["AddCommGroup"],["Eq"],["SubNegZeroMonoid","toNegZeroClass"],["IsLinearMap"],["Semiring"]],"valueReferences":[["IsLinearMap","mk'"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["LinearMap","map_neg"],["AddCommGroup","toAddCommMonoid"]]},{"isProp":false,"kind":"definition","name":["LinearMap","comp"],"typeFallback":"forall {R₁ : Type.{u_2}} {R₂ : Type.{u_3}} {R₃ : Type.{u_4}} {M₁ : Type.{u_9}} {M₂ : Type.{u_10}} {M₃ : Type.{u_11}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.21 : Semiring.{u_2} R₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.24 : Semiring.{u_3} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.27 : Semiring.{u_4} R₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.33 : AddCommMonoid.{u_9} M₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.36 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.39 : AddCommMonoid.{u_11} M₃] {module_M₁ : Module.{u_2, u_9} R₁ M₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.33} {module_M₂ : Module.{u_3, u_10} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.36} {module_M₃ : Module.{u_4, u_11} R₃ M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.39} {σ₁₂ : RingHom.{u_2, u_3} R₁ R₂ (Semiring.toNonAssocSemiring.{u_2} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.21) (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.24)} {σ₂₃ : RingHom.{u_3, u_4} R₂ R₃ (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.24) (Semiring.toNonAssocSemiring.{u_4} R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.27)} {σ₁₃ : RingHom.{u_2, u_4} R₁ R₃ (Semiring.toNonAssocSemiring.{u_2} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.21) (Semiring.toNonAssocSemiring.{u_4} R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.27)} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.66 : RingHomCompTriple.{u_2, u_3, u_4} R₁ R₂ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.27 σ₁₂ σ₂₃ σ₁₃], (LinearMap.{u_3, u_4, u_10, u_11} R₂ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.27 σ₂₃ M₂ M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.39 module_M₂ module_M₃) -> (LinearMap.{u_2, u_3, u_9, u_10} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.24 σ₁₂ M₁ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.36 module_M₁ module_M₂) -> (LinearMap.{u_2, u_4, u_9, u_11} R₁ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.27 σ₁₃ M₁ M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.39 module_M₁ module_M₃)","typeFull":"{R₁ : Type u_2} →\n  {R₂ : Type u_3} →\n    {R₃ : Type u_4} →\n      {M₁ : Type u_9} →\n        {M₂ : Type u_10} →\n          {M₃ : Type u_11} →\n            [inst : Semiring R₁] →\n              [inst_1 : Semiring R₂] →\n                [inst_2 : Semiring R₃] →\n                  [inst_3 : AddCommMonoid M₁] →\n                    [inst_4 : AddCommMonoid M₂] →\n                      [inst_5 : AddCommMonoid M₃] →\n                        {module_M₁ : Module R₁ M₁} →\n                          {module_M₂ : Module R₂ M₂} →\n                            {module_M₃ : Module R₃ M₃} →\n                              {σ₁₂ : R₁ →+* R₂} →\n                                {σ₂₃ : R₂ →+* R₃} →\n                                  {σ₁₃ : R₁ →+* R₃} →\n                                    [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] →\n                                      (M₂ →ₛₗ[σ₂₃] M₃) → (M₁ →ₛₗ[σ₁₂] M₂) → M₁ →ₛₗ[σ₁₃] M₃","typeReadable":"{R₁ : Type u_2} →\n  {R₂ : Type u_3} →\n    {R₃ : Type u_4} →\n      {M₁ : Type u_9} →\n        {M₂ : Type u_10} →\n          {M₃ : Type u_11} →\n            [inst : Semiring R₁] →\n              [inst_1 : Semiring R₂] →\n                [inst_2 : Semiring R₃] →\n                  [inst_3 : AddCommMonoid M₁] →\n                    [inst_4 : AddCommMonoid M₂] →\n                      [inst_5 : AddCommMonoid M₃] →\n                        {module_M₁ : Module R₁ M₁} →\n                          {module_M₂ : Module R₂ M₂} →\n                            {module_M₃ : Module R₃ M₃} →\n                              {σ₁₂ : R₁ →+* R₂} →\n                                {σ₂₃ : R₂ →+* R₃} →\n                                  {σ₁₃ : R₁ →+* R₃} →\n                                    [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] →\n                                      (M₂ →ₛₗ[σ₂₃] M₃) → (M₁ →ₛₗ[σ₁₂] M₂) → M₁ →ₛₗ[σ₁₃] M₃","typeReferences":[["RingHom"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["Module"],["LinearMap"],["RingHomCompTriple"],["Semiring"]],"valueReferences":[["LinearMap","comp","_proof_2"],["LinearMap","comp","_proof_3"],["LinearMap","instFunLike"],["AddCommMonoid","toAddCommSemigroup"],["LinearMap","mk"],["AddHom","mk"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Function","comp"],["LinearMap"],["DFunLike","coe"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","module","_proof_2"],"typeFallback":"forall {R : Type.{u_1}} {R₂ : Type.{u_2}} {S : Type.{u_5}} {M : Type.{u_3}} {M₂ : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 : Semiring.{u_2} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 : AddCommMonoid.{u_4} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 : Module.{u_1, u_3} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37 : Module.{u_2, u_4} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27] {σ₁₂ : RingHom.{u_1, u_2} R R₂ (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_2} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18)} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.65 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.72 : Module.{u_5, u_4} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.65 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.76 : SMulCommClass.{u_2, u_5, u_4} R₂ S M₂ (SMulZeroClass.toSMul.{u_2, u_4} R₂ M₂ (AddZero.toZero.{u_4} M₂ (AddZeroClass.toAddZero.{u_4} M₂ (AddMonoid.toAddZeroClass.{u_4} M₂ (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27)))) (DistribSMul.toSMulZeroClass.{u_2, u_4} R₂ M₂ (AddMonoid.toAddZeroClass.{u_4} M₂ (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27)) (DistribMulAction.toDistribSMul.{u_2, u_4} R₂ M₂ (MonoidWithZero.toMonoid.{u_2} R₂ (Semiring.toMonoidWithZero.{u_2} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18)) (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27) (Module.toDistribMulAction.{u_2, u_4} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37)))) (SMulZeroClass.toSMul.{u_5, u_4} S M₂ (AddZero.toZero.{u_4} M₂ (AddZeroClass.toAddZero.{u_4} M₂ (AddMonoid.toAddZeroClass.{u_4} M₂ (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27)))) (DistribSMul.toSMulZeroClass.{u_5, u_4} S M₂ (AddMonoid.toAddZeroClass.{u_4} M₂ (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27)) (DistribMulAction.toDistribSMul.{u_5, u_4} S M₂ (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.65)) (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27) (Module.toDistribMulAction.{u_5, u_4} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.65 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.72))))] (x._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.117 : LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37) (AddCommMonoid.toAddMonoid.{max u_4 u_3} (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37) 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[inst_1 : Semiring R₂] [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M]\n  [inst_5 : Module R₂ M₂] {σ₁₂ : R →+* R₂} [inst_6 : Semiring S] [inst_7 : Module S M₂] [inst_8 : SMulCommClass R₂ S M₂]\n  (x : M →ₛₗ[σ₁₂] M₂), 0 • x = 0","typeReferences":[["RingHom"],["Module"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["Zero","toOfNat0"],["Monoid","toSemigroup"],["Eq"],["DistribSMul","toSMulZeroClass"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["LinearMap","instDistribMulAction"],["AddZeroClass","toAddZero"],["LinearMap"],["OfNat","ofNat"],["LinearMap","addCommMonoid"],["SMulCommClass"],["Module","toDistribMulAction"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["HSMul","hSMul"],["AddZero","toZero"],["MulAction","toSemigroupAction"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["LinearMap","instFunLike"],["SemigroupAction","toSMul"],["MonoidWithZero","toMulZeroOneClass"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["Semiring","toNonAssocSemiring"],["LinearMap","ext"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["Zero","toOfNat0"],["Monoid","toSemigroup"],["MulActionWithZero","toSMulWithZero"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Semiring","toMonoidWithZero"],["Module","toMulActionWithZero"],["LinearMap","instDistribMulAction"],["LinearMap"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["LinearMap","addCommMonoid"],["Module","toDistribMulAction"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["HSMul","hSMul"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["zero_smul"],["MulAction","toSemigroupAction"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","toAddMonoidHom'","_proof_1"],"typeFallback":"forall {R₁ : Type.{u_3}} {R₂ : Type.{u_4}} {M : Type.{u_2}} {M₂ : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.15 : Semiring.{u_3} R₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.18 : Semiring.{u_4} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.24 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.27 : AddCommMonoid.{u_1} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.39 : Module.{u_3, u_2} R₁ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.43 : Module.{u_4, u_1} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.27] {σ₁₂ : RingHom.{u_3, u_4} R₁ R₂ 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.43 σ₁₂ (OfNat.ofNat.{max u_1 u_2} (LinearMap.{u_3, u_4, u_2, u_1} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.43) 0 (Zero.toOfNat0.{max u_1 u_2} (LinearMap.{u_3, u_4, u_2, u_1} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.24 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(AddCommMonoid.toAddMonoid.{max u_1 u_2} (AddMonoidHom.{u_2, u_1} M M₂ (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.24))) (AddZeroClass.toAddZero.{u_1} M₂ (AddMonoid.toAddZeroClass.{u_1} M₂ (AddCommMonoid.toAddMonoid.{u_1} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.27)))) (AddMonoidHom.instAddCommMonoid.{u_2, u_1} M M₂ (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.24)) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.27)))))))","typeFull":"∀ {R₁ : Type u_3} {R₂ : Type u_4} {M : Type u_2} {M₂ : Type u_1} [inst : Semiring R₁] [inst_1 : Semiring R₂]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R₁ M] [inst_5 : Module R₂ M₂]\n  {σ₁₂ : R₁ →+* R₂}, LinearMap.toAddMonoidHom 0 = 0","typeReadable":"∀ {R₁ : Type u_3} {R₂ : Type u_4} {M : Type u_2} {M₂ : Type u_1} [inst : Semiring R₁] [inst_1 : Semiring R₂]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R₁ M] [inst_5 : Module R₂ M₂]\n  {σ₁₂ : R₁ →+* R₂}, LinearMap.toAddMonoidHom 0 = 0","typeReferences":[["RingHom"],["LinearMap","toAddMonoidHom"],["Module"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["LinearMap"],["OfNat","ofNat"],["LinearMap","addCommMonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["AddMonoidHom"],["Zero","toOfNat0"],["AddMonoidHom","instAddCommMonoid"],["Eq"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["LinearMap","toAddMonoidHom"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["LinearMap"],["DFunLike","coe"],["OfNat","ofNat"],["LinearMap","addCommMonoid"],["AddMonoidHom"],["Eq","refl"],["AddMonoidHom","ext"],["AddMonoidHom","instFunLike"],["Zero","toOfNat0"],["AddMonoidHom","instAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","restrictScalars_apply"],"typeFallback":"forall (R : Type.{u_1}) {S : Type.{u_5}} {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.18 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.27 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.33 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.37 : Module.{u_1, u_10} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.27] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.71 : Module.{u_5, u_8} S M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.75 : Module.{u_5, u_10} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.27] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.79 : LinearMap.CompatibleSMul.{u_8, u_10, u_1, u_5} M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.27 R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.18 (SMulZeroClass.toSMul.{u_1, u_8} R M (AddZero.toZero.{u_8} M (AddZeroClass.toAddZero.{u_8} M (AddMonoid.toAddZeroClass.{u_8} M (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.21)))) 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.27)))) (DistribSMul.toSMulZeroClass.{u_1, u_10} R M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.27)) (DistribMulAction.toDistribSMul.{u_1, u_10} R M₂ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.15)) (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.27) (Module.toDistribMulAction.{u_1, u_10} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.37)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.75] (fₗ : LinearMap.{u_5, u_5, u_8, u_10} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.18 (RingHom.id.{u_5} S (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.18)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.71 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.75) (x : M), Eq.{succ u_10} M₂ (DFunLike.coe.{max (succ u_10) (succ u_8), succ u_8, succ u_10} (LinearMap.{u_1, u_1, u_8, u_10} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.15 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.15)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.37) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) (LinearMap.instFunLike.{u_1, u_1, u_8, u_10} R R M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.37 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.15))) 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(Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.18)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.71 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.75) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) (LinearMap.instFunLike.{u_5, u_5, u_8, u_10} S S M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.71 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.75 (RingHom.id.{u_5} S (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3399072275._hygCtx._hyg.18))) fₗ x)","typeFull":"∀ (R : Type u_1) {S : Type u_5} {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M] [inst_5 : Module R M₂]\n  [inst_6 : Module S M] [inst_7 : Module S M₂] [inst_8 : LinearMap.CompatibleSMul M M₂ R S] (fₗ : M →ₗ[S] M₂) (x : M),\n  (↑R fₗ) x = fₗ x","typeReadable":"∀ (R : Type u_1) {S : Type u_5} {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M] [inst_5 : Module R M₂]\n  [inst_6 : Module S M] [inst_7 : Module S M₂] [inst_8 : LinearMap.CompatibleSMul M M₂ R S] (fₗ : M →ₗ[S] M₂) (x : M),\n  (↑R fₗ) x = fₗ x","typeReferences":[["LinearMap","CompatibleSMul"],["LinearMap","instFunLike"],["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["LinearMap","restrictScalars"],["Eq"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["rfl"],["Semiring","toNonAssocSemiring"],["LinearMap","instFunLike"],["RingHom","id"],["LinearMap","restrictScalars"],["LinearMap"],["DFunLike","coe"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","Module","LinearMap","Defs",0,"LinearMap","ne_zero_of_injective","match_1_1"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.812052005._hygCtx._hyg.24 : AddCommMonoid.{u_1} M] (motive : (Exists.{succ u_1} M (fun (y : M) => Ne.{succ u_1} M y (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.LinearMap.Defs.812052005._hygCtx._hyg.24)))))))) -> Prop) (x._@.Mathlib.Algebra.Module.LinearMap.Defs.812052005._hygCtx._hyg.110 : Exists.{succ u_1} M (fun (y : M) => Ne.{succ u_1} M y (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.LinearMap.Defs.812052005._hygCtx._hyg.24)))))))), (forall (x : M) (ne : Ne.{succ u_1} M x (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.LinearMap.Defs.812052005._hygCtx._hyg.24))))))), motive (Exists.intro.{succ u_1} M (fun (y : M) => Ne.{succ u_1} M y (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.LinearMap.Defs.812052005._hygCtx._hyg.24))))))) x ne)) -> (motive x._@.Mathlib.Algebra.Module.LinearMap.Defs.812052005._hygCtx._hyg.110)","typeFull":"∀ {M : Type u_1} [inst : AddCommMonoid M] (motive : (∃ y, y ≠ 0) → Prop) (x : ∃ y, y ≠ 0),\n  (∀ (x : M) (ne : x ≠ 0), motive ⋯) → motive x","typeReadable":"∀ {M : Type u_1} [inst : AddCommMonoid M] (motive : (∃ y, y ≠ 0) → Prop) (x : ∃ y, y ≠ 0),\n  (∀ (x : M) (ne : x ≠ 0), motive ⋯) → motive x","typeReferences":[["AddCommMonoid"],["Exists"],["Exists","intro"],["Zero","toOfNat0"],["Ne"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toZero"],["OfNat","ofNat"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Exists","casesOn"],["Zero","toOfNat0"],["Ne"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toZero"],["OfNat","ofNat"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","toAddMonoidHom_mulRight"],"typeFallback":"forall (R : Type.{u_14}) {A : Type.{u_15}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 : Semiring.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20 : NonUnitalNonAssocSemiring.{u_15} A] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23 : Module.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20)] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.27 : IsScalarTower.{u_14, u_15, u_15} R A A (SMulZeroClass.toSMul.{u_14, u_15} R A (AddZero.toZero.{u_15} A (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_14, u_15} R A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20))) (DistribMulAction.toDistribSMul.{u_14, u_15} R A (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17)) (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20)) (Module.toDistribMulAction.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23)))) (SMulZeroClass.toSMul.{u_15, u_15} A A (AddZero.toZero.{u_15} A (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_15, u_15} A A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20))) (instDistribSMul.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20))) (SMulZeroClass.toSMul.{u_14, u_15} R A (AddZero.toZero.{u_15} A (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_14, u_15} R A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20))) (DistribMulAction.toDistribSMul.{u_14, u_15} R A (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17)) (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20)) (Module.toDistribMulAction.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23))))] (a : A), Eq.{succ u_15} (AddMonoidHom.{u_15, u_15} A A (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20)))) (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20))))) (AddMonoidHomClass.toAddMonoidHom.{u_15, u_15, u_15} A A (LinearMap.{u_14, u_14, u_15, u_15} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23) (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20)))) (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20)))) (LinearMap.instFunLike.{u_14, u_14, u_15, u_15} R R A A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17))) (DistribMulActionSemiHomClass.toAddMonoidHomClass.{u_15, u_14, u_14, u_15, u_15} (LinearMap.{u_14, u_14, u_15, u_15} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23) R R (DFunLike.coe.{succ u_14, succ u_14, succ u_14} (RingHom.{u_14, u_14} R R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17) (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17)) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => R) (RingHom.instFunLike.{u_14, u_14} R R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17) (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17)) (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17))) A A (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17)) (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17)) (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20)) (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20)) (Module.toDistribMulAction.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23) (Module.toDistribMulAction.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23) (LinearMap.instFunLike.{u_14, u_14, u_15, u_15} R R A A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17))) (SemilinearMapClass.distribMulActionSemiHomClass.{u_14, u_14, u_15, u_15, u_15} R R A A (LinearMap.{u_14, u_14, u_15, u_15} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17)) (LinearMap.instFunLike.{u_14, u_14, u_15, u_15} R R A A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17))) (LinearMap.semilinearMapClass.{u_14, u_14, u_15, u_15} R R A A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17))))) (LinearMap.mulRight.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3846967412._hygCtx._hyg.23 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[inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 : AddCommMonoid.{u_16} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 : AddCommMonoid.{u_17} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 : Module.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53 : Module.{u_15, u_17} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46] {motive : (LinearMap.{u_14, u_15, u_16, u_17} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53) -> Sort.{u}}, (forall (toAddHom : AddHom.{u_16, u_17} M M₂ (AddCommMagma.toAdd.{u_16} M (AddCommSemigroup.toAddCommMagma.{u_16} M (AddCommMonoid.toAddCommSemigroup.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43))) (AddCommMagma.toAdd.{u_17} M₂ (AddCommSemigroup.toAddCommMagma.{u_17} M₂ (AddCommMonoid.toAddCommSemigroup.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46)))) (map_smul' : forall (m : R) (x : M), Eq.{succ u_17} M₂ (AddHom.toFun.{u_16, u_17} M M₂ (AddCommMagma.toAdd.{u_16} M (AddCommSemigroup.toAddCommMagma.{u_16} M (AddCommMonoid.toAddCommSemigroup.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43))) (AddCommMagma.toAdd.{u_17} M₂ 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(Module.toDistribMulAction.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49))))) m x)) (HSMul.hSMul.{u_15, u_17, u_17} S M₂ M₂ (instHSMul.{u_15, u_17} S M₂ (SMulZeroClass.toSMul.{u_15, u_17} S M₂ (AddZero.toZero.{u_17} M₂ (AddZeroClass.toAddZero.{u_17} M₂ (AddMonoid.toAddZeroClass.{u_17} M₂ (AddCommMonoid.toAddMonoid.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46)))) (DistribSMul.toSMulZeroClass.{u_15, u_17} S M₂ (AddMonoid.toAddZeroClass.{u_17} M₂ (AddCommMonoid.toAddMonoid.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46)) (DistribMulAction.toDistribSMul.{u_15, u_17} S M₂ (MonoidWithZero.toMonoid.{u_15} S (Semiring.toMonoidWithZero.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33)) (AddCommMonoid.toAddMonoid.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46) (Module.toDistribMulAction.{u_15, u_17} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53))))) (DFunLike.coe.{max (succ u_14) (succ u_15), succ u_14, succ u_15} (RingHom.{u_14, u_15} R S (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30) (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33)) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => S) (RingHom.instFunLike.{u_14, u_15} R S (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30) (Semiring.toNonAssocSemiring.{u_15} S 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    {motive : (M →ₛₗ[σ] M₂) → Sort u} →\n                        ((toAddHom : M →ₙ+ M₂) →\n                            (map_smul' : ∀ (m : R) (x : M), toAddHom.toFun (m • x) = σ m • toAddHom.toFun x) →\n                              motive { toAddHom := toAddHom, map_smul' := map_smul' }) →\n                          (t : M →ₛₗ[σ] M₂) → motive t","typeReadable":"{R : Type u_14} →\n  {S : Type u_15} →\n    [inst : Semiring R] →\n      [inst_1 : Semiring S] →\n        {σ : R →+* S} →\n          {M : Type u_16} →\n            {M₂ : Type u_17} →\n              [inst_2 : AddCommMonoid M] →\n                [inst_3 : AddCommMonoid M₂] →\n                  [inst_4 : Module R M] →\n                    [inst_5 : Module S M₂] →\n                      {motive : (M →ₛₗ[σ] M₂) → Sort u} →\n                        ((toAddHom : M →ₙ+ M₂) →\n                            (map_smul' : ∀ (m : R) (x : M), toAddHom.toFun (m • x) = σ m • toAddHom.toFun x) →\n                              motive { toAddHom := toAddHom, map_smul' := map_smul' }) →\n                          (t : M →ₛₗ[σ] M₂) → motive t","typeReferences":[["RingHom"],["Module"],["LinearMap","mk"],["RingHom","instFunLike"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddHom","toFun"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Eq"],["DistribSMul","toSMulZeroClass"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["LinearMap"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["HSMul","hSMul"],["AddHom"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["LinearMap","evalAddMonoidHom_apply"],"typeFallback":"forall {R₁ : Type.{u_2}} {R₂ : Type.{u_3}} {M : Type.{u_8}} {M₂ : Type.{u_10}} 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.43) M₂ (AddZeroClass.toAddZero.{max u_10 u_8} (LinearMap.{u_2, u_3, u_8, u_10} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.43) (AddMonoid.toAddZeroClass.{max u_10 u_8} (LinearMap.{u_2, u_3, u_8, u_10} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.43) (AddCommMonoid.toAddMonoid.{max u_10 u_8} (LinearMap.{u_2, u_3, u_8, u_10} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.43) (LinearMap.addCommMonoid.{u_2, u_3, u_8, u_10} R₁ R₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.43 σ₁₂)))) (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.27)))) (LinearMap.evalAddMonoidHom.{u_2, u_3, u_8, u_10} R₁ R₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.43 σ₁₂ a) f) (DFunLike.coe.{max (succ u_10) (succ u_8), succ u_8, succ u_10} (LinearMap.{u_2, u_3, u_8, u_10} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.43) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) (LinearMap.instFunLike.{u_2, u_3, u_8, u_10} R₁ R₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.43 σ₁₂) f a)","typeFull":"∀ {R₁ : Type u_2} {R₂ : Type u_3} {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R₁] [inst_1 : Semiring R₂]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R₁ M] [inst_5 : Module R₂ M₂]\n  {σ₁₂ : R₁ →+* R₂} (a : M) (f : M →ₛₗ[σ₁₂] M₂), (LinearMap.evalAddMonoidHom a) f = f a","typeReadable":"∀ {R₁ : Type u_2} {R₂ : Type u_3} {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R₁] [inst_1 : Semiring R₂]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R₁ M] [inst_5 : Module R₂ M₂]\n  {σ₁₂ : R₁ →+* R₂} (a : M) (f : M →ₛₗ[σ₁₂] M₂), (LinearMap.evalAddMonoidHom a) f = f a","typeReferences":[["RingHom"],["LinearMap","instFunLike"],["Module"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["LinearMap","addCommMonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["LinearMap","evalAddMonoidHom"],["AddMonoidHom"],["AddMonoidHom","instFunLike"],["Eq"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["LinearMap","addCommMonoid"],["LinearMap","evalAddMonoidHom"],["AddMonoidHom"],["Eq","refl"],["AddMonoidHom","instFunLike"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["LinearMap"],["DFunLike","coe"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"definition","name":["LinearMap","coeIsScalarTower"],"typeFallback":"forall (R : Type.{u_1}) {S : Type.{u_5}} {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.18 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.27 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.33 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.37 : Module.{u_1, u_10} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.27] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.71 : Module.{u_5, u_8} S M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.75 : Module.{u_5, u_10} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.27] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.79 : LinearMap.CompatibleSMul.{u_8, u_10, u_1, u_5} M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.27 R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.18 (SMulZeroClass.toSMul.{u_1, u_8} R M (AddZero.toZero.{u_8} M (AddZeroClass.toAddZero.{u_8} M (AddMonoid.toAddZeroClass.{u_8} M (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.21)))) (DistribSMul.toSMulZeroClass.{u_1, u_8} R M (AddMonoid.toAddZeroClass.{u_8} M (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.21)) (DistribMulAction.toDistribSMul.{u_1, u_8} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.15)) (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.21) (Module.toDistribMulAction.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.33)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.71 (SMulZeroClass.toSMul.{u_1, u_10} R M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.27)))) (DistribSMul.toSMulZeroClass.{u_1, u_10} R M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.27)) (DistribMulAction.toDistribSMul.{u_1, u_10} R M₂ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.15)) (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.27) (Module.toDistribMulAction.{u_1, u_10} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.37)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.75], CoeHTCT.{max (succ u_10) (succ u_8), max (succ u_10) (succ u_8)} (LinearMap.{u_5, u_5, u_8, u_10} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.18 (RingHom.id.{u_5} S (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.18)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.71 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.75) (LinearMap.{u_1, u_1, u_8, u_10} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.15 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.15)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3573201982._hygCtx._hyg.37)","typeFull":"(R : Type u_1) →\n  {S : Type u_5} →\n    {M : Type u_8} →\n      {M₂ : Type u_10} →\n        [inst : Semiring R] →\n          [inst_1 : Semiring S] →\n            [inst_2 : AddCommMonoid M] →\n              [inst_3 : AddCommMonoid M₂] →\n                [inst_4 : Module R M] →\n                  [inst_5 : Module R M₂] →\n                    [inst_6 : Module S M] →\n                      [inst_7 : Module S M₂] → [LinearMap.CompatibleSMul M M₂ R S] → CoeHTCT (M →ₗ[S] M₂) (M →ₗ[R] M₂)","typeReadable":"(R : Type u_1) →\n  {S : Type u_5} →\n    {M : Type u_8} →\n      {M₂ : Type u_10} →\n        [inst : Semiring R] →\n          [inst_1 : Semiring S] →\n            [inst_2 : AddCommMonoid M] →\n              [inst_3 : AddCommMonoid M₂] →\n                [inst_4 : Module R M] →\n                  [inst_5 : Module R M₂] →\n                    [inst_6 : Module S M] →\n                      [inst_7 : Module S M₂] → [LinearMap.CompatibleSMul M M₂ R S] → CoeHTCT (M →ₗ[S] M₂) (M →ₗ[R] M₂)","typeReferences":[["LinearMap","CompatibleSMul"],["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["CoeHTCT"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["Semiring","toNonAssocSemiring"],["RingHom","id"],["CoeHTCT","mk"],["LinearMap","restrictScalars"],["LinearMap"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","copy","_proof_2"],"typeFallback":"forall {R : Type.{u_2}} {S : Type.{u_4}} {M : Type.{u_3}} {M₃ : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.15 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.18 : Semiring.{u_4} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.21 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.30 : AddCommMonoid.{u_1} M₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.33 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.41 : Module.{u_4, u_1} S M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.30] {σ : RingHom.{u_2, u_4} R S (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_4} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.18)} (f : LinearMap.{u_2, u_4, u_3, u_1} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.41) (f' : M -> M₃), (Eq.{max (succ u_1) (succ u_3)} (M -> M₃) f' (DFunLike.coe.{max (succ u_1) (succ u_3), succ u_3, succ u_1} (LinearMap.{u_2, u_4, u_3, u_1} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.41) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₃) (LinearMap.instFunLike.{u_2, u_4, u_3, u_1} R S M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.41 σ) f)) -> (forall (m : R) (x : M), Eq.{succ u_1} M₃ (f' (HSMul.hSMul.{u_2, u_3, u_3} R M M (instHSMul.{u_2, u_3} R M 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(HSMul.hSMul.{u_4, u_1, u_1} S M₃ M₃ (instHSMul.{u_4, u_1} S M₃ (SMulZeroClass.toSMul.{u_4, u_1} S M₃ (AddZero.toZero.{u_1} M₃ (AddZeroClass.toAddZero.{u_1} M₃ (AddMonoid.toAddZeroClass.{u_1} M₃ (AddCommMonoid.toAddMonoid.{u_1} M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.30)))) (DistribSMul.toSMulZeroClass.{u_4, u_1} S M₃ (AddMonoid.toAddZeroClass.{u_1} M₃ (AddCommMonoid.toAddMonoid.{u_1} M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.30)) (DistribMulAction.toDistribSMul.{u_4, u_1} S M₃ (MonoidWithZero.toMonoid.{u_4} S (Semiring.toMonoidWithZero.{u_4} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.18)) (AddCommMonoid.toAddMonoid.{u_1} M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.30) (Module.toDistribMulAction.{u_4, u_1} S M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.30 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{R : Type.{u_4}} {R₂ : Type.{u_5}} {S : Type.{u_2}} {M : Type.{u_3}} {M₂ : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1097252691._hygCtx._hyg.15 : Semiring.{u_4} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1097252691._hygCtx._hyg.18 : Semiring.{u_5} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1097252691._hygCtx._hyg.21 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1097252691._hygCtx._hyg.24 : AddCommMonoid.{u_1} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1097252691._hygCtx._hyg.27 : Module.{u_4, u_3} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1097252691._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1097252691._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1097252691._hygCtx._hyg.31 : Module.{u_5, u_1} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1097252691._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1097252691._hygCtx._hyg.24] {σ₁₂ : RingHom.{u_4, u_5} R R₂ 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[inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.18 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.27 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.33 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.37 : Module.{u_1, u_10} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.27] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.71 : Module.{u_5, u_8} S M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.18 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(DistribSMul.toSMulZeroClass.{u_1, u_10} R M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.27)) (DistribMulAction.toDistribSMul.{u_1, u_10} R M₂ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.15)) (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.27) (Module.toDistribMulAction.{u_1, u_10} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.37)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.75], Function.Injective.{max (succ u_10) (succ u_8), max (succ u_10) (succ u_8)} (LinearMap.{u_5, u_5, u_8, u_10} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.18 (RingHom.id.{u_5} S (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.18)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.71 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.75) (LinearMap.{u_1, u_1, u_8, u_10} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.15 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.15)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.37) (LinearMap.restrictScalars.{u_1, u_5, u_8, u_10} R S M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.71 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.75 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4146065061._hygCtx._hyg.79)","typeFull":"∀ (R : Type u_1) {S : Type u_5} {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M] [inst_5 : Module R M₂]\n  [inst_6 : Module S M] [inst_7 : Module S M₂] [inst_8 : LinearMap.CompatibleSMul M M₂ R S], Function.Injective ↑R","typeReadable":"∀ (R : Type u_1) {S : Type u_5} {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M] [inst_5 : Module R M₂]\n  [inst_6 : Module S M] [inst_7 : Module S M₂] [inst_8 : LinearMap.CompatibleSMul M M₂ R S], Function.Injective ↑R","typeReferences":[["LinearMap","CompatibleSMul"],["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["LinearMap","restrictScalars"],["AddZero","toZero"],["Function","Injective"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["Semiring","toNonAssocSemiring"],["RingHom","id"],["LinearMap","ext"],["LinearMap","restrictScalars"],["LinearMap","congr_fun"]]},{"isProp":false,"kind":"definition","name":["_aux_Mathlib_Algebra_Module_LinearMap_Defs___macroRules_term_→ₗ[_]__1"],"typeFallback":"Lean.Macro","typeFull":"Lean.Macro","typeReadable":"Lean.Macro","typeReferences":[["Lean","Macro"]],"valueReferences":[["Lean","Name"],["Lean","TSyntax","raw"],["Lean","Macro","State"],["Lean","MonadQuotation","getCurrMacroScope"],["Lean","MacroScope"],["Lean","SourceInfo"],["Lean","Syntax","ident"],["String"],["Lean","Macro","Context"],["Lean","Macro","Exception","unsupportedSyntax"],["Lean","MonadRef","mkInfoFromRefPos"],["ReaderT","instApplicativeOfMonad"],["Lean","Syntax","Preresolved"],["List","cons"],["Bool","true"],["Lean","Syntax","Preresolved","decl"],["MonadExcept","throw"],["Lean","Name","mkStr4"],["Lean","SyntaxNodeKind"],["Lean","TSyntax","mk"],["instMonadExceptOfMonadExceptOf"],["EStateM","instMonad"],["Lean","Macro","instMonadQuotationMacroM"],["Lean","Syntax"],["instDecidableEqBool"],["Nat"],["Monad","toBind"],["Lean","Syntax","atom"],["Lean","Name","anonymous"],["Bool"],["Applicative","toPure"],["ReaderT","instMonadExceptOf"],["EStateM"],["String","toRawSubstring'"],["Lean","Syntax","isOfKind"],["PUnit"],["instOfNatNat"],["Lean","MonadQuotation","getContext"],["Lean","addMacroScope"],["Lean","Name","mkStr2"],["Lean","MacroM"],["Eq"],["Lean","Syntax","node2"],["EStateM","instMonadExceptOfOfBacktrackable"],["Bind","bind"],["List","nil"],["Lean","Name","mkStr1"],["ite"],["Lean","Syntax","node1"],["Lean","Syntax","Preresolved","namespace"],["Lean","Macro","instMonadRefMacroM"],["Lean","Syntax","node3"],["Lean","Macro","Exception"],["ReaderT","instMonad"],["OfNat","ofNat"],["EStateM","nonBacktrackable"],["Lean","Syntax","getArg"],["Pure","pure"]]},{"isProp":false,"kind":"definition","name":["LinearMap","instDistribMulAction"],"typeFallback":"forall {R : Type.{u_1}} {R₂ : Type.{u_3}} {S : Type.{u_5}} {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 : Semiring.{u_3} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37 : Module.{u_3, u_10} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27] {σ₁₂ : RingHom.{u_1, u_3} R R₂ (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18)} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.65 : Monoid.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.68 : DistribMulAction.{u_5, u_10} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.65 (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27)] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.72 : SMulCommClass.{u_3, u_5, u_10} R₂ S M₂ (SMulZeroClass.toSMul.{u_3, u_10} R₂ M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27)))) (DistribSMul.toSMulZeroClass.{u_3, u_10} R₂ M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27)) (DistribMulAction.toDistribSMul.{u_3, u_10} R₂ M₂ (MonoidWithZero.toMonoid.{u_3} R₂ (Semiring.toMonoidWithZero.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18)) (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27) (Module.toDistribMulAction.{u_3, u_10} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37)))) (SMulZeroClass.toSMul.{u_5, u_10} S M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27)))) (DistribSMul.toSMulZeroClass.{u_5, u_10} S M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27)) (DistribMulAction.toDistribSMul.{u_5, u_10} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.65 (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.68)))], DistribMulAction.{u_5, max u_10 u_8} S (LinearMap.{u_1, u_3, u_8, u_10} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.65 (AddCommMonoid.toAddMonoid.{max u_10 u_8} (LinearMap.{u_1, u_3, u_8, u_10} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (LinearMap.addCommMonoid.{u_1, u_3, u_8, u_10} R R₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37 σ₁₂))","typeFull":"{R : Type u_1} →\n  {R₂ : Type u_3} →\n    {S : Type u_5} →\n      {M : Type u_8} →\n        {M₂ : Type u_10} →\n          [inst : Semiring R] →\n            [inst_1 : Semiring R₂] →\n              [inst_2 : AddCommMonoid M] →\n                [inst_3 : AddCommMonoid M₂] →\n                  [inst_4 : Module R M] →\n                    [inst_5 : Module R₂ M₂] →\n                      {σ₁₂ : R →+* R₂} →\n                        [inst_6 : Monoid S] →\n                          [inst_7 : DistribMulAction S M₂] →\n                            [SMulCommClass R₂ S M₂] → DistribMulAction S (M →ₛₗ[σ₁₂] M₂)","typeReadable":"{R : Type u_1} →\n  {R₂ : Type u_3} →\n    {S : Type u_5} →\n      {M : Type u_8} →\n        {M₂ : Type u_10} →\n          [inst : Semiring R] →\n            [inst_1 : Semiring R₂] →\n              [inst_2 : AddCommMonoid M] →\n                [inst_3 : AddCommMonoid M₂] →\n              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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.37 : Module.{u_1, u_10} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.27] (fₗ : LinearMap.{u_1, u_1, u_8, u_10} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.15 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.15)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.37), IsLinearMap.{u_1, u_8, u_10} R M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.37 (DFunLike.coe.{max (succ u_10) (succ u_8), succ u_8, succ u_10} (LinearMap.{u_1, u_1, u_8, u_10} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.15 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.15)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.37) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) (LinearMap.instFunLike.{u_1, u_1, u_8, u_10} R R M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.37 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4010769530._hygCtx._hyg.15))) fₗ)","typeFull":"∀ {R : Type u_1} {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M₂] [inst_3 : Module R M] [inst_4 : Module R M₂] (fₗ : M →ₗ[R] M₂), IsLinearMap R ⇑fₗ","typeReadable":"∀ {R : Type u_1} {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M₂] [inst_3 : Module R M] [inst_4 : Module R M₂] (fₗ : M →ₗ[R] M₂), IsLinearMap R ⇑fₗ","typeReferences":[["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["LinearMap","instFunLike"],["RingHom","id"],["Module"],["LinearMap"],["DFunLike","coe"],["IsLinearMap"],["Semiring"]],"valueReferences":[["AddHom","map_add'"],["IsLinearMap","mk"],["Semiring","toNonAssocSemiring"],["AddCommMonoid","toAddCommSemigroup"],["LinearMap","instFunLike"],["RingHom","id"],["LinearMap","toAddHom"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["LinearMap","map_smul'"],["LinearMap"],["DFunLike","coe"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","smul_comp"],"typeFallback":"forall {R : Type.{u_1}} {R₂ : Type.{u_3}} {R₃ : Type.{u_4}} {S₃ : Type.{u_6}} {M : Type.{u_8}} {M₂ : Type.{u_10}} {M₃ : Type.{u_11}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.18 : Semiring.{u_3} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.21 : Semiring.{u_4} R₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.24 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.27 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.30 : AddCommMonoid.{u_11} M₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.33 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.37 : Module.{u_3, u_10} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.27] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.41 : Module.{u_4, u_11} R₃ M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.30] {σ₁₂ : RingHom.{u_1, u_3} R R₂ (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.18)} {σ₂₃ : RingHom.{u_3, u_4} R₂ R₃ (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.18) (Semiring.toNonAssocSemiring.{u_4} R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.21)} {σ₁₃ : RingHom.{u_1, u_4} R R₃ (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_4} R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.21)} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.60 : RingHomCompTriple.{u_1, u_3, u_4} R R₂ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.21 σ₁₂ σ₂₃ σ₁₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.77 : Monoid.{u_6} S₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.80 : DistribMulAction.{u_6, u_11} S₃ M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.77 (AddCommMonoid.toAddMonoid.{u_11} M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.30)] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.84 : SMulCommClass.{u_4, u_6, u_11} R₃ S₃ M₃ (SMulZeroClass.toSMul.{u_4, u_11} R₃ M₃ 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u_11} S₃ M₃ (AddZero.toZero.{u_11} M₃ (AddZeroClass.toAddZero.{u_11} M₃ (AddMonoid.toAddZeroClass.{u_11} M₃ (AddCommMonoid.toAddMonoid.{u_11} M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.30)))) (DistribSMul.toSMulZeroClass.{u_6, u_11} S₃ M₃ (AddMonoid.toAddZeroClass.{u_11} M₃ (AddCommMonoid.toAddMonoid.{u_11} M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.30)) (DistribMulAction.toDistribSMul.{u_6, u_11} S₃ M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.77 (AddCommMonoid.toAddMonoid.{u_11} M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.30) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.80)))] (a : S₃) (g : LinearMap.{u_3, u_4, u_10, u_11} R₂ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.21 σ₂₃ M₂ M₃ 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.41) (LinearMap.comp.{u_1, u_3, u_4, u_8, u_10, u_11} R R₂ R₃ M M₂ M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.37 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.41 σ₂₃ (DistribMulAction.toDistribSMul.{u_6, u_11} S₃ M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.77 (AddCommMonoid.toAddMonoid.{u_11} M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.30) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.80) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.84)) a g) f) (HSMul.hSMul.{u_6, max u_11 u_8, max u_11 u_8} S₃ (LinearMap.{u_1, u_4, u_8, u_11} R R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.21 σ₁₃ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.41) (LinearMap.{u_1, u_4, u_8, u_11} R R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.21 σ₁₃ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.41) (instHSMul.{u_6, max u_11 u_8} S₃ (LinearMap.{u_1, u_4, u_8, u_11} R R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.21 σ₁₃ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.41) (LinearMap.instSMul.{u_1, u_4, u_6, u_8, u_11} R R₃ S₃ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.41 σ₁₃ (DistribMulAction.toDistribSMul.{u_6, u_11} S₃ M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.77 (AddCommMonoid.toAddMonoid.{u_11} M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.30) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.80) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.84)) a (LinearMap.comp.{u_1, u_3, u_4, u_8, u_10, u_11} R R₂ R₃ M M₂ M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.41 σ₁₂ σ₂₃ σ₁₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.379536305._hygCtx._hyg.60 g f))","typeFull":"∀ {R : Type u_1} {R₂ : Type u_3} {R₃ : Type u_4} {S₃ : Type u_6} {M : Type u_8} {M₂ : Type u_10} {M₃ : Type u_11}\n  [inst : Semiring R] [inst_1 : Semiring R₂] [inst_2 : Semiring R₃] [inst_3 : AddCommMonoid M]\n  [inst_4 : AddCommMonoid M₂] [inst_5 : AddCommMonoid M₃] [inst_6 : Module R M] [inst_7 : Module R₂ M₂]\n  [inst_8 : Module R₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [inst_9 : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]\n  [inst_10 : Monoid S₃] [inst_11 : DistribMulAction S₃ M₃] [inst_12 : SMulCommClass R₃ S₃ M₃] (a : S₃)\n  (g : M₂ →ₛₗ[σ₂₃] M₃) (f : M →ₛₗ[σ₁₂] M₂), (a • g) ∘ₛₗ f = a • g ∘ₛₗ f","typeReadable":"∀ {R : Type u_1} {R₂ : Type u_3} {R₃ : Type u_4} {S₃ : Type u_6} {M : Type u_8} {M₂ : Type u_10} {M₃ : Type u_11}\n  [inst : Semiring R] [inst_1 : Semiring R₂] [inst_2 : Semiring R₃] [inst_3 : AddCommMonoid M]\n  [inst_4 : AddCommMonoid M₂] [inst_5 : AddCommMonoid M₃] [inst_6 : Module R M] [inst_7 : Module R₂ M₂]\n  [inst_8 : Module R₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [inst_9 : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]\n  [inst_10 : Monoid S₃] [inst_11 : DistribMulAction S₃ M₃] [inst_12 : SMulCommClass R₃ S₃ M₃] (a : S₃)\n  (g : M₂ →ₛₗ[σ₂₃] M₃) (f : M →ₛₗ[σ₁₂] M₂), (a • g) ∘ₛₗ f = a • g ∘ₛₗ f","typeReferences":[["RingHom"],["Module"],["LinearMap","comp"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["RingHomCompTriple"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["LinearMap"],["SMulCommClass"],["Module","toDistribMulAction"],["HSMul","hSMul"],["Monoid"],["LinearMap","instSMul"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["rfl"],["HSMul","hSMul"],["DistribMulAction","toDistribSMul"],["LinearMap","instSMul"],["instHSMul"],["LinearMap","comp"],["AddCommMonoid","toAddMonoid"],["LinearMap"]]},{"isProp":false,"kind":"definition","name":["IsLinearMap","casesOn"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} {M₂ : Type.{w}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.31 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.34 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.37 : AddCommMonoid.{w} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.40 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.34] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.44 : Module.{u, w} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.37] {f : M -> M₂} {motive : (IsLinearMap.{u, v, w} R M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.44 f) -> Sort.{u_1}} (t : IsLinearMap.{u, v, w} R M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.44 f), (forall (map_add : forall (x : M) (y : M), Eq.{succ w} M₂ (f (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.LinearMap.Defs.470043658._hygCtx._hyg.34)))) x y)) (HAdd.hAdd.{w, w, w} M₂ M₂ M₂ (instHAdd.{w} M₂ (AddCommMagma.toAdd.{w} M₂ (AddCommSemigroup.toAddCommMagma.{w} M₂ (AddCommMonoid.toAddCommSemigroup.{w} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.37)))) (f x) (f y))) (map_smul : forall (c : R) (x : M), Eq.{succ w} M₂ (f (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.LinearMap.Defs.470043658._hygCtx._hyg.34)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.34)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.31)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.34) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.40))))) c x)) (HSMul.hSMul.{u, w, w} R M₂ M₂ (instHSMul.{u, w} R M₂ (SMulZeroClass.toSMul.{u, w} R M₂ (AddZero.toZero.{w} M₂ (AddZeroClass.toAddZero.{w} M₂ (AddMonoid.toAddZeroClass.{w} M₂ (AddCommMonoid.toAddMonoid.{w} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.37)))) (DistribSMul.toSMulZeroClass.{u, w} R M₂ (AddMonoid.toAddZeroClass.{w} M₂ (AddCommMonoid.toAddMonoid.{w} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.37)) (DistribMulAction.toDistribSMul.{u, w} R M₂ (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.31)) (AddCommMonoid.toAddMonoid.{w} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.37) (Module.toDistribMulAction.{u, w} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.44))))) c (f x))), motive (IsLinearMap.mk.{u, v, w} R M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.44 f map_add map_smul)) -> (motive t)","typeFull":"{R : Type u} →\n  {M : Type v} →\n    {M₂ : Type w} →\n      [inst : Semiring R] →\n        [inst_1 : AddCommMonoid M] →\n          [inst_2 : AddCommMonoid M₂] →\n            [inst_3 : Module R M] →\n              [inst_4 : Module R M₂] →\n                {f : M → M₂} →\n                  {motive : IsLinearMap R f → Sort u_1} →\n                    (t : IsLinearMap R f) →\n                      ((map_add : ∀ (x y : M), f (x + y) = f x + f y) →\n                          (map_smul : ∀ (c : R) (x : M), f (c • x) = c • f x) → motive ⋯) →\n                        motive t","typeReadable":"{R : Type u} →\n  {M : Type v} →\n    {M₂ : Type w} →\n      [inst : Semiring R] →\n        [inst_1 : AddCommMonoid M] →\n          [inst_2 : AddCommMonoid M₂] →\n            [inst_3 : Module R M] →\n              [inst_4 : Module R M₂] →\n                {f : M → M₂} →\n                  {motive : IsLinearMap R f → Sort u_1} →\n                    (t : IsLinearMap R f) →\n                      ((map_add : ∀ (x y : M), f (x + y) = f x + f y) →\n                          (map_smul : ∀ (c : R) (x : M), f (c • x) = c • f x) → motive ⋯) →\n                        motive t","typeReferences":[["IsLinearMap","mk"],["Module"],["instHAdd"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["HAdd","hAdd"],["Module","toDistribMulAction"],["AddCommMonoid"],["AddCommMonoid","toAddCommSemigroup"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["instHSMul"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Eq"],["AddZero","toZero"],["IsLinearMap"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["IsLinearMap","rec"]]},{"isProp":true,"kind":"theorem","name":["LinearMapClass","map_smul"],"typeFallback":"forall {R : outParam.{succ (succ u_14)} Type.{u_14}} {M : outParam.{succ (succ u_15)} Type.{u_15}} {M₂ : outParam.{succ (succ u_16)} Type.{u_16}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404420._hygCtx._hyg.21 : Semiring.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404420._hygCtx._hyg.24 : AddCommMonoid.{u_15} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404420._hygCtx._hyg.27 : AddCommMonoid.{u_16} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404420._hygCtx._hyg.30 : Module.{u_14, u_15} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404420._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404420._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404420._hygCtx._hyg.34 : Module.{u_14, u_16} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404420._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404420._hygCtx._hyg.27] {F : Type.{u_17}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404420._hygCtx._hyg.39 : FunLike.{succ u_17, succ u_15, succ u_16} F M M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404420._hygCtx._hyg.44 : LinearMapClass.{u_17, u_14, u_15, u_16} F R M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404420._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404420._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404420._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404420._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404420._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404420._hygCtx._hyg.39] (f : F) (r : R) (x : M), Eq.{succ u_16} M₂ (DFunLike.coe.{succ u_17, succ u_15, succ u_16} F M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404420._hygCtx._hyg.39 f (HSMul.hSMul.{u_14, u_15, u_15} R M M (instHSMul.{u_14, u_15} R M (SMulZeroClass.toSMul.{u_14, u_15} R M (AddZero.toZero.{u_15} M (AddZeroClass.toAddZero.{u_15} M (AddMonoid.toAddZeroClass.{u_15} M (AddCommMonoid.toAddMonoid.{u_15} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404420._hygCtx._hyg.24)))) (DistribSMul.toSMulZeroClass.{u_14, u_15} R M (AddMonoid.toAddZeroClass.{u_15} M (AddCommMonoid.toAddMonoid.{u_15} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404420._hygCtx._hyg.24)) (DistribMulAction.toDistribSMul.{u_14, u_15} R M (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404420._hygCtx._hyg.21)) (AddCommMonoid.toAddMonoid.{u_15} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404420._hygCtx._hyg.24) (Module.toDistribMulAction.{u_14, u_15} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404420._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404420._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404420._hygCtx._hyg.30))))) r x)) (HSMul.hSMul.{u_14, u_16, u_16} R M₂ M₂ (instHSMul.{u_14, u_16} R M₂ (SMulZeroClass.toSMul.{u_14, u_16} R M₂ (AddZero.toZero.{u_16} M₂ (AddZeroClass.toAddZero.{u_16} M₂ (AddMonoid.toAddZeroClass.{u_16} M₂ 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(x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404420._hygCtx._hyg.39 f x))","typeFull":"∀ {R : outParam (Type u_14)} {M : outParam (Type u_15)} {M₂ : outParam (Type u_16)} [inst : Semiring R]\n  [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M₂] [inst_3 : Module R M] [inst_4 : Module R M₂] {F : Type u_17}\n  [inst_5 : FunLike F M M₂] [LinearMapClass F R M M₂] (f : F) (r : R) (x : M), f (r • x) = r • f x","typeReadable":"∀ {R : outParam (Type u_14)} {M : outParam (Type u_15)} {M₂ : outParam (Type u_16)} [inst : Semiring R]\n  [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M₂] [inst_3 : Module R M] [inst_4 : Module R M₂] {F : Type u_17}\n  [inst_5 : FunLike F M M₂] [LinearMapClass F R M M₂] (f : F) (r : R) (x : M), f (r • x) = r • f 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{R : Type.{u_1}} {S : Type.{u_5}} {M : Type.{u_8}} {M₃ : Type.{u_11}} {F : Type.{u_14}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.16 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.19 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.22 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.31 : AddCommMonoid.{u_11} M₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.34 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.22] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.42 : Module.{u_5, u_11} S M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.31] {σ : RingHom.{u_1, u_5} R S (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.16) (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.19)}, F -> (forall [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.52 : FunLike.{succ u_14, succ u_8, succ u_11} F M M₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.57 : SemilinearMapClass.{u_14, u_1, u_5, u_8, u_11} F R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.19 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.42 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.52], LinearMap.{u_1, u_5, u_8, u_11} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.19 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.42)","typeFull":"{R : Type u_1} →\n  {S : Type u_5} →\n    {M : Type u_8} →\n      {M₃ : Type u_11} →\n        {F : Type u_14} →\n          [inst : Semiring R] →\n            [inst_1 : Semiring S] →\n              [inst_2 : AddCommMonoid M] →\n                [inst_3 : AddCommMonoid M₃] →\n                  [inst_4 : Module R M] →\n                    [inst_5 : Module S M₃] →\n                      {σ : R →+* S} → F → [inst_6 : FunLike F M M₃] → [SemilinearMapClass F σ M M₃] → M →ₛₗ[σ] M₃","typeReadable":"{R : Type u_1} →\n  {S : Type u_5} →\n    {M : Type u_8} →\n      {M₃ : Type u_11} →\n        {F : Type u_14} →\n          [inst : Semiring R] →\n            [inst_1 : Semiring S] →\n              [inst_2 : AddCommMonoid M] →\n                [inst_3 : AddCommMonoid M₃] →\n                  [inst_4 : Module R M] →\n                    [inst_5 : Module S M₃] →\n                      {σ : R →+* S} → F → [inst_6 : FunLike F M M₃] → [SemilinearMapClass F σ M M₃] → M →ₛₗ[σ] M₃","typeReferences":[["FunLike"],["RingHom"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["SemilinearMapClass"],["Module"],["LinearMap"],["Semiring"]],"valueReferences":[["AddCommMonoid","toAddCommSemigroup"],["LinearMap","mk"],["AddHom","mk"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["SemilinearMapClass","semilinearMap","_proof_2"],["SemilinearMapClass","semilinearMap","_proof_1"],["DFunLike","coe"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","instZero","_proof_2"],"typeFallback":"forall {R₁ : Type.{u_3}} {R₂ : Type.{u_4}} {M : Type.{u_2}} {M₂ : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.15 : Semiring.{u_3} R₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.18 : Semiring.{u_4} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.24 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.27 : AddCommMonoid.{u_1} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.39 : Module.{u_3, u_2} R₁ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.43 : Module.{u_4, u_1} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.27] {σ₁₂ : RingHom.{u_3, u_4} R₁ R₂ (Semiring.toNonAssocSemiring.{u_3} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_4} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.18)} (m : R₁) (x : M), Eq.{succ u_1} M₂ (OfNat.ofNat.{max u_1 u_2} (M -> M₂) 0 (Zero.toOfNat0.{max u_1 u_2} (M -> M₂) (Pi.instZero.{u_2, u_1} M (fun (a._@._internal._hyg.0 : M) => M₂) (fun (i : M) => AddZero.toZero.{u_1} M₂ (AddZeroClass.toAddZero.{u_1} M₂ (AddMonoid.toAddZeroClass.{u_1} M₂ (AddCommMonoid.toAddMonoid.{u_1} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.27)))))) (HSMul.hSMul.{u_3, u_2, u_2} R₁ M M (instHSMul.{u_3, u_2} R₁ M (SMulZeroClass.toSMul.{u_3, u_2} R₁ M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.24)))) (DistribSMul.toSMulZeroClass.{u_3, u_2} R₁ M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.24)) (DistribMulAction.toDistribSMul.{u_3, u_2} R₁ M (MonoidWithZero.toMonoid.{u_3} R₁ (Semiring.toMonoidWithZero.{u_3} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.15)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.24) (Module.toDistribMulAction.{u_3, u_2} R₁ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.39))))) m x)) (HSMul.hSMul.{u_4, u_1, u_1} R₂ M₂ M₂ (instHSMul.{u_4, u_1} R₂ M₂ (SMulZeroClass.toSMul.{u_4, u_1} R₂ M₂ (AddZero.toZero.{u_1} M₂ (AddZeroClass.toAddZero.{u_1} M₂ (AddMonoid.toAddZeroClass.{u_1} M₂ (AddCommMonoid.toAddMonoid.{u_1} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.27)))) (DistribSMul.toSMulZeroClass.{u_4, u_1} R₂ M₂ (AddMonoid.toAddZeroClass.{u_1} M₂ (AddCommMonoid.toAddMonoid.{u_1} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.27)) (DistribMulAction.toDistribSMul.{u_4, u_1} R₂ M₂ (MonoidWithZero.toMonoid.{u_4} R₂ (Semiring.toMonoidWithZero.{u_4} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.18)) (AddCommMonoid.toAddMonoid.{u_1} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.27) (Module.toDistribMulAction.{u_4, u_1} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.43))))) (DFunLike.coe.{max (succ u_3) (succ u_4), succ u_3, succ u_4} (RingHom.{u_3, u_4} R₁ R₂ (Semiring.toNonAssocSemiring.{u_3} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_4} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.18)) R₁ (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R₁) => R₂) (RingHom.instFunLike.{u_3, u_4} R₁ R₂ (Semiring.toNonAssocSemiring.{u_3} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_4} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.18)) σ₁₂ m) (OfNat.ofNat.{max u_1 u_2} (M -> M₂) 0 (Zero.toOfNat0.{max u_1 u_2} (M -> M₂) (Pi.instZero.{u_2, u_1} M (fun (a._@._internal._hyg.0 : M) => M₂) (fun (i : M) => AddZero.toZero.{u_1} M₂ (AddZeroClass.toAddZero.{u_1} M₂ (AddMonoid.toAddZeroClass.{u_1} M₂ (AddCommMonoid.toAddMonoid.{u_1} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.27)))))) x))","typeFull":"∀ {R₁ : Type u_3} {R₂ : Type u_4} {M : Type u_2} {M₂ : Type u_1} [inst : Semiring R₁] [inst_1 : Semiring R₂]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R₁ M] [inst_5 : Module R₂ M₂]\n  {σ₁₂ : R₁ →+* R₂} (m : R₁) (x : M), 0 (m • x) = σ₁₂ m • 0 x","typeReadable":"∀ {R₁ : Type u_3} {R₂ : Type u_4} {M : Type u_2} {M₂ : Type u_1} [inst : Semiring R₁] [inst_1 : Semiring R₂]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R₁ M] [inst_5 : Module R₂ M₂]\n  {σ₁₂ : R₁ →+* R₂} (m : R₁) (x : M), 0 (m • x) = σ₁₂ m • 0 x","typeReferences":[["RingHom"],["Module"],["DistribMulAction","toDistribSMul"],["RingHom","instFunLike"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["OfNat","ofNat"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["instHSMul"],["Zero","toOfNat0"],["Eq"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["Semiring"],["Pi","instZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["smul_zero"],["RingHom"],["Eq","trans"],["RingHom","instFunLike"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["congrArg"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["forall_congr"],["instHSMul"],["Zero","toOfNat0"],["Eq"],["DistribSMul","toSMulZeroClass"],["True"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["implies_true"],["eq_self"],["Module","toDistribMulAction"],["of_eq_true"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Pi","instZero"]]},{"isProp":false,"kind":"definition","name":["term_→ₛₗ[_]_"],"typeFallback":"Lean.TrailingParserDescr","typeFull":"Lean.TrailingParserDescr","typeReadable":"Lean.TrailingParserDescr","typeReferences":[["Lean","TrailingParserDescr"]],"valueReferences":[["Nat"],["Lean","ParserDescr","cat"],["Lean","Name","mkStr1"],["instOfNatNat"],["Lean","ParserDescr","trailingNode"],["Lean","ParserDescr","symbol"],["OfNat","ofNat"],["Lean","ParserDescr","binary"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","instIsCentralScalar"],"typeFallback":"forall {R : Type.{u_1}} {R₂ : Type.{u_3}} {S : Type.{u_5}} {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.18 : Semiring.{u_3} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.24 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.27 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.31 : Module.{u_3, u_10} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.24] {σ₁₂ : RingHom.{u_1, u_3} R R₂ (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.18)} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.40 : DistribSMul.{u_5, u_10} S M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.24))] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.44 : SMulCommClass.{u_3, u_5, u_10} R₂ S M₂ (SMulZeroClass.toSMul.{u_3, u_10} R₂ M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.24)))) (DistribSMul.toSMulZeroClass.{u_3, u_10} R₂ M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.24)) (DistribMulAction.toDistribSMul.{u_3, u_10} R₂ M₂ (MonoidWithZero.toMonoid.{u_3} R₂ (Semiring.toMonoidWithZero.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.18)) (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.24) (Module.toDistribMulAction.{u_3, u_10} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.31)))) (SMulZeroClass.toSMul.{u_5, u_10} S M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.24)))) (DistribSMul.toSMulZeroClass.{u_5, u_10} S M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.24)) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.40))] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.58 : DistribSMul.{u_5, u_10} (MulOpposite.{u_5} S) M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.24))] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.65 : SMulCommClass.{u_3, u_5, u_10} R₂ (MulOpposite.{u_5} S) M₂ (SMulZeroClass.toSMul.{u_3, u_10} R₂ M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.24)))) (DistribSMul.toSMulZeroClass.{u_3, u_10} R₂ M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.24)) (DistribMulAction.toDistribSMul.{u_3, u_10} R₂ M₂ (MonoidWithZero.toMonoid.{u_3} R₂ (Semiring.toMonoidWithZero.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.18)) (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.24) (Module.toDistribMulAction.{u_3, u_10} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.31)))) (SMulZeroClass.toSMul.{u_5, u_10} (MulOpposite.{u_5} S) M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.24)))) (DistribSMul.toSMulZeroClass.{u_5, u_10} (MulOpposite.{u_5} S) M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.24)) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.58))] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.73 : IsCentralScalar.{u_5, u_10} S M₂ (SMulZeroClass.toSMul.{u_5, u_10} S M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.24)))) (DistribSMul.toSMulZeroClass.{u_5, u_10} S M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.24)) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.40)) (SMulZeroClass.toSMul.{u_5, u_10} (MulOpposite.{u_5} S) M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ 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u_8, u_10} R R₂ S M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.31 σ₁₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.44) (LinearMap.instSMul.{u_1, u_3, u_5, u_8, u_10} R R₂ (MulOpposite.{u_5} S) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.31 σ₁₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.58 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1836316326._hygCtx._hyg.65)","typeFull":"∀ {R : Type u_1} {R₂ : Type u_3} {S : Type u_5} {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R]\n  [inst_1 : Semiring R₂] [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M]\n  [inst_5 : Module R₂ M₂] {σ₁₂ : R →+* R₂} [inst_6 : DistribSMul S M₂] [inst_7 : SMulCommClass R₂ S M₂]\n  [inst_8 : DistribSMul Sᵐᵒᵖ M₂] [inst_9 : SMulCommClass R₂ Sᵐᵒᵖ M₂] [IsCentralScalar S M₂],\n  IsCentralScalar S (M →ₛₗ[σ₁₂] M₂)","typeReadable":"∀ {R : Type u_1} {R₂ : Type u_3} {S : Type u_5} {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R]\n  [inst_1 : Semiring R₂] [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M]\n  [inst_5 : Module R₂ M₂] {σ₁₂ : R →+* R₂} [inst_6 : DistribSMul S M₂] [inst_7 : SMulCommClass R₂ S M₂]\n  [inst_8 : DistribSMul Sᵐᵒᵖ M₂] [inst_9 : SMulCommClass R₂ Sᵐᵒᵖ M₂] [IsCentralScalar S M₂],\n  IsCentralScalar S (M →ₛₗ[σ₁₂] M₂)","typeReferences":[["RingHom"],["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["DistribSMul"],["SMulCommClass"],["MulOpposite"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["IsCentralScalar"],["MonoidWithZero","toMonoid"],["LinearMap","instSMul"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["LinearMap","instFunLike"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["MulOpposite"],["LinearMap","ext"],["MulOpposite","op"],["HSMul","hSMul"],["IsCentralScalar","mk"],["LinearMap","instSMul"],["instHSMul"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["IsCentralScalar","op_smul_eq_smul"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"definition","name":["LinearMap","toAddMonoidHom'"],"typeFallback":"forall {R₁ : Type.{u_2}} {R₂ : Type.{u_3}} {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.15 : Semiring.{u_2} R₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.18 : Semiring.{u_3} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.24 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.27 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.39 : Module.{u_2, u_8} R₁ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.43 : Module.{u_3, u_10} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.27] {σ₁₂ : RingHom.{u_2, u_3} R₁ R₂ 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(AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.27)))) (AddZeroClass.toAddZero.{max u_10 u_8} (LinearMap.{u_2, u_3, u_8, u_10} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.43) (AddMonoid.toAddZeroClass.{max u_10 u_8} (LinearMap.{u_2, u_3, u_8, u_10} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.43) (AddCommMonoid.toAddMonoid.{max u_10 u_8} (LinearMap.{u_2, u_3, u_8, u_10} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.43) (LinearMap.addCommMonoid.{u_2, u_3, u_8, u_10} R₁ R₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.43 σ₁₂)))) (AddZeroClass.toAddZero.{max u_10 u_8} (AddMonoidHom.{u_8, u_10} M M₂ (AddZeroClass.toAddZero.{u_8} M (AddMonoid.toAddZeroClass.{u_8} M (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.24))) (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.27)))) (AddMonoid.toAddZeroClass.{max u_10 u_8} (AddMonoidHom.{u_8, u_10} M M₂ (AddZeroClass.toAddZero.{u_8} M (AddMonoid.toAddZeroClass.{u_8} M (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.24))) 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Semiring R₁] →\n          [inst_1 : Semiring R₂] →\n            [inst_2 : AddCommMonoid M] →\n              [inst_3 : AddCommMonoid M₂] →\n                [inst_4 : Module R₁ M] → [inst_5 : Module R₂ M₂] → {σ₁₂ : R₁ →+* R₂} → (M →ₛₗ[σ₁₂] M₂) →+ M →+ M₂","typeReadable":"{R₁ : Type u_2} →\n  {R₂ : Type u_3} →\n    {M : Type u_8} →\n      {M₂ : Type u_10} →\n        [inst : Semiring R₁] →\n          [inst_1 : Semiring R₂] →\n            [inst_2 : AddCommMonoid M] →\n              [inst_3 : AddCommMonoid M₂] →\n                [inst_4 : Module R₁ M] → [inst_5 : Module R₂ M₂] → {σ₁₂ : R₁ →+* R₂} → (M →ₛₗ[σ₁₂] M₂) →+ M →+ M₂","typeReferences":[["LinearMap","addCommMonoid"],["RingHom"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["Module"],["AddMonoidHom"],["AddMonoidHom","instAddCommMonoid"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["LinearMap"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["LinearMap","toAddMonoidHom'","_proof_2"],["LinearMap","toAddMonoidHom"],["AddMonoidHom","mk"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["LinearMap","addCommMonoid"],["LinearMap","toAddMonoidHom'","_proof_1"],["AddMonoidHom"],["AddMonoidHom","instAddCommMonoid"],["ZeroHom","mk"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","mulLeft","_proof_2"],"typeFallback":"forall (R : Type.{u_2}) {A : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.17 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.20 : NonUnitalNonAssocSemiring.{u_1} A] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.23 : Module.{u_2, u_1} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.20)] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.27 : SMulCommClass.{u_2, u_1, u_1} R A A (SMulZeroClass.toSMul.{u_2, u_1} R A (AddZero.toZero.{u_1} A (AddZeroClass.toAddZero.{u_1} A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.20))) (DistribMulAction.toDistribSMul.{u_2, u_1} R A (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.17)) (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.20)) (Module.toDistribMulAction.{u_2, u_1} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.23)))) (SMulZeroClass.toSMul.{u_1, u_1} A A (AddZero.toZero.{u_1} A (AddZeroClass.toAddZero.{u_1} A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} A A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.20))) (instDistribSMul.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.20)))] (a : A) (x._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.57 : R) (y : A), Eq.{succ u_1} A (HMul.hMul.{u_1, u_1, u_1} A A A (instHMul.{u_1} A (Distrib.toMul.{u_1} A (NonUnitalNonAssocSemiring.toDistrib.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.20))) a (HSMul.hSMul.{u_2, u_1, u_1} R A A (instHSMul.{u_2, u_1} R A (SMulZeroClass.toSMul.{u_2, u_1} R A (AddZero.toZero.{u_1} A (AddZeroClass.toAddZero.{u_1} A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.20))) (DistribMulAction.toDistribSMul.{u_2, u_1} R A (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.17)) (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.20)) (Module.toDistribMulAction.{u_2, u_1} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.23))))) x._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.57 y)) (HSMul.hSMul.{u_2, u_1, u_1} R A A (instHSMul.{u_2, u_1} R 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(NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.23))))) x._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.57 (HMul.hMul.{u_1, u_1, u_1} A A A (instHMul.{u_1} A (Distrib.toMul.{u_1} A (NonUnitalNonAssocSemiring.toDistrib.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.20))) a y))","typeFull":"∀ (R : Type u_2) {A : Type u_1} [inst : Semiring R] [inst_1 : NonUnitalNonAssocSemiring A] [inst_2 : Module R A]\n  [SMulCommClass R A A] (a : A) (x : R) (y : A), a * x • y = x • (a * y)","typeReadable":"∀ (R : Type u_2) {A : Type u_1} [inst : Semiring R] [inst_1 : NonUnitalNonAssocSemiring A] [inst_2 : Module R A]\n  [SMulCommClass R A A] (a : A) (x : R) (y : A), a * x • y = x • (a * y)","typeReferences":[["NonUnitalNonAssocSemiring","toDistrib"],["Module"],["Distrib","toMul"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["HMul","hMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["instDistribSMul"],["AddZeroClass","toAddZero"],["SMulCommClass"],["Module","toDistribMulAction"],["HSMul","hSMul"],["MonoidWithZero","toMonoid"],["instHSMul"],["instHMul"],["Eq"],["AddZero","toZero"],["NonUnitalNonAssocSemiring"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["NonUnitalNonAssocSemiring","toDistrib"],["Distrib","toMul"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["mul_smul_comm"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["MonoidWithZero","toMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["IsLinearMap","isLinearMap_smul"],"typeFallback":"forall {R : Type.{u_14}} {M : Type.{u_15}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1659520248._hygCtx._hyg.34 : CommSemiring.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1659520248._hygCtx._hyg.37 : AddCommMonoid.{u_15} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1659520248._hygCtx._hyg.40 : Module.{u_14, u_15} R M (CommSemiring.toSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1659520248._hygCtx._hyg.34) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1659520248._hygCtx._hyg.37] (c : R), IsLinearMap.{u_14, u_15, u_15} R M M (CommSemiring.toSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1659520248._hygCtx._hyg.34) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1659520248._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1659520248._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1659520248._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1659520248._hygCtx._hyg.40 (fun (z : M) => HSMul.hSMul.{u_14, 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1659520248._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1659520248._hygCtx._hyg.40))))) c z)","typeFull":"∀ {R : Type u_14} {M : Type u_15} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (c : R),\n  IsLinearMap R fun z => c • z","typeReadable":"∀ {R : Type u_14} {M : Type u_15} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (c : R),\n  IsLinearMap R fun z => c • z","typeReferences":[["Module"],["CommSemiring"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["instHSMul"],["IsLinearMap"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["MulOneClass","toMulOne"],["IsLinearMap","mk"],["Eq","trans"],["NonUnitalNonAssocCommSemiring","toCommMagma"],["SemigroupAction","toSMul"],["HMul","hMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["congrArg"],["NonUnitalCommSemiring","toNonUnitalNonAssocCommSemiring"],["MulOne","toMul"],["CommSemiring","toNonUnitalCommSemiring"],["congr"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["mul_comm"],["DistribMulAction","toMulAction"],["instHSMul"],["congrFun'"],["Monoid","toSemigroup"],["Eq"],["DistribSMul","toSMulZeroClass"],["True"],["CommMagma","toMul"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["eq_self"],["Module","toDistribMulAction"],["of_eq_true"],["smul_smul"],["HSMul","hSMul"],["smul_add"],["instHMul"],["AddZero","toZero"],["MulAction","toSemigroupAction"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","comp_neg"],"typeFallback":"forall {R₁ : Type.{u_2}} {R₂ : Type.{u_3}} {R₃ : Type.{u_4}} {M : Type.{u_8}} {N₂ : Type.{u_12}} {N₃ : Type.{u_13}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.15 : Semiring.{u_2} R₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.18 : Semiring.{u_3} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.21 : Semiring.{u_4} R₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.24 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.33 : AddCommGroup.{u_12} N₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.36 : AddCommGroup.{u_13} N₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.39 : Module.{u_2, u_8} R₁ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.51 : Module.{u_3, u_12} R₂ N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.18 (AddCommGroup.toAddCommMonoid.{u_12} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.33)] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.55 : Module.{u_4, u_13} R₃ N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.21 (AddCommGroup.toAddCommMonoid.{u_13} N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.36)] {σ₁₂ : RingHom.{u_2, u_3} R₁ R₂ (Semiring.toNonAssocSemiring.{u_2} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.18)} {σ₂₃ : RingHom.{u_3, u_4} R₂ R₃ (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.18) (Semiring.toNonAssocSemiring.{u_4} R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.21)} {σ₁₃ : RingHom.{u_2, u_4} R₁ R₃ (Semiring.toNonAssocSemiring.{u_2} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_4} R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.21)} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.74 : RingHomCompTriple.{u_2, u_3, u_4} R₁ R₂ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.21 σ₁₂ σ₂₃ σ₁₃] (f : LinearMap.{u_2, u_3, u_8, u_12} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.18 σ₁₂ M N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_12} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.33) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.51) (g : LinearMap.{u_3, u_4, u_12, u_13} R₂ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.21 σ₂₃ N₂ N₃ (AddCommGroup.toAddCommMonoid.{u_12} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.33) (AddCommGroup.toAddCommMonoid.{u_13} N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.36) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.51 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.55), Eq.{max (succ u_13) (succ u_8)} (LinearMap.{u_2, u_4, u_8, u_13} R₁ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.21 σ₁₃ M N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_13} N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.36) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.55) (LinearMap.comp.{u_2, u_3, u_4, u_8, u_12, u_13} R₁ R₂ R₃ M N₂ N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_12} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.33) (AddCommGroup.toAddCommMonoid.{u_13} N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.36) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.51 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.55 σ₁₂ σ₂₃ σ₁₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.74 g (Neg.neg.{max u_12 u_8} (LinearMap.{u_2, u_3, u_8, u_12} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.18 σ₁₂ M N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_12} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.33) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.51) (LinearMap.instNeg.{u_2, u_3, u_8, u_12} R₁ R₂ M N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.51 σ₁₂) f)) (Neg.neg.{max u_13 u_8} (LinearMap.{u_2, u_4, u_8, u_13} R₁ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.21 σ₁₃ M N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_13} N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.36) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.55) (LinearMap.instNeg.{u_2, u_4, u_8, u_13} R₁ R₃ M N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.55 σ₁₃) (LinearMap.comp.{u_2, u_3, u_4, u_8, u_12, u_13} R₁ R₂ R₃ M N₂ N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_12} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.33) (AddCommGroup.toAddCommMonoid.{u_13} N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.36) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.51 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.55 σ₁₂ σ₂₃ σ₁₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4170893916._hygCtx._hyg.74 g f))","typeFull":"∀ {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_8} {N₂ : Type u_12} {N₃ : Type u_13} [inst : Semiring R₁]\n  [inst_1 : Semiring R₂] [inst_2 : Semiring R₃] [inst_3 : AddCommMonoid M] [inst_4 : AddCommGroup N₂]\n  [inst_5 : AddCommGroup N₃] [inst_6 : Module R₁ M] [inst_7 : Module R₂ N₂] [inst_8 : Module R₃ N₃] {σ₁₂ : R₁ →+* R₂}\n  {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [inst_9 : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (f : M →ₛₗ[σ₁₂] N₂) (g : N₂ →ₛₗ[σ₂₃] N₃),\n  g ∘ₛₗ (-f) = -g ∘ₛₗ f","typeReadable":"∀ {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_8} {N₂ : Type u_12} {N₃ : Type u_13} [inst : Semiring R₁]\n  [inst_1 : Semiring R₂] [inst_2 : Semiring R₃] [inst_3 : AddCommMonoid M] [inst_4 : AddCommGroup N₂]\n  [inst_5 : AddCommGroup N₃] [inst_6 : Module R₁ M] [inst_7 : Module R₂ N₂] [inst_8 : Module R₃ N₃] {σ₁₂ : R₁ →+* R₂}\n  {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [inst_9 : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (f : M →ₛₗ[σ₁₂] N₂) (g : N₂ →ₛₗ[σ₂₃] N₃),\n  g ∘ₛₗ (-f) = -g ∘ₛₗ f","typeReferences":[["RingHom"],["Module"],["Neg","neg"],["LinearMap","comp"],["AddCommGroup"],["LinearMap"],["RingHomCompTriple"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["LinearMap","instNeg"],["AddCommGroup","toAddCommMonoid"],["Eq"],["Semiring"]],"valueReferences":[["LinearMap","instNeg"],["LinearMap","instFunLike"],["Neg","neg"],["LinearMap","ext"],["LinearMap","map_neg"],["AddCommGroup","toAddCommMonoid"],["LinearMap","comp"],["LinearMap"],["DFunLike","coe"]]},{"isProp":false,"kind":"definition","name":["LinearMap","casesOn"],"typeFallback":"forall {R : Type.{u_14}} {S : Type.{u_15}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 : Semiring.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 : Semiring.{u_15} S] {σ : RingHom.{u_14, u_15} R S (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30) (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33)} {M : Type.{u_16}} {M₂ : Type.{u_17}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 : AddCommMonoid.{u_16} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 : AddCommMonoid.{u_17} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 : Module.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53 : Module.{u_15, u_17} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46] {motive : (LinearMap.{u_14, u_15, u_16, u_17} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53) -> Sort.{u}} (t : LinearMap.{u_14, u_15, u_16, u_17} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53), (forall (toAddHom : AddHom.{u_16, u_17} M M₂ (AddCommMagma.toAdd.{u_16} M (AddCommSemigroup.toAddCommMagma.{u_16} M (AddCommMonoid.toAddCommSemigroup.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43))) (AddCommMagma.toAdd.{u_17} M₂ (AddCommSemigroup.toAddCommMagma.{u_17} M₂ (AddCommMonoid.toAddCommSemigroup.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46)))) (map_smul' : forall (m : R) (x : M), Eq.{succ u_17} M₂ (AddHom.toFun.{u_16, u_17} M M₂ (AddCommMagma.toAdd.{u_16} M (AddCommSemigroup.toAddCommMagma.{u_16} M (AddCommMonoid.toAddCommSemigroup.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43))) (AddCommMagma.toAdd.{u_17} M₂ (AddCommSemigroup.toAddCommMagma.{u_17} M₂ (AddCommMonoid.toAddCommSemigroup.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46))) toAddHom (HSMul.hSMul.{u_14, u_16, u_16} R M M (instHSMul.{u_14, u_16} R M (SMulZeroClass.toSMul.{u_14, u_16} R M (AddZero.toZero.{u_16} M (AddZeroClass.toAddZero.{u_16} M (AddMonoid.toAddZeroClass.{u_16} M (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43)))) (DistribSMul.toSMulZeroClass.{u_14, u_16} R M (AddMonoid.toAddZeroClass.{u_16} M (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43)) (DistribMulAction.toDistribSMul.{u_14, u_16} R M (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30)) (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43) (Module.toDistribMulAction.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49))))) m x)) (HSMul.hSMul.{u_15, u_17, u_17} S M₂ M₂ (instHSMul.{u_15, u_17} S M₂ (SMulZeroClass.toSMul.{u_15, u_17} S M₂ (AddZero.toZero.{u_17} M₂ (AddZeroClass.toAddZero.{u_17} M₂ (AddMonoid.toAddZeroClass.{u_17} M₂ (AddCommMonoid.toAddMonoid.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46)))) (DistribSMul.toSMulZeroClass.{u_15, u_17} S M₂ (AddMonoid.toAddZeroClass.{u_17} M₂ (AddCommMonoid.toAddMonoid.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46)) (DistribMulAction.toDistribSMul.{u_15, u_17} S M₂ (MonoidWithZero.toMonoid.{u_15} S (Semiring.toMonoidWithZero.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33)) (AddCommMonoid.toAddMonoid.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46) (Module.toDistribMulAction.{u_15, u_17} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53))))) (DFunLike.coe.{max (succ u_14) (succ u_15), succ u_14, succ u_15} (RingHom.{u_14, u_15} R S (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30) (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33)) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => S) (RingHom.instFunLike.{u_14, u_15} R S (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30) (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33)) σ m) (AddHom.toFun.{u_16, u_17} M M₂ (AddCommMagma.toAdd.{u_16} M (AddCommSemigroup.toAddCommMagma.{u_16} M (AddCommMonoid.toAddCommSemigroup.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43))) (AddCommMagma.toAdd.{u_17} M₂ (AddCommSemigroup.toAddCommMagma.{u_17} M₂ (AddCommMonoid.toAddCommSemigroup.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46))) toAddHom x))), motive (LinearMap.mk.{u_14, u_15, u_16, u_17} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53 toAddHom map_smul')) -> (motive t)","typeFull":"{R : Type u_14} →\n  {S : Type u_15} →\n    [inst : Semiring R] →\n      [inst_1 : Semiring S] →\n        {σ : R →+* S} →\n          {M : Type u_16} →\n            {M₂ : Type u_17} →\n              [inst_2 : AddCommMonoid M] →\n                [inst_3 : AddCommMonoid M₂] →\n                  [inst_4 : Module R M] →\n                    [inst_5 : Module S M₂] →\n                      {motive : (M →ₛₗ[σ] M₂) → Sort u} →\n                        (t : M →ₛₗ[σ] M₂) →\n                          ((toAddHom : M →ₙ+ M₂) →\n                              (map_smul' : ∀ (m : R) (x : M), toAddHom.toFun (m • x) = σ m • toAddHom.toFun x) →\n                                motive { toAddHom := toAddHom, map_smul' := map_smul' }) →\n                            motive t","typeReadable":"{R : Type u_14} →\n  {S : Type u_15} →\n    [inst : Semiring R] →\n      [inst_1 : Semiring S] →\n        {σ : R →+* S} →\n          {M : Type u_16} →\n            {M₂ : Type u_17} →\n              [inst_2 : AddCommMonoid M] →\n                [inst_3 : AddCommMonoid M₂] →\n                  [inst_4 : Module R M] →\n                    [inst_5 : Module S M₂] →\n                      {motive : (M →ₛₗ[σ] M₂) → Sort u} →\n                        (t : M →ₛₗ[σ] M₂) →\n                          ((toAddHom : M →ₙ+ M₂) →\n                              (map_smul' : ∀ (m : R) (x : M), toAddHom.toFun (m • x) = σ m • toAddHom.toFun x) →\n                                motive { toAddHom := toAddHom, map_smul' := map_smul' }) →\n                            motive t","typeReferences":[["RingHom"],["Module"],["LinearMap","mk"],["RingHom","instFunLike"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddHom","toFun"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Eq"],["DistribSMul","toSMulZeroClass"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["LinearMap"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["HSMul","hSMul"],["AddHom"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["LinearMap","rec"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","instDistribMulAction","_proof_1"],"typeFallback":"forall {R : Type.{u_1}} {R₂ : Type.{u_2}} {S : Type.{u_5}} {M : Type.{u_3}} {M₂ : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 : Semiring.{u_2} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 : AddCommMonoid.{u_4} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 : Module.{u_1, u_3} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37 : Module.{u_2, u_4} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27] {σ₁₂ : RingHom.{u_1, u_2} R R₂ (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_2} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18)} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.65 : Monoid.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.68 : DistribMulAction.{u_5, u_4} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.65 (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27)] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.72 : SMulCommClass.{u_2, u_5, u_4} R₂ S M₂ (SMulZeroClass.toSMul.{u_2, u_4} R₂ M₂ (AddZero.toZero.{u_4} M₂ (AddZeroClass.toAddZero.{u_4} M₂ (AddMonoid.toAddZeroClass.{u_4} M₂ (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27)))) (DistribSMul.toSMulZeroClass.{u_2, u_4} R₂ M₂ (AddMonoid.toAddZeroClass.{u_4} M₂ (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27)) (DistribMulAction.toDistribSMul.{u_2, u_4} R₂ M₂ (MonoidWithZero.toMonoid.{u_2} R₂ (Semiring.toMonoidWithZero.{u_2} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18)) (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27) (Module.toDistribMulAction.{u_2, u_4} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37)))) (SMulZeroClass.toSMul.{u_5, u_4} S M₂ (AddZero.toZero.{u_4} M₂ (AddZeroClass.toAddZero.{u_4} M₂ (AddMonoid.toAddZeroClass.{u_4} M₂ (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27)))) (DistribSMul.toSMulZeroClass.{u_5, u_4} S M₂ (AddMonoid.toAddZeroClass.{u_4} M₂ (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27)) (DistribMulAction.toDistribSMul.{u_5, u_4} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.65 (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.68)))] (x._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.108 : S) (x._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.110 : S) (x._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.112 : LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 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u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (instHSMul.{u_5, max u_4 u_3} S (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (LinearMap.instSMul.{u_1, u_2, u_5, 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (instHSMul.{u_5, max u_4 u_3} S (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (LinearMap.instSMul.{u_1, u_2, u_5, u_3, u_4} R R₂ S M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37 σ₁₂ (DistribMulAction.toDistribSMul.{u_5, u_4} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.65 (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.68) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.72)) x._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.110 x._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.112))","typeFull":"∀ {R : Type u_1} {R₂ : Type u_2} {S : Type u_5} {M : Type u_3} {M₂ : Type u_4} [inst : Semiring R]\n  [inst_1 : Semiring R₂] [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M]\n  [inst_5 : Module R₂ M₂] {σ₁₂ : R →+* R₂} [inst_6 : Monoid S] [inst_7 : DistribMulAction S M₂]\n  [inst_8 : SMulCommClass R₂ S M₂] (x x_1 : S) (x_2 : M →ₛₗ[σ₁₂] M₂), (x * x_1) • x_2 = x • x_1 • x_2","typeReadable":"∀ {R : Type u_1} {R₂ : Type u_2} {S : Type u_5} {M : Type u_3} {M₂ : Type u_4} [inst : Semiring R]\n  [inst_1 : Semiring R₂] [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M]\n  [inst_5 : Module R₂ M₂] {σ₁₂ : R →+* R₂} [inst_6 : Monoid S] [inst_7 : DistribMulAction S M₂]\n  [inst_8 : SMulCommClass R₂ S M₂] (x x_1 : S) (x_2 : M →ₛₗ[σ₁₂] M₂), (x * x_1) • x_2 = x • x_1 • x_2","typeReferences":[["RingHom"],["Module"],["HMul","hMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Semigroup","toMul"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["Monoid","toSemigroup"],["Eq"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["LinearMap"],["SMulCommClass"],["Module","toDistribMulAction"],["HSMul","hSMul"],["Monoid"],["LinearMap","instSMul"],["instHMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["LinearMap","instFunLike"],["DistribMulAction","toDistribSMul"],["HMul","hMul"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["DFunLike","coe"],["Semigroup","toMul"],["LinearMap","ext"],["HSMul","hSMul"],["DistribMulAction","toMulAction"],["instHSMul"],["LinearMap","instSMul"],["instHMul"],["Monoid","toSemigroup"],["MulAction","toSemigroupAction"],["SemigroupAction","mul_smul"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","mulRight","_proof_1"],"typeFallback":"forall {A : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20 : NonUnitalNonAssocSemiring.{u_1} A] (b : A) (x : A) (y : A), Eq.{succ u_1} A (ZeroHom.toFun.{u_1, u_1} A A (AddZero.toZero.{u_1} A (AddZeroClass.toAddZero.{u_1} A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20))))) (AddZero.toZero.{u_1} A (AddZeroClass.toAddZero.{u_1} A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20))))) (AddMonoidHom.toZeroHom.{u_1, u_1} A A (AddZeroClass.toAddZero.{u_1} A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20)))) (AddZeroClass.toAddZero.{u_1} A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20)))) (AddMonoidHom.mulRight.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20 b)) (HAdd.hAdd.{u_1, u_1, u_1} A A A (instHAdd.{u_1} A (AddZero.toAdd.{u_1} A (AddZeroClass.toAddZero.{u_1} A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20)))))) x y)) (HAdd.hAdd.{u_1, u_1, u_1} A A A (instHAdd.{u_1} A (AddZero.toAdd.{u_1} A (AddZeroClass.toAddZero.{u_1} A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20)))))) (ZeroHom.toFun.{u_1, u_1} A A (AddZero.toZero.{u_1} A (AddZeroClass.toAddZero.{u_1} A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20))))) (AddZero.toZero.{u_1} A (AddZeroClass.toAddZero.{u_1} A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20))))) (AddMonoidHom.toZeroHom.{u_1, u_1} A A (AddZeroClass.toAddZero.{u_1} A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20)))) (AddZeroClass.toAddZero.{u_1} A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20)))) (AddMonoidHom.mulRight.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20 b)) x) (ZeroHom.toFun.{u_1, u_1} A A (AddZero.toZero.{u_1} A (AddZeroClass.toAddZero.{u_1} A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20))))) (AddZero.toZero.{u_1} A (AddZeroClass.toAddZero.{u_1} A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20))))) (AddMonoidHom.toZeroHom.{u_1, u_1} A A (AddZeroClass.toAddZero.{u_1} A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20)))) (AddZeroClass.toAddZero.{u_1} A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20)))) (AddMonoidHom.mulRight.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20 b)) y))","typeFull":"∀ {A : Type u_1} [inst : NonUnitalNonAssocSemiring A] (b x y : A),\n  (↑(AddMonoidHom.mulRight b)).toFun (x + y) =\n    (↑(AddMonoidHom.mulRight b)).toFun x + (↑(AddMonoidHom.mulRight b)).toFun y","typeReadable":"∀ {A : Type u_1} [inst : NonUnitalNonAssocSemiring A] (b x y : A),\n  (↑(AddMonoidHom.mulRight b)).toFun (x + y) =\n    (↑(AddMonoidHom.mulRight b)).toFun x + (↑(AddMonoidHom.mulRight b)).toFun y","typeReferences":[["instHAdd"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["AddMonoidHom","toZeroHom"],["ZeroHom","toFun"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["HAdd","hAdd"],["AddMonoidHom","mulRight"],["Eq"],["AddZero","toZero"],["NonUnitalNonAssocSemiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["AddMonoidHom","map_add'"],["AddMonoidHom","mulRight"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","mulRight","_proof_2"],"typeFallback":"forall (R : Type.{u_2}) {A : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.17 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20 : NonUnitalNonAssocSemiring.{u_1} A] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.23 : Module.{u_2, u_1} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20)] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.27 : IsScalarTower.{u_2, u_1, u_1} R A A (SMulZeroClass.toSMul.{u_2, u_1} R A (AddZero.toZero.{u_1} A (AddZeroClass.toAddZero.{u_1} A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20))) (DistribMulAction.toDistribSMul.{u_2, u_1} R A (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.17)) (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20)) (Module.toDistribMulAction.{u_2, u_1} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.23)))) (SMulZeroClass.toSMul.{u_1, u_1} A A (AddZero.toZero.{u_1} A (AddZeroClass.toAddZero.{u_1} A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} A A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20))) (instDistribSMul.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20))) (SMulZeroClass.toSMul.{u_2, u_1} R A (AddZero.toZero.{u_1} A (AddZeroClass.toAddZero.{u_1} A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20))) (DistribMulAction.toDistribSMul.{u_2, u_1} R A (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.17)) (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20)) (Module.toDistribMulAction.{u_2, u_1} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.23))))] (b : A) (x._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.57 : R) (x._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.59 : A), Eq.{succ u_1} A (HMul.hMul.{u_1, u_1, u_1} A A A (instHMul.{u_1} A (Distrib.toMul.{u_1} A (NonUnitalNonAssocSemiring.toDistrib.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20))) (HSMul.hSMul.{u_2, u_1, u_1} R A A (instHSMul.{u_2, u_1} R A (SMulZeroClass.toSMul.{u_2, u_1} R A (AddZero.toZero.{u_1} A (AddZeroClass.toAddZero.{u_1} A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20))) (DistribMulAction.toDistribSMul.{u_2, u_1} R A (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.17)) (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20)) (Module.toDistribMulAction.{u_2, u_1} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.23))))) x._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.57 x._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.59) b) (HSMul.hSMul.{u_2, u_1, u_1} R A A (instHSMul.{u_2, u_1} R A (SMulZeroClass.toSMul.{u_2, u_1} R A (AddZero.toZero.{u_1} A (AddZeroClass.toAddZero.{u_1} A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R A (AddMonoid.toAddZeroClass.{u_1} A (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20))) (DistribMulAction.toDistribSMul.{u_2, u_1} R A (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.17)) (AddCommMonoid.toAddMonoid.{u_1} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20)) (Module.toDistribMulAction.{u_2, u_1} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.23))))) x._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.57 (HMul.hMul.{u_1, u_1, u_1} A A A (instHMul.{u_1} A (Distrib.toMul.{u_1} A (NonUnitalNonAssocSemiring.toDistrib.{u_1} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20))) x._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.59 b))","typeFull":"∀ (R : Type u_2) {A : Type u_1} [inst : Semiring R] [inst_1 : NonUnitalNonAssocSemiring A] [inst_2 : Module R A]\n  [IsScalarTower R A A] (b : A) (x : R) (x_1 : A), x • x_1 * b = x • (x_1 * b)","typeReadable":"∀ (R : Type u_2) {A : Type u_1} [inst : Semiring R] [inst_1 : NonUnitalNonAssocSemiring A] [inst_2 : Module R A]\n  [IsScalarTower R A A] (b : A) (x : R) (x_1 : A), x • x_1 * b = x • (x_1 * b)","typeReferences":[["NonUnitalNonAssocSemiring","toDistrib"],["Module"],["IsScalarTower"],["Distrib","toMul"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["HMul","hMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["instDistribSMul"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["HSMul","hSMul"],["MonoidWithZero","toMonoid"],["instHSMul"],["instHMul"],["Eq"],["AddZero","toZero"],["NonUnitalNonAssocSemiring"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["smul_mul_assoc"],["NonUnitalNonAssocSemiring","toDistrib"],["Distrib","toMul"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["MonoidWithZero","toMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","comp_assoc"],"typeFallback":"forall {R₁ : Type.{u_2}} {R₂ : Type.{u_3}} {R₃ : Type.{u_4}} {M₁ : Type.{u_9}} {M₂ : Type.{u_10}} {M₃ : Type.{u_11}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.21 : Semiring.{u_2} R₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.24 : Semiring.{u_3} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.27 : Semiring.{u_4} R₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.33 : AddCommMonoid.{u_9} M₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.36 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.39 : AddCommMonoid.{u_11} M₃] {module_M₁ : Module.{u_2, u_9} R₁ M₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.33} {module_M₂ : Module.{u_3, u_10} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.24 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.27)} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.66 : RingHomCompTriple.{u_2, u_3, u_4} R₁ R₂ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.27 σ₁₂ σ₂₃ σ₁₃] {R₄ : Type.{u_14}} {M₄ : Type.{u_15}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.85 : Semiring.{u_14} R₄] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.88 : AddCommMonoid.{u_15} M₄] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.91 : Module.{u_14, u_15} R₄ M₄ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.85 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.88] {σ₃₄ : RingHom.{u_4, u_14} R₃ R₄ (Semiring.toNonAssocSemiring.{u_4} R₃ 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.85 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.88 module_M₁ module_M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.91 σ₁₃ σ₃₄ σ₁₄ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.115 h (LinearMap.comp.{u_2, u_3, u_4, u_9, u_10, u_11} R₁ R₂ R₃ M₁ M₂ M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.39 module_M₁ module_M₂ module_M₃ σ₁₂ σ₂₃ σ₁₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.66 g f))","typeFull":"∀ {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_9} {M₂ : Type u_10} {M₃ : Type u_11} [inst : Semiring R₁]\n  [inst_1 : Semiring R₂] [inst_2 : Semiring R₃] [inst_3 : AddCommMonoid M₁] [inst_4 : AddCommMonoid M₂]\n  [inst_5 : AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃}\n  {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [inst_6 : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {R₄ : Type u_14}\n  {M₄ : Type u_15} [inst_7 : Semiring R₄] [inst_8 : AddCommMonoid M₄] [inst_9 : Module R₄ M₄] {σ₃₄ : R₃ →+* R₄}\n  {σ₂₄ : R₂ →+* R₄} {σ₁₄ : R₁ →+* R₄} [inst_10 : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄]\n  [inst_11 : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄] [inst_12 : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄] (f : M₁ →ₛₗ[σ₁₂] M₂)\n  (g : M₂ →ₛₗ[σ₂₃] M₃) (h : M₃ →ₛₗ[σ₃₄] M₄), (h ∘ₛₗ g) ∘ₛₗ f = h ∘ₛₗ g ∘ₛₗ f","typeReadable":"∀ {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_9} {M₂ : Type u_10} {M₃ : Type u_11} [inst : Semiring R₁]\n  [inst_1 : Semiring R₂] [inst_2 : Semiring R₃] [inst_3 : AddCommMonoid M₁] [inst_4 : AddCommMonoid M₂]\n  [inst_5 : AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃}\n  {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [inst_6 : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {R₄ : Type u_14}\n  {M₄ : Type u_15} [inst_7 : Semiring R₄] [inst_8 : AddCommMonoid M₄] [inst_9 : Module R₄ M₄] {σ₃₄ : R₃ →+* R₄}\n  {σ₂₄ : R₂ →+* R₄} {σ₁₄ : R₁ →+* R₄} [inst_10 : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄]\n  [inst_11 : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄] [inst_12 : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄] (f : M₁ →ₛₗ[σ₁₂] M₂)\n  (g : M₂ →ₛₗ[σ₂₃] M₃) (h : M₃ →ₛₗ[σ₃₄] M₄), (h ∘ₛₗ g) ∘ₛₗ f = h ∘ₛₗ g ∘ₛₗ f","typeReferences":[["RingHom"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["Module"],["LinearMap","comp"],["Eq"],["LinearMap"],["RingHomCompTriple"],["Semiring"]],"valueReferences":[["rfl"],["LinearMap","comp"],["LinearMap"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","map_smulₛₗ"],"typeFallback":"forall {R : Type.{u_1}} {S : Type.{u_5}} {M : Type.{u_8}} {M₃ : Type.{u_11}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1358843655._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1358843655._hygCtx._hyg.18 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1358843655._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1358843655._hygCtx._hyg.30 : AddCommMonoid.{u_11} M₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1358843655._hygCtx._hyg.33 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1358843655._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1358843655._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1358843655._hygCtx._hyg.41 : Module.{u_5, u_11} S M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1358843655._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1358843655._hygCtx._hyg.30] {σ : RingHom.{u_1, u_5} R S (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1358843655._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1358843655._hygCtx._hyg.18)} (f : LinearMap.{u_1, u_5, u_8, u_11} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1358843655._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1358843655._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1358843655._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1358843655._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1358843655._hygCtx._hyg.33 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x._@.Mathlib.Algebra.Module.LinearMap.Defs.4245081335._hygCtx._hyg.97))","typeFull":"∀ {R₁ : Type u_1} {R₂ : Type u_2} {M : Type u_3} {M₂ : Type u_4} [inst : Semiring R₁] [inst_1 : Semiring R₂]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R₁ M] [inst_5 : Module R₂ M₂]\n  {σ₁₂ : R₁ →+* R₂} (x x_1 : M →ₛₗ[σ₁₂] M₂), ⇑(x + x_1) = ⇑(x + x_1)","typeReadable":"∀ {R₁ : Type u_1} {R₂ : Type u_2} {M : Type u_3} {M₂ : Type u_4} [inst : Semiring R₁] [inst_1 : Semiring R₂]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R₁ M] [inst_5 : Module R₂ M₂]\n  {σ₁₂ : R₁ →+* R₂} (x x_1 : M →ₛₗ[σ₁₂] M₂), ⇑(x + x_1) = ⇑(x + x_1)","typeReferences":[["HAdd","hAdd"],["RingHom"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["LinearMap","instFunLike"],["instHAdd"],["Module"],["LinearMap","instAdd"],["Eq"],["LinearMap"],["DFunLike","coe"],["Semiring"]],"valueReferences":[["HAdd","hAdd"],["rfl"],["LinearMap","instFunLike"],["instHAdd"],["LinearMap","instAdd"],["LinearMap"],["DFunLike","coe"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","mulLeft_toAddMonoidHom"],"typeFallback":"forall (R : Type.{u_14}) {A : Type.{u_15}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17 : Semiring.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20 : NonUnitalNonAssocSemiring.{u_15} A] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23 : Module.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20)] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.27 : SMulCommClass.{u_14, u_15, u_15} R A A (SMulZeroClass.toSMul.{u_14, u_15} R A (AddZero.toZero.{u_15} A (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_14, u_15} R A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20))) (DistribMulAction.toDistribSMul.{u_14, u_15} R A (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17)) (AddCommMonoid.toAddMonoid.{u_15} A 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(instDistribSMul.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20)))] (a : A), Eq.{succ u_15} (AddMonoidHom.{u_15, u_15} A A (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20)))) (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20))))) (AddMonoidHomClass.toAddMonoidHom.{u_15, u_15, u_15} A A (LinearMap.{u_14, u_14, u_15, u_15} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23) (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20)))) (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20)))) (LinearMap.instFunLike.{u_14, u_14, u_15, u_15} R R A A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17))) (DistribMulActionSemiHomClass.toAddMonoidHomClass.{u_15, u_14, u_14, u_15, u_15} (LinearMap.{u_14, u_14, u_15, u_15} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23) R R (DFunLike.coe.{succ u_14, succ u_14, succ u_14} (RingHom.{u_14, u_14} R R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17) (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17)) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => R) (RingHom.instFunLike.{u_14, u_14} R R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17) (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17)) (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17))) A A (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17)) (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17)) (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20)) (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20)) (Module.toDistribMulAction.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23) (Module.toDistribMulAction.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23) (LinearMap.instFunLike.{u_14, u_14, u_15, u_15} R R A A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17))) (SemilinearMapClass.distribMulActionSemiHomClass.{u_14, u_14, u_15, u_15, u_15} R R A A (LinearMap.{u_14, u_14, u_15, u_15} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17)) (LinearMap.instFunLike.{u_14, u_14, u_15, u_15} R R A A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17))) (LinearMap.semilinearMapClass.{u_14, u_14, u_15, u_15} R R A A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17))))) (LinearMap.mulLeft.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.27 a)) (AddMonoidHom.mulLeft.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3193787028._hygCtx._hyg.20 a)","typeFull":"∀ (R : Type u_14) {A : Type u_15} [inst : Semiring R] [inst_1 : NonUnitalNonAssocSemiring A] [inst_2 : Module R A]\n  [inst_3 : SMulCommClass R A A] (a : A), ↑(LinearMap.mulLeft R a) = AddMonoidHom.mulLeft a","typeReadable":"∀ (R : Type u_14) {A : Type u_15} [inst : Semiring R] [inst_1 : NonUnitalNonAssocSemiring A] [inst_2 : Module R A]\n  [inst_3 : SMulCommClass R A A] (a : A), ↑(LinearMap.mulLeft R a) = AddMonoidHom.mulLeft a","typeReferences":[["RingHom"],["AddMonoidHomClass","toAddMonoidHom"],["LinearMap","instFunLike"],["Module"],["RingHom","instFunLike"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddMonoidHom","mulLeft"],["instDistribSMul"],["DFunLike","coe"],["LinearMap","semilinearMapClass"],["DistribMulActionSemiHomClass","toAddMonoidHomClass"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["AddMonoidHom"],["MonoidWithZero","toMonoid"],["LinearMap","mulLeft"],["Eq"],["DistribSMul","toSMulZeroClass"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["LinearMap"],["SMulCommClass"],["Module","toDistribMulAction"],["AddZero","toZero"],["SemilinearMapClass","distribMulActionSemiHomClass"],["NonUnitalNonAssocSemiring"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["LinearMap","toAddMonoidHom_mulLeft"]]},{"isProp":false,"kind":"definition","name":["LinearMap","toDistribMulActionHom"],"typeFallback":"forall {R : Type.{u_1}} {S : Type.{u_5}} {M : Type.{u_8}} {M₃ : Type.{u_11}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.18 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.30 : AddCommMonoid.{u_11} M₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.33 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.41 : Module.{u_5, u_11} S M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.30] {σ : RingHom.{u_1, u_5} R S (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.18)}, (LinearMap.{u_1, u_5, u_8, u_11} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.41) -> (DistribMulActionHom.{u_1, u_5, u_8, u_11} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.15)) S (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.18)) (RingHom.toMonoidHom.{u_1, u_5} R S (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.18) σ) M (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.21) (Module.toDistribMulAction.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.33) M₃ (AddCommMonoid.toAddMonoid.{u_11} M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.30) (Module.toDistribMulAction.{u_5, u_11} S M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.41))","typeFull":"{R : Type u_1} →\n  {S : Type u_5} →\n    {M : Type u_8} →\n      {M₃ : Type u_11} →\n        [inst : Semiring R] →\n          [inst_1 : Semiring S] →\n            [inst_2 : AddCommMonoid M] →\n              [inst_3 : AddCommMonoid M₃] →\n                [inst_4 : Module R M] → [inst_5 : Module S M₃] → {σ : R →+* S} → (M →ₛₗ[σ] M₃) → M →ₑ+[↑σ] M₃","typeReadable":"{R : Type u_1} →\n  {S : Type u_5} →\n    {M : Type u_8} →\n      {M₃ : Type u_11} →\n        [inst : Semiring R] →\n          [inst_1 : Semiring S] →\n            [inst_2 : AddCommMonoid M] →\n              [inst_3 : AddCommMonoid M₃] →\n                [inst_4 : Module R M] → [inst_5 : Module S M₃] → {σ : R →+* S} → (M →ₛₗ[σ] M₃) → M →ₑ+[↑σ] M₃","typeReferences":[["RingHom","toMonoidHom"],["RingHom"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["Module"],["MonoidWithZero","toMonoid"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["DistribMulActionHom"],["Semiring"]],"valueReferences":[["MulOneClass","toMulOne"],["MonoidHom"],["MonoidHom","instFunLike"],["LinearMap","toAddHom"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["RingHom","toMonoidHom"],["Semiring","toNonAssocSemiring"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["AddHom","toFun"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["LinearMap","toDistribMulActionHom","_proof_2"],["LinearMap","map_smul'"],["LinearMap","toDistribMulActionHom","_proof_1"],["DistribSMul","toSMulZeroClass"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["MulActionHom","mk"],["DistribMulActionHom","mk"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","add_comp"],"typeFallback":"forall {R₁ : Type.{u_2}} {R₂ : Type.{u_3}} {R₃ : Type.{u_4}} {M : Type.{u_8}} {M₂ : Type.{u_10}} {M₃ : Type.{u_11}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.15 : Semiring.{u_2} R₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.18 : Semiring.{u_3} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.21 : Semiring.{u_4} R₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.24 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.27 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.30 : AddCommMonoid.{u_11} M₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.39 : Module.{u_2, u_8} R₁ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.43 : Module.{u_3, u_10} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.27] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.47 : Module.{u_4, u_11} R₃ M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.30] {σ₁₂ : RingHom.{u_2, u_3} R₁ R₂ (Semiring.toNonAssocSemiring.{u_2} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.18)} {σ₂₃ : RingHom.{u_3, u_4} R₂ R₃ (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.18) (Semiring.toNonAssocSemiring.{u_4} R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.21)} {σ₁₃ : RingHom.{u_2, u_4} R₁ R₃ (Semiring.toNonAssocSemiring.{u_2} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_4} R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.21)} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.74 : RingHomCompTriple.{u_2, u_3, u_4} R₁ R₂ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.21 σ₁₂ σ₂₃ σ₁₃] (f : LinearMap.{u_2, u_3, u_8, u_10} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.24 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.47 σ₁₂ σ₂₃ σ₁₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.74 h f) (LinearMap.comp.{u_2, u_3, u_4, u_8, u_10, u_11} R₁ R₂ R₃ M M₂ M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2243394639._hygCtx._hyg.24 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Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_8} {M₂ : Type u_10} {M₃ : Type u_11} [inst : Semiring R₁]\n  [inst_1 : Semiring R₂] [inst_2 : Semiring R₃] [inst_3 : AddCommMonoid M] [inst_4 : AddCommMonoid M₂]\n  [inst_5 : AddCommMonoid M₃] [inst_6 : Module R₁ M] [inst_7 : Module R₂ M₂] [inst_8 : Module R₃ M₃] {σ₁₂ : R₁ →+* R₂}\n  {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [inst_9 : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (f : M →ₛₗ[σ₁₂] M₂)\n  (g h : M₂ →ₛₗ[σ₂₃] M₃), (h + g) ∘ₛₗ f = h ∘ₛₗ f + g ∘ₛₗ f","typeReferences":[["HAdd","hAdd"],["RingHom"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["instHAdd"],["Module"],["LinearMap","instAdd"],["LinearMap","comp"],["Eq"],["LinearMap"],["RingHomCompTriple"],["Semiring"]],"valueReferences":[["HAdd","hAdd"],["rfl"],["instHAdd"],["LinearMap","instAdd"],["LinearMap","comp"],["LinearMap"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","identityMapOfZeroModuleIsZero"],"typeFallback":"forall {R₁ : Type.{u_2}} {M : Type.{u_8}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.15 : Semiring.{u_2} R₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.24 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.39 : Module.{u_2, u_8} R₁ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.79 : Subsingleton.{succ u_8} M], Eq.{succ u_8} (LinearMap.{u_2, u_2, u_8, u_8} R₁ R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.15 (RingHom.id.{u_2} R₁ (Semiring.toNonAssocSemiring.{u_2} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.15)) M M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.39) (LinearMap.id.{u_2, u_8} R₁ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.39) (OfNat.ofNat.{u_8} (LinearMap.{u_2, u_2, u_8, u_8} R₁ R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.15 (RingHom.id.{u_2} R₁ (Semiring.toNonAssocSemiring.{u_2} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.15)) M M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.39) 0 (Zero.toOfNat0.{u_8} (LinearMap.{u_2, u_2, u_8, u_8} R₁ R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.15 (RingHom.id.{u_2} R₁ (Semiring.toNonAssocSemiring.{u_2} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.15)) M M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.39) (LinearMap.instZero.{u_2, u_2, u_8, u_8} R₁ R₁ M M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.39 (RingHom.id.{u_2} R₁ (Semiring.toNonAssocSemiring.{u_2} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2390717725._hygCtx._hyg.15)))))","typeFull":"∀ {R₁ : Type u_2} {M : Type u_8} [inst : Semiring R₁] [inst_1 : AddCommMonoid M] [inst_2 : Module R₁ M]\n  [Subsingleton M], LinearMap.id = 0","typeReadable":"∀ {R₁ : Type u_2} {M : Type u_8} [inst : Semiring R₁] [inst_1 : AddCommMonoid M] [inst_2 : Module R₁ M]\n  [Subsingleton M], LinearMap.id = 0","typeReferences":[["LinearMap","instZero"],["Subsingleton"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Module"],["Zero","toOfNat0"],["LinearMap"],["Eq"],["LinearMap","id"],["OfNat","ofNat"],["Semiring"]],"valueReferences":[["LinearMap","uniqueOfRight"],["LinearMap","instZero"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Unique","instSubsingleton"],["LinearMap"],["LinearMap","id"],["Subsingleton","eq_zero"]]},{"isProp":false,"kind":"definition","name":["LinearMap","module"],"typeFallback":"forall {R : Type.{u_1}} {R₂ : Type.{u_3}} {S : Type.{u_5}} {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 : Semiring.{u_3} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37 : Module.{u_3, u_10} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27] {σ₁₂ : RingHom.{u_1, u_3} R R₂ (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18)} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.65 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.72 : Module.{u_5, u_10} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.65 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.76 : SMulCommClass.{u_3, u_5, u_10} R₂ S M₂ (SMulZeroClass.toSMul.{u_3, u_10} R₂ M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27)))) (DistribSMul.toSMulZeroClass.{u_3, u_10} R₂ M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27)) (DistribMulAction.toDistribSMul.{u_3, u_10} R₂ M₂ (MonoidWithZero.toMonoid.{u_3} R₂ (Semiring.toMonoidWithZero.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18)) (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27) (Module.toDistribMulAction.{u_3, u_10} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37)))) (SMulZeroClass.toSMul.{u_5, u_10} S M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27)))) (DistribSMul.toSMulZeroClass.{u_5, u_10} S M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27)) (DistribMulAction.toDistribSMul.{u_5, u_10} S M₂ (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.65)) (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27) (Module.toDistribMulAction.{u_5, u_10} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.65 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.72))))], Module.{u_5, max u_10 u_8} S (LinearMap.{u_1, u_3, u_8, u_10} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.65 (LinearMap.addCommMonoid.{u_1, u_3, u_8, u_10} R R₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37 σ₁₂)","typeFull":"{R : Type u_1} →\n  {R₂ : Type u_3} →\n    {S : Type u_5} →\n      {M : Type u_8} →\n        {M₂ : Type u_10} →\n          [inst : Semiring R] →\n            [inst_1 : Semiring R₂] →\n              [inst_2 : AddCommMonoid M] →\n                [inst_3 : AddCommMonoid M₂] →\n                  [inst_4 : Module R M] →\n                    [inst_5 : Module R₂ M₂] →\n                      {σ₁₂ : R →+* R₂} →\n                        [inst_6 : Semiring S] →\n                          [inst_7 : Module S M₂] → [SMulCommClass R₂ S M₂] → Module S (M →ₛₗ[σ₁₂] M₂)","typeReadable":"{R : Type u_1} →\n  {R₂ : Type u_3} →\n    {S : Type u_5} →\n      {M : Type u_8} →\n        {M₂ : Type u_10} →\n          [inst : Semiring R] →\n            [inst_1 : Semiring R₂] →\n              [inst_2 : AddCommMonoid M] →\n                [inst_3 : AddCommMonoid M₂] →\n                  [inst_4 : Module R M] →\n                    [inst_5 : Module R₂ M₂] →\n                      {σ₁₂ : R →+* R₂} →\n                        [inst_6 : Semiring S] →\n                          [inst_7 : Module S M₂] → [SMulCommClass R₂ S M₂] → Module S (M →ₛₗ[σ₁₂] M₂)","typeReferences":[["RingHom"],["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["LinearMap","addCommMonoid"],["SMulCommClass"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["LinearMap","addCommMonoid"],["LinearMap","module","_proof_1"],["Module","toDistribMulAction"],["Module","mk"],["MonoidWithZero","toMonoid"],["Semiring","toMonoidWithZero"],["LinearMap","module","_proof_2"],["LinearMap","instDistribMulAction"],["LinearMap"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","default_def"],"typeFallback":"forall {R₁ : Type.{u_2}} {R₂ : Type.{u_3}} {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.730117555._hygCtx._hyg.15 : Semiring.{u_2} R₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.730117555._hygCtx._hyg.18 : Semiring.{u_3} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.730117555._hygCtx._hyg.24 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.730117555._hygCtx._hyg.27 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.730117555._hygCtx._hyg.39 : Module.{u_2, u_8} R₁ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.730117555._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.730117555._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.730117555._hygCtx._hyg.43 : Module.{u_3, u_10} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.730117555._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.730117555._hygCtx._hyg.27] {σ₁₂ : RingHom.{u_2, u_3} R₁ R₂ (Semiring.toNonAssocSemiring.{u_2} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.730117555._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_3} R₂ 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Semiring R₂]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommGroup N₂] [inst_4 : Module R₁ M] [inst_5 : Module R₂ N₂] {σ₁₂ : R₁ →+* R₂}\n  (x x_1 : M →ₛₗ[σ₁₂] N₂), ⇑(x - x_1) = ⇑(x - x_1)","typeReferences":[["RingHom"],["LinearMap","instFunLike"],["Module"],["AddCommGroup"],["LinearMap"],["DFunLike","coe"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["LinearMap","instSub"],["AddCommGroup","toAddCommMonoid"],["HSub","hSub"],["Eq"],["instHSub"],["Semiring"]],"valueReferences":[["rfl"],["LinearMap","instFunLike"],["LinearMap","instSub"],["HSub","hSub"],["AddCommGroup","toAddCommMonoid"],["instHSub"],["LinearMap"],["DFunLike","coe"]]},{"isProp":true,"kind":"theorem","name":["AddMonoidHom","toNatLinearMap","_proof_2"],"typeFallback":"forall {M : Type.{u_2}} {M₂ : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3419663415._hygCtx._hyg.15 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3419663415._hygCtx._hyg.18 : AddCommMonoid.{u_1} M₂] (f : AddMonoidHom.{u_2, u_1} M M₂ (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3419663415._hygCtx._hyg.15))) (AddZeroClass.toAddZero.{u_1} M₂ (AddMonoid.toAddZeroClass.{u_1} M₂ (AddCommMonoid.toAddMonoid.{u_1} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3419663415._hygCtx._hyg.18)))) (n : Nat) (a : M), Eq.{succ u_1} M₂ (DFunLike.coe.{succ (max u_1 u_2), succ u_2, succ u_1} (AddMonoidHom.{u_2, u_1} M M₂ (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3419663415._hygCtx._hyg.15))) (AddZeroClass.toAddZero.{u_1} M₂ (AddMonoid.toAddZeroClass.{u_1} M₂ (AddCommMonoid.toAddMonoid.{u_1} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3419663415._hygCtx._hyg.18)))) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) (AddMonoidHom.instFunLike.{u_2, u_1} M M₂ 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: M →+ M₂) (n : ℕ) (a : M),\n  f (n • a) = n • f a","typeReferences":[["Nat"],["AddCommMonoid"],["AddMonoid","toNSMul"],["AddMonoidHom"],["HSMul","hSMul"],["instHSMul"],["AddMonoidHom","instFunLike"],["AddCommMonoid","toAddMonoid"],["Eq"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["map_nsmul"],["AddMonoidHom"],["AddMonoidHom","instFunLike"],["AddMonoidHom","instAddMonoidHomClass"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","restrictScalars_neg"],"typeFallback":"forall {R : Type.{u_14}} {S : Type.{u_15}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1839422664._hygCtx._hyg.20 : Semiring.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1839422664._hygCtx._hyg.23 : Semiring.{u_15} S] {M : Type.{u_19}} {N : Type.{u_20}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1839422664._hygCtx._hyg.56 : AddCommMonoid.{u_19} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1839422664._hygCtx._hyg.59 : AddCommGroup.{u_20} N] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1839422664._hygCtx._hyg.62 : Module.{u_14, u_19} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1839422664._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1839422664._hygCtx._hyg.56] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1839422664._hygCtx._hyg.66 : Module.{u_14, u_20} R N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1839422664._hygCtx._hyg.20 (AddCommGroup.toAddCommMonoid.{u_20} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1839422664._hygCtx._hyg.59)] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1839422664._hygCtx._hyg.70 : Module.{u_15, u_19} S M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1839422664._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1839422664._hygCtx._hyg.56] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1839422664._hygCtx._hyg.74 : Module.{u_15, u_20} S N 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(AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43)) (DistribMulAction.toDistribSMul.{u_14, u_16} R M (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30)) (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43) (Module.toDistribMulAction.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49))))) m x)) (HSMul.hSMul.{u_15, u_17, u_17} S M₂ M₂ (instHSMul.{u_15, u_17} S M₂ (SMulZeroClass.toSMul.{u_15, u_17} S M₂ (AddZero.toZero.{u_17} M₂ (AddZeroClass.toAddZero.{u_17} M₂ (AddMonoid.toAddZeroClass.{u_17} M₂ (AddCommMonoid.toAddMonoid.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46)))) (DistribSMul.toSMulZeroClass.{u_15, u_17} S M₂ (AddMonoid.toAddZeroClass.{u_17} M₂ (AddCommMonoid.toAddMonoid.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46)) (DistribMulAction.toDistribSMul.{u_15, u_17} S M₂ (MonoidWithZero.toMonoid.{u_15} S (Semiring.toMonoidWithZero.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33)) (AddCommMonoid.toAddMonoid.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46) (Module.toDistribMulAction.{u_15, u_17} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53))))) (DFunLike.coe.{max (succ u_14) (succ u_15), succ u_14, succ u_15} (RingHom.{u_14, u_15} R S (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30) (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33)) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => S) (RingHom.instFunLike.{u_14, u_15} R S (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30) (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33)) σ m) (toFun x))) -> (LinearMap.{u_14, u_15, u_16, u_17} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53)","typeFull":"{R : Type u_14} →\n  {S : Type u_15} →\n    [inst : Semiring R] →\n      [inst_1 : Semiring S] →\n        {σ : R →+* S} →\n          {M : Type u_16} →\n            {M₂ : Type u_17} →\n              [inst_2 : AddCommMonoid M] →\n                [inst_3 : AddCommMonoid M₂] →\n                  [inst_4 : Module R M] →\n                    [inst_5 : Module S M₂] →\n                      (toFun : M → M₂) →\n                        (∀ (x y : M), toFun (x + y) = toFun x + toFun y) →\n                          (∀ (m : R) (x : M), toFun (m • x) = σ m • toFun x) → M →ₛₗ[σ] M₂","typeReadable":"{R : Type u_14} →\n  {S : Type u_15} →\n    [inst : Semiring R] →\n      [inst_1 : Semiring S] →\n        {σ : R →+* S} →\n          {M : Type u_16} →\n            {M₂ : Type u_17} →\n              [inst_2 : AddCommMonoid M] →\n                [inst_3 : AddCommMonoid M₂] →\n                  [inst_4 : Module R M] →\n                    [inst_5 : Module S M₂] →\n                      (toFun : M → M₂) →\n                        (∀ (x y : M), toFun (x + y) = toFun x + toFun y) →\n                          (∀ (m : R) (x : M), toFun (m • x) = σ m • toFun x) → M →ₛₗ[σ] M₂","typeReferences":[["RingHom"],["Module"],["RingHom","instFunLike"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Eq"],["DistribSMul","toSMulZeroClass"],["instHAdd"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["LinearMap"],["AddZeroClass","toAddZero"],["HAdd","hAdd"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["HSMul","hSMul"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["AddCommMonoid","toAddCommSemigroup"],["LinearMap","mk"],["AddHom","mk"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","restrictScalarsₗ_apply"],"typeFallback":"forall (R : Type.{u_14}) (S : Type.{u_15}) (M : Type.{u_16}) (N : Type.{u_17}) [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 : Semiring.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 : Semiring.{u_15} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26 : AddCommMonoid.{u_16} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 : AddCommMonoid.{u_17} N] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.32 : Module.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.36 : Module.{u_14, u_17} R N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.40 : Module.{u_15, u_16} S M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.44 : Module.{u_15, u_17} S N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.48 : LinearMap.CompatibleSMul.{u_16, u_17, u_14, u_15} M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 (SMulZeroClass.toSMul.{u_14, u_16} R M (AddZero.toZero.{u_16} M (AddZeroClass.toAddZero.{u_16} M (AddMonoid.toAddZeroClass.{u_16} M (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26)))) (DistribSMul.toSMulZeroClass.{u_14, u_16} R M (AddMonoid.toAddZeroClass.{u_16} M (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26)) (DistribMulAction.toDistribSMul.{u_14, u_16} R M (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20)) (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26) (Module.toDistribMulAction.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.32)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.40 (SMulZeroClass.toSMul.{u_14, u_17} R N (AddZero.toZero.{u_17} N (AddZeroClass.toAddZero.{u_17} N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29)))) (DistribSMul.toSMulZeroClass.{u_14, u_17} R N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29)) (DistribMulAction.toDistribSMul.{u_14, u_17} R N (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20)) (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29) (Module.toDistribMulAction.{u_14, u_17} R N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.36)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.44] (R₁ : Type.{u_19}) [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.55 : Semiring.{u_19} R₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.58 : Module.{u_19, u_17} R₁ N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.55 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.62 : SMulCommClass.{u_15, u_19, u_17} S R₁ N (SMulZeroClass.toSMul.{u_15, u_17} S N (AddZero.toZero.{u_17} N (AddZeroClass.toAddZero.{u_17} N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29)))) (DistribSMul.toSMulZeroClass.{u_15, u_17} S N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29)) (DistribMulAction.toDistribSMul.{u_15, u_17} S N (MonoidWithZero.toMonoid.{u_15} S (Semiring.toMonoidWithZero.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23)) (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29) (Module.toDistribMulAction.{u_15, u_17} S N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.44)))) (SMulZeroClass.toSMul.{u_19, u_17} R₁ N (AddZero.toZero.{u_17} N (AddZeroClass.toAddZero.{u_17} N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29)))) (DistribSMul.toSMulZeroClass.{u_19, u_17} R₁ N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29)) (DistribMulAction.toDistribSMul.{u_19, u_17} R₁ N (MonoidWithZero.toMonoid.{u_19} R₁ (Semiring.toMonoidWithZero.{u_19} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.55)) (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29) (Module.toDistribMulAction.{u_19, u_17} R₁ N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.55 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.58))))] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.67 : SMulCommClass.{u_14, u_19, u_17} R R₁ N (SMulZeroClass.toSMul.{u_14, u_17} R N (AddZero.toZero.{u_17} N (AddZeroClass.toAddZero.{u_17} N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29)))) (DistribSMul.toSMulZeroClass.{u_14, u_17} R N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29)) (DistribMulAction.toDistribSMul.{u_14, u_17} R N (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20)) (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29) (Module.toDistribMulAction.{u_14, u_17} R N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.36)))) (SMulZeroClass.toSMul.{u_19, u_17} R₁ N (AddZero.toZero.{u_17} N (AddZeroClass.toAddZero.{u_17} N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29)))) (DistribSMul.toSMulZeroClass.{u_19, u_17} R₁ N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29)) (DistribMulAction.toDistribSMul.{u_19, u_17} R₁ N (MonoidWithZero.toMonoid.{u_19} R₁ (Semiring.toMonoidWithZero.{u_19} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.55)) (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29) (Module.toDistribMulAction.{u_19, u_17} R₁ N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.55 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.58))))] (fₗ : LinearMap.{u_15, u_15, u_16, u_17} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 (RingHom.id.{u_15} S (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.44), Eq.{max (succ u_16) (succ u_17)} (LinearMap.{u_14, u_14, u_16, u_17} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.32 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.36 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20)) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.55 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.58 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.67)) (LinearMap.{u_15, u_15, u_16, u_17} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 (RingHom.id.{u_15} S (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.44) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : LinearMap.{u_15, u_15, u_16, u_17} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 (RingHom.id.{u_15} S (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.44) => LinearMap.{u_14, u_14, u_16, u_17} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.36) (LinearMap.instFunLike.{u_19, u_19, max u_17 u_16, max u_17 u_16} R₁ R₁ (LinearMap.{u_15, u_15, u_16, u_17} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 (RingHom.id.{u_15} S (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.44) (LinearMap.{u_14, u_14, u_16, u_17} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.36) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.55 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.55 (LinearMap.addCommMonoid.{u_15, u_15, u_16, u_17} S S M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.44 (RingHom.id.{u_15} S (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23))) (LinearMap.addCommMonoid.{u_14, u_14, u_16, u_17} R R M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.36 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20))) (LinearMap.module.{u_15, u_15, u_19, u_16, u_17} S S R₁ M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.44 (RingHom.id.{u_15} S (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23)) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.55 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.58 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.62) (LinearMap.module.{u_14, u_14, u_19, u_16, u_17} R R R₁ M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.36 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20)) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.55 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.58 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.67) (RingHom.id.{u_19} R₁ (Semiring.toNonAssocSemiring.{u_19} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.55))) (LinearMap.restrictScalarsₗ.{u_14, u_15, u_16, u_17, u_19} R S M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.44 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.48 R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.55 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.58 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.62 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.67) fₗ) (LinearMap.restrictScalars.{u_14, u_15, u_16, u_17} R S M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.32 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[inst_4 : Module R M] [inst_5 : Module R N]\n  [inst_6 : Module S M] [inst_7 : Module S N] [inst_8 : LinearMap.CompatibleSMul M N R S] (R₁ : Type u_19)\n  [inst_9 : Semiring R₁] [inst_10 : Module R₁ N] [inst_11 : SMulCommClass S R₁ N] [inst_12 : SMulCommClass R R₁ N]\n  (fₗ : M →ₗ[S] N), (LinearMap.restrictScalarsₗ R S M N R₁) fₗ = ↑R fₗ","typeReferences":[["LinearMap","CompatibleSMul"],["LinearMap","instFunLike"],["Module"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["LinearMap","restrictScalarsₗ"],["Eq"],["LinearMap","restrictScalars"],["DistribSMul","toSMulZeroClass"],["LinearMap","module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["LinearMap"],["LinearMap","addCommMonoid"],["SMulCommClass"],["Module","toDistribMulAction"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["LinearMap","addCommMonoid"],["Semiring","toNonAssocSemiring"],["LinearMap","instFunLike"],["RingHom","id"],["Eq","refl"],["LinearMap","module"],["LinearMap","restrictScalarsₗ"],["LinearMap"],["DFunLike","coe"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","instSMulCommClass"],"typeFallback":"forall {R : Type.{u_1}} {R₂ : Type.{u_3}} {S : Type.{u_5}} {T : Type.{u_7}} {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.18 : Semiring.{u_3} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.24 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.27 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.31 : Module.{u_3, u_10} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.24] {σ₁₂ : RingHom.{u_1, u_3} R R₂ (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.18)} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.40 : DistribSMul.{u_5, u_10} S M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.24))] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.44 : SMulCommClass.{u_3, u_5, u_10} R₂ S M₂ (SMulZeroClass.toSMul.{u_3, u_10} R₂ M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.24)))) (DistribSMul.toSMulZeroClass.{u_3, u_10} R₂ M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.24)) (DistribMulAction.toDistribSMul.{u_3, u_10} R₂ M₂ (MonoidWithZero.toMonoid.{u_3} R₂ (Semiring.toMonoidWithZero.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.18)) (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.24) (Module.toDistribMulAction.{u_3, u_10} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.31)))) (SMulZeroClass.toSMul.{u_5, u_10} S M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.24)))) (DistribSMul.toSMulZeroClass.{u_5, u_10} S M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.24)) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.40))] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.49 : DistribSMul.{u_7, u_10} T M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.24))] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.53 : SMulCommClass.{u_3, u_7, u_10} R₂ T M₂ (SMulZeroClass.toSMul.{u_3, u_10} R₂ M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.24)))) (DistribSMul.toSMulZeroClass.{u_3, u_10} R₂ M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.24)) (DistribMulAction.toDistribSMul.{u_3, u_10} R₂ M₂ (MonoidWithZero.toMonoid.{u_3} R₂ (Semiring.toMonoidWithZero.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.18)) (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.24) (Module.toDistribMulAction.{u_3, u_10} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.31)))) (SMulZeroClass.toSMul.{u_7, u_10} T M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.24)))) (DistribSMul.toSMulZeroClass.{u_7, u_10} T M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.24)) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.49))] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.58 : SMulCommClass.{u_5, u_7, u_10} S T M₂ (SMulZeroClass.toSMul.{u_5, u_10} S M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.24)))) (DistribSMul.toSMulZeroClass.{u_5, u_10} S M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.24)) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.40)) (SMulZeroClass.toSMul.{u_7, u_10} T M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.24)))) (DistribSMul.toSMulZeroClass.{u_7, u_10} T M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.24)) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.49))], SMulCommClass.{u_5, u_7, max u_10 u_8} S T (LinearMap.{u_1, u_3, u_8, u_10} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.31) (LinearMap.instSMul.{u_1, u_3, u_5, u_8, u_10} R R₂ S M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.31 σ₁₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.44) (LinearMap.instSMul.{u_1, u_3, u_7, u_8, u_10} R R₂ T M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.31 σ₁₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.49 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1193137464._hygCtx._hyg.53)","typeFull":"∀ {R : Type u_1} {R₂ : Type u_3} {S : Type u_5} {T : Type u_7} {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R]\n  [inst_1 : Semiring R₂] [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M]\n  [inst_5 : Module R₂ M₂] {σ₁₂ : R →+* R₂} [inst_6 : DistribSMul S M₂] [inst_7 : SMulCommClass R₂ S M₂]\n  [inst_8 : DistribSMul T M₂] [inst_9 : SMulCommClass R₂ T M₂] [SMulCommClass S T M₂], SMulCommClass S T (M →ₛₗ[σ₁₂] M₂)","typeReadable":"∀ {R : Type u_1} {R₂ : Type u_3} {S : Type u_5} {T : Type u_7} {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R]\n  [inst_1 : Semiring R₂] [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M]\n  [inst_5 : Module R₂ M₂] {σ₁₂ : R →+* R₂} [inst_6 : DistribSMul S M₂] [inst_7 : SMulCommClass R₂ S M₂]\n  [inst_8 : DistribSMul T M₂] [inst_9 : SMulCommClass R₂ T M₂] [SMulCommClass S T M₂], SMulCommClass S T (M →ₛₗ[σ₁₂] M₂)","typeReferences":[["RingHom"],["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["DistribSMul"],["SMulCommClass"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["LinearMap","instSMul"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["LinearMap","instFunLike"],["SMulCommClass","mk"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["SMulCommClass","smul_comm"],["LinearMap","ext"],["HSMul","hSMul"],["LinearMap","instSMul"],["instHSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":false,"kind":"inductive","name":["IsLinearMap"],"typeFallback":"forall (R : Type.{u}) {M : Type.{v}} {M₂ : Type.{w}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.31 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.34 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.37 : AddCommMonoid.{w} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.40 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.34] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.44 : Module.{u, w} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.37], (M -> M₂) -> Prop","typeFull":"(R : Type u) →\n  {M : Type v} →\n    {M₂ : Type w} →\n      [inst : Semiring R] →\n        [inst_1 : AddCommMonoid M] → [inst_2 : AddCommMonoid M₂] → [Module R M] → [Module R M₂] → (M → M₂) → Prop","typeReadable":"(R : Type u) →\n  {M : Type v} →\n    {M₂ : Type w} →\n      [inst : Semiring R] →\n        [inst_1 : AddCommMonoid M] → [inst_2 : AddCommMonoid M₂] → [Module R M] → [Module R M₂] → (M → M₂) → Prop","typeReferences":[["AddCommMonoid"],["Module"],["Semiring"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["LinearMap","sub_comp"],"typeFallback":"forall {R₁ : Type.{u_2}} {R₂ : Type.{u_3}} {R₃ : Type.{u_4}} {M : Type.{u_8}} {M₂ : Type.{u_10}} {N₃ : Type.{u_13}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.15 : Semiring.{u_2} R₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.18 : Semiring.{u_3} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.21 : Semiring.{u_4} R₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.24 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.27 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.36 : AddCommGroup.{u_13} N₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.39 : Module.{u_2, u_8} R₁ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.43 : Module.{u_3, u_10} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.27] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.55 : Module.{u_4, u_13} R₃ N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.21 (AddCommGroup.toAddCommMonoid.{u_13} N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.36)] {σ₁₂ : RingHom.{u_2, u_3} R₁ R₂ (Semiring.toNonAssocSemiring.{u_2} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.18)} {σ₂₃ : RingHom.{u_3, u_4} R₂ R₃ (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.18) (Semiring.toNonAssocSemiring.{u_4} R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.21)} {σ₁₃ : RingHom.{u_2, u_4} R₁ R₃ (Semiring.toNonAssocSemiring.{u_2} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_4} R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.21)} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.74 : RingHomCompTriple.{u_2, u_3, u_4} R₁ R₂ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.21 σ₁₂ σ₂₃ σ₁₃] (f : LinearMap.{u_2, u_3, u_8, u_10} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.43) (g : LinearMap.{u_3, u_4, u_10, u_13} R₂ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.21 σ₂₃ M₂ N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.27 (AddCommGroup.toAddCommMonoid.{u_13} N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.36) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.55) (h : LinearMap.{u_3, u_4, u_10, u_13} R₂ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.21 σ₂₃ M₂ N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.27 (AddCommGroup.toAddCommMonoid.{u_13} N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.36) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.55), Eq.{max (succ u_13) (succ u_8)} (LinearMap.{u_2, u_4, u_8, u_13} R₁ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.21 σ₁₃ M N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_13} N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.36) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.55) (LinearMap.comp.{u_2, u_3, u_4, u_8, u_10, u_13} R₁ R₂ R₃ M M₂ N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.27 (AddCommGroup.toAddCommMonoid.{u_13} N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.36) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.55 σ₁₂ σ₂₃ σ₁₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.74 (HSub.hSub.{max u_10 u_13, max u_10 u_13, max u_10 u_13} (LinearMap.{u_3, u_4, u_10, u_13} R₂ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.21 σ₂₃ M₂ N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.27 (AddCommGroup.toAddCommMonoid.{u_13} N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.36) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.55) (LinearMap.{u_3, u_4, u_10, u_13} R₂ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.21 σ₂₃ M₂ N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.27 (AddCommGroup.toAddCommMonoid.{u_13} N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.36) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.55) (LinearMap.{u_3, u_4, u_10, u_13} R₂ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.21 σ₂₃ M₂ N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.27 (AddCommGroup.toAddCommMonoid.{u_13} N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.36) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.55) (instHSub.{max u_10 u_13} (LinearMap.{u_3, u_4, u_10, u_13} R₂ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.21 σ₂₃ M₂ N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.27 (AddCommGroup.toAddCommMonoid.{u_13} N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.36) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.55) (LinearMap.instSub.{u_3, u_4, u_10, u_13} R₂ R₃ M₂ N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.55 σ₂₃)) g h) f) (HSub.hSub.{max u_13 u_8, max u_13 u_8, max u_13 u_8} (LinearMap.{u_2, u_4, u_8, u_13} R₁ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.21 σ₁₃ M N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_13} N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.36) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.55) (LinearMap.{u_2, u_4, u_8, u_13} R₁ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.21 σ₁₃ M N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_13} N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.36) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.55) (LinearMap.{u_2, u_4, u_8, u_13} R₁ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.21 σ₁₃ M N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_13} N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.36) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.55) (instHSub.{max u_13 u_8} (LinearMap.{u_2, u_4, u_8, u_13} R₁ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.21 σ₁₃ M N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_13} N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.36) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.55) (LinearMap.instSub.{u_2, u_4, u_8, u_13} R₁ R₃ M N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.55 σ₁₃)) (LinearMap.comp.{u_2, u_3, u_4, u_8, u_10, u_13} R₁ R₂ R₃ M M₂ N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.27 (AddCommGroup.toAddCommMonoid.{u_13} N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.36) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.55 σ₁₂ σ₂₃ σ₁₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.74 g f) (LinearMap.comp.{u_2, u_3, u_4, u_8, u_10, u_13} R₁ R₂ R₃ M M₂ N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.27 (AddCommGroup.toAddCommMonoid.{u_13} N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.36) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.55 σ₁₂ σ₂₃ σ₁₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.75639284._hygCtx._hyg.74 h f))","typeFull":"∀ {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_8} {M₂ : Type u_10} {N₃ : Type u_13} [inst : Semiring R₁]\n  [inst_1 : Semiring R₂] [inst_2 : Semiring R₃] [inst_3 : AddCommMonoid M] [inst_4 : AddCommMonoid M₂]\n  [inst_5 : AddCommGroup N₃] [inst_6 : Module R₁ M] [inst_7 : Module R₂ M₂] [inst_8 : Module R₃ N₃] {σ₁₂ : R₁ →+* R₂}\n  {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [inst_9 : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (f : M →ₛₗ[σ₁₂] M₂)\n  (g h : M₂ →ₛₗ[σ₂₃] N₃), (g - h) ∘ₛₗ f = g ∘ₛₗ f - h ∘ₛₗ f","typeReadable":"∀ {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_8} {M₂ : Type u_10} {N₃ : Type u_13} [inst : Semiring R₁]\n  [inst_1 : Semiring R₂] [inst_2 : Semiring R₃] [inst_3 : AddCommMonoid M] [inst_4 : AddCommMonoid M₂]\n  [inst_5 : AddCommGroup N₃] [inst_6 : Module R₁ M] [inst_7 : Module R₂ M₂] [inst_8 : Module R₃ N₃] {σ₁₂ : R₁ →+* R₂}\n  {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [inst_9 : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (f : M →ₛₗ[σ₁₂] M₂)\n  (g h : M₂ →ₛₗ[σ₂₃] N₃), (g - h) ∘ₛₗ f = g ∘ₛₗ f - h ∘ₛₗ f","typeReferences":[["RingHom"],["Module"],["LinearMap","comp"],["AddCommGroup"],["LinearMap"],["RingHomCompTriple"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["LinearMap","instSub"],["AddCommGroup","toAddCommMonoid"],["HSub","hSub"],["Eq"],["instHSub"],["Semiring"]],"valueReferences":[["rfl"],["LinearMap","instSub"],["HSub","hSub"],["AddCommGroup","toAddCommMonoid"],["LinearMap","comp"],["instHSub"],["LinearMap"]]},{"isProp":false,"kind":"definition","name":["LinearMap","restrictScalars"],"typeFallback":"forall (R : Type.{u_1}) {S : Type.{u_5}} {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.33 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.37 : Module.{u_1, u_10} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.71 : Module.{u_5, u_8} S M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.75 : Module.{u_5, u_10} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.79 : LinearMap.CompatibleSMul.{u_8, u_10, u_1, u_5} M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27 R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 (SMulZeroClass.toSMul.{u_1, u_8} R M (AddZero.toZero.{u_8} M (AddZeroClass.toAddZero.{u_8} M (AddMonoid.toAddZeroClass.{u_8} M (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21)))) (DistribSMul.toSMulZeroClass.{u_1, u_8} R M (AddMonoid.toAddZeroClass.{u_8} M (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21)) (DistribMulAction.toDistribSMul.{u_1, u_8} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15)) (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21) (Module.toDistribMulAction.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.33)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.71 (SMulZeroClass.toSMul.{u_1, u_10} R M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27)))) (DistribSMul.toSMulZeroClass.{u_1, u_10} R M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27)) (DistribMulAction.toDistribSMul.{u_1, u_10} R M₂ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15)) (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27) (Module.toDistribMulAction.{u_1, u_10} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.37)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.75], (LinearMap.{u_5, u_5, u_8, u_10} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18 (RingHom.id.{u_5} S (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.18)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.71 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.75) -> (LinearMap.{u_1, u_1, u_8, u_10} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.15)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3287639652._hygCtx._hyg.37)","typeFull":"(R : Type u_1) →\n  {S : Type u_5} →\n    {M : Type u_8} →\n      {M₂ : Type u_10} →\n        [inst : Semiring R] →\n          [inst_1 : Semiring S] →\n            [inst_2 : AddCommMonoid M] →\n              [inst_3 : AddCommMonoid M₂] →\n                [inst_4 : Module R M] →\n                  [inst_5 : Module R M₂] →\n                    [inst_6 : Module S M] →\n                      [inst_7 : Module S M₂] → [LinearMap.CompatibleSMul M M₂ R S] → (M →ₗ[S] M₂) → M →ₗ[R] M₂","typeReadable":"(R : Type u_1) →\n  {S : Type u_5} →\n    {M : Type u_8} →\n      {M₂ : Type u_10} →\n        [inst : Semiring R] →\n          [inst_1 : Semiring S] →\n            [inst_2 : AddCommMonoid M] →\n              [inst_3 : AddCommMonoid M₂] →\n                [inst_4 : Module R M] →\n                  [inst_5 : Module R M₂] →\n                    [inst_6 : Module S M] →\n                      [inst_7 : Module S M₂] → [LinearMap.CompatibleSMul M M₂ R S] → (M →ₗ[S] M₂) → M →ₗ[R] M₂","typeReferences":[["LinearMap","CompatibleSMul"],["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["LinearMap","restrictScalars","_proof_1"],["Semiring","toNonAssocSemiring"],["LinearMap","instFunLike"],["AddCommMonoid","toAddCommSemigroup"],["RingHom","id"],["LinearMap","mk"],["AddHom","mk"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["LinearMap","restrictScalars","_proof_2"],["LinearMap"],["DFunLike","coe"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","id","_proof_1"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1952539257._hygCtx._hyg.15 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1952539257._hygCtx._hyg.21 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1952539257._hygCtx._hyg.33 : Module.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1952539257._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1952539257._hygCtx._hyg.21] (x : M) (y : M), Eq.{succ u_1} M (MulActionHom.toFun.{u_2, u_2, u_1, u_1} R R (DFunLike.coe.{succ u_2, succ u_2, succ u_2} (MonoidHom.{u_2, u_2} R R (MulOneClass.toMulOne.{u_2} R (Monoid.toMulOneClass.{u_2} R (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1952539257._hygCtx._hyg.15)))) (MulOneClass.toMulOne.{u_2} R (Monoid.toMulOneClass.{u_2} R (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1952539257._hygCtx._hyg.15))))) R 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{R : Type.{u_1}} {S : Type.{u_5}} {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1636155313._hygCtx._hyg.15 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1636155313._hygCtx._hyg.18 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1636155313._hygCtx._hyg.24 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1636155313._hygCtx._hyg.27 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1636155313._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1636155313._hygCtx._hyg.15] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1636155313._hygCtx._hyg.31 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1636155313._hygCtx._hyg.34 : Module.{u_5, u_10} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1636155313._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1636155313._hygCtx._hyg.18] {σ : RingHom.{u_1, u_5} R S 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(Module.toDistribMulAction.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1636155313._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1636155313._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1636155313._hygCtx._hyg.27) M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1636155313._hygCtx._hyg.18) (Module.toDistribMulAction.{u_5, u_10} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1636155313._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1636155313._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1636155313._hygCtx._hyg.34)) R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1636155313._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1636155313._hygCtx._hyg.31 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1636155313._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1636155313._hygCtx._hyg.18 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Semiring S] [inst_5 : Module S M₂] {σ : R →+* S},\n  SemilinearMapClass (M →ₑ+[↑σ] M₂) σ M M₂","typeReferences":[["RingHom"],["SemilinearMapClass"],["Module"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"],["RingHom","toMonoidHom"],["AddCommMonoid"],["Module","toDistribMulAction"],["Semiring","toNonAssocSemiring"],["DistribMulActionHom","instFunLike"],["MonoidWithZero","toMonoid"],["Semiring"],["DistribMulActionHom"]],"valueReferences":[["MulOneClass","toMulOne"],["DistribMulActionHom","instDistribMulActionSemiHomClassCoeMonoidHom"],["MonoidHom"],["MonoidHom","instFunLike"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["SemilinearMapClass","mk"],["DistribMulActionSemiHomClass","toAddMonoidHomClass"],["RingHom","toMonoidHom"],["Module","toDistribMulAction"],["Semiring","toNonAssocSemiring"],["DistribMulActionSemiHomClass","toMulActionSemiHomClass"],["DistribMulActionHom","instFunLike"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["AddMonoidHomClass","toAddHomClass"],["DistribMulActionHom"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","id'","_proof_1"],"typeFallback":"forall 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Module R M] {σ : R →+* R}\n  [RingHomId σ] (r : R) (x : M), r • x = σ r • 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{M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.27 : AddCommMonoid.{u_10} M₂] (R : Type.{u_14}) {S : Type.{u_15}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.73 : Semiring.{u_15} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.76 : SMul.{u_14, u_8} R M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.80 : Module.{u_15, u_8} S M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.84 : SMul.{u_14, u_10} R M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.88 : Module.{u_15, u_10} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.27] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.92 : SMul.{u_14, u_15} R S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.96 : LinearMap.CompatibleSMul.{u_8, u_10, u_14, u_15} M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.27 R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.76 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.84 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.88] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.102 : IsScalarTower.{u_14, u_15, u_10} R S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.92 (SMulZeroClass.toSMul.{u_15, u_10} S M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.27)))) (DistribSMul.toSMulZeroClass.{u_15, u_10} S M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.27)) (DistribMulAction.toDistribSMul.{u_15, u_10} S M₂ (MonoidWithZero.toMonoid.{u_15} S (Semiring.toMonoidWithZero.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.73)) (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.27) (Module.toDistribMulAction.{u_15, u_10} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.88)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.84] (f : LinearMap.{u_15, u_15, u_8, u_10} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.73 (RingHom.id.{u_15} S (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.73)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.88), (Function.Injective.{succ u_8, succ u_10} M M₂ (DFunLike.coe.{max (succ u_10) (succ u_8), succ u_8, succ u_10} (LinearMap.{u_15, u_15, u_8, u_10} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.73 (RingHom.id.{u_15} S (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.73)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.88) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) (LinearMap.instFunLike.{u_15, u_15, u_8, u_10} S S M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.80 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.88 (RingHom.id.{u_15} S (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.73))) f)) -> (IsScalarTower.{u_14, u_15, u_8} R S M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.92 (SMulZeroClass.toSMul.{u_15, u_8} S M (AddZero.toZero.{u_8} M (AddZeroClass.toAddZero.{u_8} M (AddMonoid.toAddZeroClass.{u_8} M (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.21)))) (DistribSMul.toSMulZeroClass.{u_15, u_8} S M (AddMonoid.toAddZeroClass.{u_8} M (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.21)) (DistribMulAction.toDistribSMul.{u_15, u_8} S M (MonoidWithZero.toMonoid.{u_15} S (Semiring.toMonoidWithZero.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.73)) (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.21) (Module.toDistribMulAction.{u_15, u_8} S M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.80)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1188063837._hygCtx._hyg.76)","typeFull":"∀ {M : Type u_8} {M₂ : Type u_10} [inst : AddCommMonoid M] [inst_1 : AddCommMonoid M₂] (R : Type u_14) {S : Type u_15}\n  [inst_2 : Semiring S] [inst_3 : SMul R M] [inst_4 : Module S M] [inst_5 : SMul R M₂] [inst_6 : Module S M₂]\n  [inst_7 : SMul R S] [LinearMap.CompatibleSMul M M₂ R S] [IsScalarTower R S M₂] (f : M →ₗ[S] M₂),\n  Function.Injective ⇑f → IsScalarTower R S M","typeReadable":"∀ {M : Type u_8} {M₂ : Type u_10} [inst : AddCommMonoid M] [inst_1 : AddCommMonoid M₂] (R : Type u_14) {S : Type u_15}\n  [inst_2 : Semiring S] [inst_3 : SMul R M] [inst_4 : Module S M] [inst_5 : SMul R M₂] [inst_6 : Module S M₂]\n  [inst_7 : SMul R S] [LinearMap.CompatibleSMul M M₂ R S] [IsScalarTower R S M₂] (f : M →ₗ[S] M₂),\n  Function.Injective ⇑f → IsScalarTower R S 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{R : Type.{u_14}} {S : Type.{u_15}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 : Semiring.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 : Semiring.{u_15} S] {σ : RingHom.{u_14, u_15} R S (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30) (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33)} {M : Type.{u_16}} {M₂ : Type.{u_17}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 : AddCommMonoid.{u_16} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 : AddCommMonoid.{u_17} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 : Module.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43] 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.36)} {M : Type.{u_17}} {M₂ : Type.{u_18}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.51 : AddCommMonoid.{u_17} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54 : AddCommMonoid.{u_18} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.57 : Module.{u_15, u_17} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.51] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.61 : Module.{u_16, u_18} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.65 : FunLike.{succ u_14, succ u_17, succ u_18} F M M₂] {motive : (SemilinearMapClass.{u_14, u_15, u_16, u_17, u_18} F R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.36 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.51 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.57 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.61 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.65) -> Sort.{u}} (t : SemilinearMapClass.{u_14, u_15, u_16, u_17, u_18} F R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.36 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.51 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.57 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.61 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.65), (forall [toAddHomClass : AddHomClass.{u_14, u_17, u_18} F M M₂ (AddCommMagma.toAdd.{u_17} M (AddCommSemigroup.toAddCommMagma.{u_17} M (AddCommMonoid.toAddCommSemigroup.{u_17} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.51))) (AddCommMagma.toAdd.{u_18} M₂ (AddCommSemigroup.toAddCommMagma.{u_18} M₂ (AddCommMonoid.toAddCommSemigroup.{u_18} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.65] [toMulActionSemiHomClass : MulActionSemiHomClass.{u_14, u_15, u_16, u_17, u_18} F R S (DFunLike.coe.{max (succ u_15) (succ u_16), succ u_15, succ u_16} (outParam.{max (succ u_16) (succ u_15)} (RingHom.{u_15, u_16} R S (Semiring.toNonAssocSemiring.{u_15} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.33) 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[toMulActionSemiHomClass : MulActionSemiHomClass F (⇑σ) M M₂] → motive ⋯) →\n                                motive t","typeReadable":"{F : Type u_14} →\n  {R : Type u_15} →\n    {S : Type u_16} →\n      [inst : Semiring R] →\n        [inst_1 : Semiring S] →\n          {σ : R →+* S} →\n            {M : Type u_17} →\n              {M₂ : Type u_18} →\n                [inst_2 : AddCommMonoid M] →\n                  [inst_3 : AddCommMonoid M₂] →\n                    [inst_4 : Module R M] →\n                      [inst_5 : Module S M₂] →\n                        [inst_6 : FunLike F M M₂] →\n                          {motive : SemilinearMapClass F σ M M₂ → Sort u} →\n                            (t : SemilinearMapClass F σ M M₂) →\n                              ([toAddHomClass : AddHomClass F M M₂] →\n                                  [toMulActionSemiHomClass : MulActionSemiHomClass F (⇑σ) M M₂] → motive ⋯) →\n                                motive 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{R : Type.{u}} {M : Type.{v}} {M₂ : Type.{w}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.31 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.34 : AddCommMonoid.{v} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.37 : AddCommMonoid.{w} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.40 : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.34] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.44 : Module.{u, w} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.37] {f : M -> M₂}, (IsLinearMap.{u, v, w} R M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.44 f) -> (forall (c : R) (x : M), Eq.{succ w} M₂ (f (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.LinearMap.Defs.470043658._hygCtx._hyg.34)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.34)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.31)) (AddCommMonoid.toAddMonoid.{v} M 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(AddCommMonoid.toAddMonoid.{w} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.37) (Module.toDistribMulAction.{u, w} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.470043658._hygCtx._hyg.44))))) c (f x)))","typeFull":"∀ {R : Type u} {M : Type v} {M₂ : Type w} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M₂]\n  [inst_3 : Module R M] [inst_4 : Module R M₂] {f : M → M₂}, IsLinearMap R f → ∀ (c : R) (x : M), f (c • x) = c • f x","typeReadable":"∀ {R : Type u} {M : Type v} {M₂ : Type w} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M₂]\n  [inst_3 : Module R M] [inst_4 : Module R M₂] {f : M → M₂}, IsLinearMap R f → ∀ (c : R) (x : M), f (c • x) = c • f 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4162764142._hygCtx._hyg.27)))))))) -> Prop) (x._@.Mathlib.Algebra.Module.LinearMap.Defs.4162764142._hygCtx._hyg.123 : Exists.{succ u_1} M₂ (fun (y : M₂) => Ne.{succ u_1} M₂ y (OfNat.ofNat.{u_1} M₂ 0 (Zero.toOfNat0.{u_1} M₂ (AddZero.toZero.{u_1} M₂ (AddZeroClass.toAddZero.{u_1} M₂ (AddMonoid.toAddZeroClass.{u_1} M₂ (AddCommMonoid.toAddMonoid.{u_1} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4162764142._hygCtx._hyg.27)))))))), (forall (y : M₂) (ne : Ne.{succ u_1} M₂ y (OfNat.ofNat.{u_1} M₂ 0 (Zero.toOfNat0.{u_1} M₂ (AddZero.toZero.{u_1} M₂ (AddZeroClass.toAddZero.{u_1} M₂ (AddMonoid.toAddZeroClass.{u_1} M₂ (AddCommMonoid.toAddMonoid.{u_1} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4162764142._hygCtx._hyg.27))))))), motive (Exists.intro.{succ u_1} M₂ (fun (y : M₂) => Ne.{succ u_1} M₂ y (OfNat.ofNat.{u_1} M₂ 0 (Zero.toOfNat0.{u_1} M₂ (AddZero.toZero.{u_1} M₂ (AddZeroClass.toAddZero.{u_1} M₂ (AddMonoid.toAddZeroClass.{u_1} M₂ 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x","typeReferences":[["AddCommMonoid"],["Exists"],["Exists","intro"],["Zero","toOfNat0"],["Ne"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toZero"],["OfNat","ofNat"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Exists","casesOn"],["Zero","toOfNat0"],["Ne"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toZero"],["OfNat","ofNat"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","mk","inj"],"typeFallback":"forall {R : Type.{u_14}} {S : Type.{u_15}} {inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 : Semiring.{u_14} R} {inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 : Semiring.{u_15} S} {σ : RingHom.{u_14, u_15} R S (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30) (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33)} {M : Type.{u_16}} {M₂ : Type.{u_17}} {inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 : AddCommMonoid.{u_16} M} {inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 : AddCommMonoid.{u_17} M₂} {inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 : Module.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43} {inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53 : Module.{u_15, u_17} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46} {toAddHom : AddHom.{u_16, u_17} M M₂ (AddCommMagma.toAdd.{u_16} M (AddCommSemigroup.toAddCommMagma.{u_16} M (AddCommMonoid.toAddCommSemigroup.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43))) (AddCommMagma.toAdd.{u_17} M₂ 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(DistribMulAction.toDistribSMul.{u_1, u_8} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404421._hygCtx._hyg.15)) (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404421._hygCtx._hyg.21) (Module.toDistribMulAction.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404421._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404421._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404421._hygCtx._hyg.33))))) c x)) (HSMul.hSMul.{u_1, u_10, u_10} R M₂ M₂ (instHSMul.{u_1, u_10} R M₂ (SMulZeroClass.toSMul.{u_1, u_10} R M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404421._hygCtx._hyg.27)))) (DistribSMul.toSMulZeroClass.{u_1, u_10} R M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404421._hygCtx._hyg.27)) (DistribMulAction.toDistribSMul.{u_1, u_10} R M₂ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404421._hygCtx._hyg.15)) (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404421._hygCtx._hyg.27) (Module.toDistribMulAction.{u_1, u_10} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404421._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404421._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404421._hygCtx._hyg.37))))) c (DFunLike.coe.{max (succ u_10) (succ u_8), succ u_8, succ u_10} (LinearMap.{u_1, u_1, u_8, u_10} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404421._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404421._hygCtx._hyg.15 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404421._hygCtx._hyg.15)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404421._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404421._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404421._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404421._hygCtx._hyg.37) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) (LinearMap.instFunLike.{u_1, u_1, u_8, u_10} R R M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404421._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404421._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404421._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404421._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404421._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404421._hygCtx._hyg.37 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4062404421._hygCtx._hyg.15))) fₗ x))","typeFull":"∀ {R : Type u_1} {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M₂] [inst_3 : Module R M] [inst_4 : Module R M₂] (fₗ : M →ₗ[R] M₂) (c : R) (x : M),\n  fₗ (c • x) = c • fₗ x","typeReadable":"∀ {R : Type u_1} {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M₂] [inst_3 : Module R M] [inst_4 : Module R M₂] (fₗ : M →ₗ[R] M₂) (c : R) (x : M),\n  fₗ (c • x) = c • fₗ x","typeReferences":[["LinearMap","instFunLike"],["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["LinearMap"],["DFunLike","coe"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["instHSMul"],["Eq"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["SemilinearMapClass","toMulActionSemiHomClass"],["LinearMap","instFunLike"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["map_smul"],["LinearMap","semilinearMapClass"],["Module","toDistribMulAction"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","module","_proof_1"],"typeFallback":"forall {R : Type.{u_1}} {R₂ : Type.{u_2}} {S : Type.{u_5}} {M : Type.{u_3}} {M₂ : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 : Semiring.{u_2} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 : AddCommMonoid.{u_4} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 : Module.{u_1, u_3} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37 : Module.{u_2, u_4} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27] {σ₁₂ : RingHom.{u_1, u_2} R R₂ (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_2} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18)} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.65 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.72 : Module.{u_5, u_4} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.65 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.76 : SMulCommClass.{u_2, u_5, u_4} R₂ S M₂ (SMulZeroClass.toSMul.{u_2, u_4} R₂ M₂ (AddZero.toZero.{u_4} M₂ (AddZeroClass.toAddZero.{u_4} M₂ (AddMonoid.toAddZeroClass.{u_4} M₂ (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27)))) (DistribSMul.toSMulZeroClass.{u_2, u_4} R₂ M₂ (AddMonoid.toAddZeroClass.{u_4} M₂ (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27)) (DistribMulAction.toDistribSMul.{u_2, u_4} R₂ M₂ (MonoidWithZero.toMonoid.{u_2} R₂ (Semiring.toMonoidWithZero.{u_2} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18)) (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27) (Module.toDistribMulAction.{u_2, u_4} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37)))) (SMulZeroClass.toSMul.{u_5, u_4} S M₂ (AddZero.toZero.{u_4} M₂ (AddZeroClass.toAddZero.{u_4} M₂ (AddMonoid.toAddZeroClass.{u_4} M₂ (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27)))) (DistribSMul.toSMulZeroClass.{u_5, u_4} S M₂ 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u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37) (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.65)) (AddCommMonoid.toAddMonoid.{max u_4 u_3} (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37) (LinearMap.addCommMonoid.{u_1, u_2, u_3, u_4} R R₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37 σ₁₂)) (LinearMap.instDistribMulAction.{u_1, u_2, u_5, u_3, u_4} R R₂ S M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37 σ₁₂ (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.65)) (Module.toDistribMulAction.{u_5, u_4} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.65 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.76))))) (HAdd.hAdd.{u_5, u_5, u_5} S S S (instHAdd.{u_5} S (Distrib.toAdd.{u_5} S (NonUnitalNonAssocSemiring.toDistrib.{u_5} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_5} S (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.65))))) x._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.100 x._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.102) x._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.104) (HAdd.hAdd.{max u_4 u_3, max u_4 u_3, max u_4 u_3} (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37) (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37) (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37) (instHAdd.{max u_4 u_3} (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37) (AddCommMagma.toAdd.{max u_4 u_3} (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37) (AddCommSemigroup.toAddCommMagma.{max u_4 u_3} (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37) (AddCommMonoid.toAddCommSemigroup.{max u_4 u_3} (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37) (LinearMap.addCommMonoid.{u_1, u_2, u_3, u_4} R R₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37 σ₁₂))))) (HSMul.hSMul.{u_5, max u_4 u_3, max u_4 u_3} S (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37) (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37) (instHSMul.{u_5, max u_4 u_3} S (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37) (SemigroupAction.toSMul.{u_5, max u_4 u_3} S (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37) (Monoid.toSemigroup.{u_5} S (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.65))) (MulAction.toSemigroupAction.{u_5, max u_4 u_3} S (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37) (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.65)) (DistribMulAction.toMulAction.{u_5, max u_4 u_3} S (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37) (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.65)) (AddCommMonoid.toAddMonoid.{max u_4 u_3} (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37) (LinearMap.addCommMonoid.{u_1, u_2, u_3, u_4} R R₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37 σ₁₂)) (LinearMap.instDistribMulAction.{u_1, u_2, u_5, u_3, u_4} R R₂ S M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37 σ₁₂ (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.65)) (Module.toDistribMulAction.{u_5, u_4} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.65 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.72) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.76))))) x._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.100 x._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.104) (HSMul.hSMul.{u_5, max u_4 u_3, max u_4 u_3} S (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37) (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37) (instHSMul.{u_5, max u_4 u_3} S (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37) (SemigroupAction.toSMul.{u_5, max u_4 u_3} S (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37) (Monoid.toSemigroup.{u_5} S (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.65))) (MulAction.toSemigroupAction.{u_5, max u_4 u_3} S (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37) (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.65)) (DistribMulAction.toMulAction.{u_5, max u_4 u_3} S (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1565340096._hygCtx._hyg.37) (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S 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[inst_7 : Module S M₂] [inst_8 : SMulCommClass R₂ S M���]\n  (x x_1 : S) (x_2 : M →ₛₗ[σ₁₂] M₂), (x + x_1) • x_2 = x • x_2 + x_1 • x_2","typeReferences":[["RingHom"],["Module"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Monoid","toSemigroup"],["Eq"],["DistribSMul","toSMulZeroClass"],["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["LinearMap","instDistribMulAction"],["AddZeroClass","toAddZero"],["LinearMap"],["HAdd","hAdd"],["LinearMap","addCommMonoid"],["SMulCommClass"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["HSMul","hSMul"],["AddZero","toZero"],["MulAction","toSemigroupAction"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["LinearMap","instFunLike"],["SemigroupAction","toSMul"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["Semiring","toNonAssocSemiring"],["LinearMap","ext"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Monoid","toSemigroup"],["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Semiring","toMonoidWithZero"],["LinearMap","instDistribMulAction"],["LinearMap"],["LinearMap","addCommMonoid"],["HAdd","hAdd"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["HSMul","hSMul"],["MulAction","toSemigroupAction"],["add_smul"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","restrictScalars_self"],"typeFallback":"forall (R : Type.{u_1}) {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1866830379._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1866830379._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1866830379._hygCtx._hyg.27 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1866830379._hygCtx._hyg.33 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1866830379._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1866830379._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1866830379._hygCtx._hyg.37 : Module.{u_1, u_10} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1866830379._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1866830379._hygCtx._hyg.27] (f : LinearMap.{u_1, u_1, u_8, u_10} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1866830379._hygCtx._hyg.15 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AddCommMonoid M]\n  [inst_2 : AddCommMonoid M₂] [inst_3 : Module R M] [inst_4 : Module R M₂] (f : M →ₗ[R] M₂), ↑R f = f","typeReadable":"∀ (R : Type u_1) {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid M₂] [inst_3 : Module R M] [inst_4 : Module R M₂] (f : M →ₗ[R] M₂), ↑R f = 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{R : Type.{u_1}} {S : Type.{u_5}} {M : Type.{u_8}} {M₃ : Type.{u_11}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3878078184._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3878078184._hygCtx._hyg.18 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3878078184._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3878078184._hygCtx._hyg.30 : AddCommMonoid.{u_11} M₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3878078184._hygCtx._hyg.33 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3878078184._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3878078184._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3878078184._hygCtx._hyg.41 : Module.{u_5, u_11} S M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3878078184._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3878078184._hygCtx._hyg.30] {σ : RingHom.{u_1, u_5} R S 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3878078184._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3878078184._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3878078184._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3878078184._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3878078184._hygCtx._hyg.41 σ)) (LinearMap.mk.{u_1, u_5, u_8, u_11} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3878078184._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3878078184._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3878078184._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3878078184._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3878078184._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3878078184._hygCtx._hyg.41 f h)) f","typeFull":"∀ {R : Type u_1} {S : Type u_5} {M : Type u_8} {M₃ : Type u_11} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₃] [inst_4 : Module R M] [inst_5 : Module S M₃] {σ : R →+* S}\n  (f : M →ₙ+ M₃) (h : ∀ (m : R) (x : M), f.toFun (m • x) = σ m • f.toFun x), ↑{ toAddHom := f, map_smul' := h } = f","typeReadable":"∀ {R : Type u_1} {S : Type u_5} {M : Type u_8} {M₃ : Type u_11} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₃] [inst_4 : Module R M] [inst_5 : Module S M₃] {σ : R →+* S}\n  (f : M →ₙ+ M₃) (h : ∀ (m : R) (x : M), f.toFun (m • x) = σ m • f.toFun x), ↑{ toAddHom := f, map_smul' := h } = f","typeReferences":[["RingHom"],["LinearMap","instFunLike"],["Module"],["LinearMap","mk"],["RingHom","instFunLike"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["LinearMap","semilinearMapClass"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddHom","toFun"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Eq"],["DistribSMul","toSMulZeroClass"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SemilinearMapClass","toAddHomClass"],["LinearMap"],["AddZeroClass","toAddZero"],["AddHomClass","toAddHom"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["HSMul","hSMul"],["AddHom"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["LinearMap","semilinearMapClass"],["rfl"],["AddHomClass","toAddHom"],["LinearMap","instFunLike"],["AddCommMonoid","toAddCommSemigroup"],["LinearMap","mk"],["SemilinearMapClass","toAddHomClass"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["LinearMap"],["AddHom"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","map_smul'"],"typeFallback":"forall {R : Type.{u_14}} {S : Type.{u_15}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 : Semiring.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 : Semiring.{u_15} S] {σ : RingHom.{u_14, u_15} R S (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30) (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33)} {M : Type.{u_16}} {M₂ : Type.{u_17}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 : AddCommMonoid.{u_16} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 : AddCommMonoid.{u_17} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 : Module.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43] 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⇑f.toAddMonoidHom = ⇑f","typeReferences":[["RingHom"],["LinearMap","toAddMonoidHom"],["LinearMap","instFunLike"],["Module"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["AddMonoidHom"],["AddMonoidHom","instFunLike"],["Eq"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["rfl"],["LinearMap","toAddMonoidHom"],["AddMonoidHom"],["AddMonoidHom","instFunLike"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"definition","name":["LinearMapClass","linearMap"],"typeFallback":"forall {R : Type.{u_1}} {M₁ : Type.{u_9}} {M₂ : Type.{u_10}} {F : Type.{u_14}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3923832418._hygCtx._hyg.16 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3923832418._hygCtx._hyg.19 : AddCommMonoid.{u_9} M₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3923832418._hygCtx._hyg.22 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3923832418._hygCtx._hyg.25 : Module.{u_1, u_9} R M₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3923832418._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3923832418._hygCtx._hyg.19] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3923832418._hygCtx._hyg.29 : Module.{u_1, u_10} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3923832418._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3923832418._hygCtx._hyg.22], F -> (forall [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3923832418._hygCtx._hyg.34 : FunLike.{succ u_14, succ u_9, succ u_10} F M₁ M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3923832418._hygCtx._hyg.39 : LinearMapClass.{u_14, u_1, u_9, u_10} F R M₁ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3923832418._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3923832418._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3923832418._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3923832418._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3923832418._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3923832418._hygCtx._hyg.34], LinearMap.{u_1, u_1, u_9, u_10} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3923832418._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3923832418._hygCtx._hyg.16 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3923832418._hygCtx._hyg.16)) M₁ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3923832418._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3923832418._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3923832418._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3923832418._hygCtx._hyg.29)","typeFull":"{R : Type u_1} →\n  {M₁ : Type u_9} →\n    {M₂ : Type u_10} →\n      {F : Type u_14} →\n        [inst : Semiring R] →\n          [inst_1 : AddCommMonoid M₁] →\n            [inst_2 : AddCommMonoid M₂] →\n              [inst_3 : Module R M₁] →\n                [inst_4 : Module R M₂] → F → [inst_5 : FunLike F M₁ M₂] → [LinearMapClass F R M₁ M₂] → M₁ →ₗ[R] M₂","typeReadable":"{R : Type u_1} →\n  {M₁ : Type u_9} →\n    {M₂ : Type u_10} →\n      {F : Type u_14} →\n        [inst : Semiring R] →\n          [inst_1 : AddCommMonoid M₁] →\n            [inst_2 : AddCommMonoid M₂] →\n              [inst_3 : Module R M₁] →\n                [inst_4 : Module R M₂] → F → [inst_5 : FunLike F M₁ M₂] → [LinearMapClass F R M₁ M₂] → M₁ →ₗ[R] M₂","typeReferences":[["FunLike"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Module"],["LinearMap"],["LinearMapClass"],["Semiring"]],"valueReferences":[["Semiring","toNonAssocSemiring"],["RingHom","id"],["SemilinearMapClass","semilinearMap"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","instNeg","_proof_2"],"typeFallback":"forall {R₁ : Type.{u_3}} {R₂ : Type.{u_4}} {M : Type.{u_2}} {N₂ : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.15 : Semiring.{u_3} R₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.18 : Semiring.{u_4} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.24 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.33 : AddCommGroup.{u_1} N₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.39 : Module.{u_3, u_2} R₁ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.51 : Module.{u_4, u_1} R₂ N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.18 (AddCommGroup.toAddCommMonoid.{u_1} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.33)] {σ₁₂ : RingHom.{u_3, u_4} R₁ R₂ (Semiring.toNonAssocSemiring.{u_3} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_4} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.18)} (f : LinearMap.{u_3, u_4, u_2, u_1} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.18 σ₁₂ M N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_1} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.33) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.51) (m : R₁) (x : M), Eq.{succ u_1} N₂ (Neg.neg.{max u_1 u_2} (M -> N₂) (Pi.instNeg.{u_2, u_1} M (fun (a : M) => N₂) (fun (i : M) => NegZeroClass.toNeg.{u_1} N₂ (SubNegZeroMonoid.toNegZeroClass.{u_1} N₂ (SubtractionMonoid.toSubNegZeroMonoid.{u_1} N₂ (SubtractionCommMonoid.toSubtractionMonoid.{u_1} N₂ (AddCommGroup.toDivisionAddCommMonoid.{u_1} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.33)))))) (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_2, succ u_1} (LinearMap.{u_3, u_4, u_2, u_1} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.18 σ₁₂ M N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_1} N₂ 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   [inst_4 : Module R₁ M] → [inst_5 : Module R₂ M₂] → {σ₁₂ : R₁ →+* R₂} → Add (M →ₛₗ[σ₁₂] M₂)","typeReadable":"{R₁ : Type u_2} →\n  {R₂ : Type u_3} →\n    {M : Type u_8} →\n      {M₂ : Type u_10} →\n        [inst : Semiring R₁] →\n          [inst_1 : Semiring R₂] →\n            [inst_2 : AddCommMonoid M] →\n              [inst_3 : AddCommMonoid M₂] →\n                [inst_4 : Module R₁ M] → [inst_5 : Module R₂ M₂] → {σ₁₂ : R₁ →+* R₂} → Add (M →ₛₗ[σ₁₂] 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(Semiring.toNonAssocSemiring.{u_14} R₄ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.85)} {σ₂₄ : RingHom.{u_3, u_14} R₂ R₄ (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.24) (Semiring.toNonAssocSemiring.{u_14} R₄ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.85)} {σ₁₄ : RingHom.{u_2, u_14} R₁ R₄ (Semiring.toNonAssocSemiring.{u_2} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.21) (Semiring.toNonAssocSemiring.{u_14} R₄ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.85)} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.110 : RingHomCompTriple.{u_3, u_4, u_14} R₂ R₃ R₄ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.85 σ₂₃ σ₃₄ σ₂₄] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.115 : RingHomCompTriple.{u_2, u_4, u_14} R₁ R₃ R₄ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.85 σ₁₃ σ₃₄ σ₁₄] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.120 : RingHomCompTriple.{u_2, u_3, u_14} R₁ R₂ R₄ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.85 σ₁₂ σ₂₄ σ₁₄] (f : LinearMap.{u_2, u_3, u_9, u_10} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.24 σ₁₂ M₁ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.33 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.115 h (LinearMap.comp.{u_2, u_3, u_4, u_9, u_10, u_11} R₁ R₂ R₃ M₁ M₂ M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.39 module_M₁ module_M₂ module_M₃ σ₁₂ σ₂₃ σ₁₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.66 g f)) (LinearMap.comp.{u_2, u_3, u_14, u_9, u_10, u_15} R₁ R₂ R₄ M₁ M₂ M₄ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3289002342._hygCtx._hyg.85 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Type u_4} {M₁ : Type u_9} {M₂ : Type u_10} {M₃ : Type u_11} [inst : Semiring R₁]\n  [inst_1 : Semiring R₂] [inst_2 : Semiring R₃] [inst_3 : AddCommMonoid M₁] [inst_4 : AddCommMonoid M₂]\n  [inst_5 : AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃}\n  {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [inst_6 : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {R₄ : Type u_14}\n  {M₄ : Type u_15} [inst_7 : Semiring R₄] [inst_8 : AddCommMonoid M₄] [inst_9 : Module R₄ M₄] {σ₃₄ : R₃ →+* R₄}\n  {σ₂₄ : R₂ →+* R₄} {σ₁₄ : R₁ →+* R₄} [inst_10 : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄]\n  [inst_11 : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄] [inst_12 : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄] (f : M₁ →ₛₗ[σ₁₂] M₂)\n  (g : M₂ →ₛₗ[σ₂₃] M₃) (h : M₃ →ₛₗ[σ₃₄] M₄), h ∘ₛₗ g ∘ₛₗ f = (h ∘ₛₗ g) ∘ₛₗ f","typeReferences":[["RingHom"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["Module"],["LinearMap","comp"],["Eq"],["LinearMap"],["RingHomCompTriple"],["Semiring"]],"valueReferences":[["LinearMap","comp_assoc"],["Eq","symm"],["LinearMap","comp"],["LinearMap"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","injective_of_comp_eq_id"],"typeFallback":"forall {R : Type.{u_1}} {S : Type.{u_5}} {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.18 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.24 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.30 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.34 : Module.{u_5, u_10} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.24] {σ : RingHom.{u_1, u_5} R S (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.18)} {σ' : RingHom.{u_5, u_1} S R (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.18) (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.15)} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.48 : RingHomInvPair.{u_1, u_5} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.18 σ σ'] (f : LinearMap.{u_1, u_5, u_8, u_10} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.18 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.34) (g : LinearMap.{u_5, u_1, u_10, u_8} S R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.15 σ' M₂ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.30), (Eq.{succ u_8} (LinearMap.{u_1, u_1, u_8, u_8} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.15 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.15)) M M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.30) (LinearMap.comp.{u_1, u_5, u_1, u_8, u_10, u_8} R S R M M₂ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.30 σ σ' (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.15)) (RingHomInvPair.triples.{u_1, u_5} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.18 σ σ' inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.48) g f) (LinearMap.id.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.30)) -> (Function.Injective.{succ u_8, succ u_10} M M₂ (DFunLike.coe.{max (succ u_10) (succ u_8), succ u_8, succ u_10} (LinearMap.{u_1, u_5, u_8, u_10} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.18 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.34) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) (LinearMap.instFunLike.{u_1, u_5, u_8, u_10} R S M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3139790652._hygCtx._hyg.34 σ) f))","typeFull":"∀ {R : Type u_1} {S : Type u_5} {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M] [inst_5 : Module S M₂] {σ : R →+* S}\n  {σ' : S →+* R} [inst_6 : RingHomInvPair σ σ'] (f : M →ₛₗ[σ] M₂) (g : M₂ →ₛₗ[σ'] M),\n  g ∘ₛₗ f = LinearMap.id → Function.Injective ⇑f","typeReadable":"∀ {R : Type u_1} {S : Type u_5} {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M] [inst_5 : Module S M₂] {σ : R →+* S}\n  {σ' : S →+* R} [inst_6 : RingHomInvPair σ σ'] (f : M →ₛₗ[σ] M₂) (g : M₂ →ₛₗ[σ'] M),\n  g ∘ₛₗ f = LinearMap.id → Function.Injective ⇑f","typeReferences":[["RingHom"],["LinearMap","instFunLike"],["Module"],["RingHomInvPair","triples"],["LinearMap","comp"],["LinearMap"],["DFunLike","coe"],["LinearMap","id"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["RingHomInvPair"],["Eq"],["Semiring"],["Function","Injective"]],"valueReferences":[["Function","bijective_id"],["LinearMap","instFunLike"],["Function","Injective","of_comp"],["RingHomInvPair","triples"],["eq_true"],["LinearMap","comp"],["Function","comp"],["LinearMap"],["Function","Surjective"],["DFunLike","coe"],["LinearMap","id"],["congrArg"],["Semiring","toNonAssocSemiring"],["of_eq_true"],["RingHom","id"],["id"],["Eq","mpr"],["Eq"],["And","left"],["Function","Injective"]]},{"isProp":false,"kind":"definition","name":["LinearMap","mulLeftRight"],"typeFallback":"forall (R : Type.{u_14}) {A : Type.{u_15}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.17 : Semiring.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.20 : NonUnitalNonAssocSemiring.{u_15} A] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.23 : Module.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.20)] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.27 : SMulCommClass.{u_14, u_15, u_15} R A A (SMulZeroClass.toSMul.{u_14, u_15} R A (AddZero.toZero.{u_15} A (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_14, u_15} R A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.20))) (DistribMulAction.toDistribSMul.{u_14, u_15} R A (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.17)) (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.20)) (Module.toDistribMulAction.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.23)))) (SMulZeroClass.toSMul.{u_15, u_15} A A (AddZero.toZero.{u_15} A (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_15, u_15} A A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.20))) (instDistribSMul.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.20)))] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.32 : IsScalarTower.{u_14, u_15, u_15} R A A (SMulZeroClass.toSMul.{u_14, u_15} R A (AddZero.toZero.{u_15} A (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_14, u_15} R A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.20))) (DistribMulAction.toDistribSMul.{u_14, u_15} R A (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.17)) (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.20)) (Module.toDistribMulAction.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.23)))) (SMulZeroClass.toSMul.{u_15, u_15} A A (AddZero.toZero.{u_15} A (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_15, u_15} A A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.20))) (instDistribSMul.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.20))) (SMulZeroClass.toSMul.{u_14, u_15} R A (AddZero.toZero.{u_15} A (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_14, u_15} R A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.20))) (DistribMulAction.toDistribSMul.{u_14, u_15} R A (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.17)) (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.20)) (Module.toDistribMulAction.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.23))))], (Prod.{u_15, u_15} A A) -> (LinearMap.{u_14, u_14, u_15, u_15} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.17 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.17)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.23)","typeFull":"(R : Type u_14) →\n  {A : Type u_15} →\n    [inst : Semiring R] →\n      [inst_1 : NonUnitalNonAssocSemiring A] →\n        [inst_2 : Module R A] → [SMulCommClass R A A] → [IsScalarTower R A A] → A × A → A →ₗ[R] A","typeReadable":"(R : Type u_14) →\n  {A : Type u_15} →\n    [inst : Semiring R] →\n      [inst_1 : NonUnitalNonAssocSemiring A] →\n        [inst_2 : Module R A] → [SMulCommClass R A A] → [IsScalarTower R A A] → A × A → A →ₗ[R] A","typeReferences":[["IsScalarTower"],["Module"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["instDistribSMul"],["AddZeroClass","toAddZero"],["Prod"],["SMulCommClass"],["Module","toDistribMulAction"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["AddZero","toZero"],["NonUnitalNonAssocSemiring"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["Semiring","toNonAssocSemiring"],["RingHom","id"],["LinearMap","mulLeftRight","_proof_1"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["LinearMap","comp"],["LinearMap","mulRight"],["Prod","snd"],["LinearMap","mulLeft"],["Prod","fst"]]},{"isProp":false,"kind":"definition","name":["LinearMap","evalAddMonoidHom"],"typeFallback":"forall {R₁ : Type.{u_2}} {R₂ : Type.{u_3}} {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.15 : Semiring.{u_2} R₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.18 : Semiring.{u_3} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.24 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.27 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.39 : Module.{u_2, u_8} R₁ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.43 : Module.{u_3, u_10} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.27] {σ₁₂ : RingHom.{u_2, u_3} R₁ R₂ (Semiring.toNonAssocSemiring.{u_2} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.18)}, M -> (AddMonoidHom.{max u_10 u_8, u_10} (LinearMap.{u_2, u_3, u_8, u_10} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.43) M₂ (AddZeroClass.toAddZero.{max u_10 u_8} (LinearMap.{u_2, u_3, u_8, u_10} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.43) (AddMonoid.toAddZeroClass.{max u_10 u_8} (LinearMap.{u_2, u_3, u_8, u_10} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.43) (AddCommMonoid.toAddMonoid.{max u_10 u_8} (LinearMap.{u_2, u_3, u_8, u_10} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.43) (LinearMap.addCommMonoid.{u_2, u_3, u_8, u_10} R₁ R₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.43 σ₁₂)))) (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2270188090._hygCtx._hyg.27))))","typeFull":"{R₁ : Type u_2} →\n  {R₂ : Type u_3} →\n    {M : Type u_8} →\n      {M₂ : Type u_10} →\n        [inst : Semiring R₁] →\n          [inst_1 : Semiring R₂] →\n            [inst_2 : AddCommMonoid M] →\n              [inst_3 : AddCommMonoid M₂] →\n                [inst_4 : Module R₁ M] → [inst_5 : Module R₂ M₂] → {σ₁₂ : R₁ →+* R₂} → M → (M →ₛₗ[σ₁₂] M₂) →+ M₂","typeReadable":"{R₁ : Type u_2} →\n  {R₂ : Type u_3} →\n    {M : Type u_8} →\n      {M₂ : Type u_10} →\n        [inst : Semiring R₁] →\n          [inst_1 : Semiring R₂] →\n            [inst_2 : AddCommMonoid M] →\n              [inst_3 : AddCommMonoid M₂] →\n                [inst_4 : Module R₁ M] → [inst_5 : Module R₂ M₂] → {σ₁₂ : R₁ →+* R₂} → M → (M →ₛₗ[σ₁₂] M₂) →+ M₂","typeReferences":[["LinearMap","addCommMonoid"],["RingHom"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["Module"],["AddMonoidHom"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["LinearMap"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["LinearMap","addCommMonoid"],["LinearMap","instFunLike"],["LinearMap","add_apply"],["AddCommMonoid","toAddMonoid"],["AddMonoidHom","mk"],["LinearMap","evalAddMonoidHom","_proof_1"],["AddZeroClass","toAddZero"],["LinearMap"],["DFunLike","coe"],["AddZero","toZero"],["ZeroHom","mk"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","instDistribMulAction","_proof_4"],"typeFallback":"forall {R : Type.{u_1}} {R₂ : Type.{u_2}} {S : Type.{u_5}} {M : Type.{u_3}} {M₂ : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 : Semiring.{u_2} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 : AddCommMonoid.{u_4} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 : Module.{u_1, u_3} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37 : Module.{u_2, u_4} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27] {σ₁₂ : RingHom.{u_1, u_2} R R₂ (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_2} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18)} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.65 : Monoid.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.68 : DistribMulAction.{u_5, u_4} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.65 (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27)] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.72 : SMulCommClass.{u_2, u_5, u_4} R₂ S M₂ (SMulZeroClass.toSMul.{u_2, u_4} R₂ M₂ (AddZero.toZero.{u_4} M₂ (AddZeroClass.toAddZero.{u_4} M₂ (AddMonoid.toAddZeroClass.{u_4} M₂ (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27)))) (DistribSMul.toSMulZeroClass.{u_2, u_4} R₂ M₂ (AddMonoid.toAddZeroClass.{u_4} M₂ (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27)) (DistribMulAction.toDistribSMul.{u_2, u_4} R₂ M₂ (MonoidWithZero.toMonoid.{u_2} R₂ (Semiring.toMonoidWithZero.{u_2} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18)) (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27) (Module.toDistribMulAction.{u_2, u_4} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37)))) (SMulZeroClass.toSMul.{u_5, u_4} S M₂ (AddZero.toZero.{u_4} M₂ (AddZeroClass.toAddZero.{u_4} M₂ (AddMonoid.toAddZeroClass.{u_4} M₂ (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27)))) (DistribSMul.toSMulZeroClass.{u_5, u_4} S M₂ (AddMonoid.toAddZeroClass.{u_4} M₂ (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27)) (DistribMulAction.toDistribSMul.{u_5, u_4} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.65 (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.68)))] (x._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.148 : S) (x._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.150 : LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (x._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.152 : LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37), Eq.{max (succ u_4) (succ u_3)} (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (HSMul.hSMul.{u_5, max u_4 u_3, max u_4 u_3} S (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (instHSMul.{u_5, max u_4 u_3} S (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (LinearMap.instSMul.{u_1, u_2, u_5, u_3, u_4} R R₂ S M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37 σ₁₂ (DistribMulAction.toDistribSMul.{u_5, u_4} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.65 (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.68) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.72)) x._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.148 (HAdd.hAdd.{max u_4 u_3, max u_4 u_3, max u_4 u_3} (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (instHAdd.{max u_4 u_3} (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (AddSemigroup.toAdd.{max u_4 u_3} (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (AddMonoid.toAddSemigroup.{max u_4 u_3} (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (AddCommMonoid.toAddMonoid.{max u_4 u_3} (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (LinearMap.addCommMonoid.{u_1, u_2, u_3, u_4} R R₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37 σ₁₂))))) x._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.150 x._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.152)) (HAdd.hAdd.{max u_4 u_3, max u_4 u_3, max u_4 u_3} (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (instHAdd.{max u_4 u_3} (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (AddSemigroup.toAdd.{max u_4 u_3} (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (AddMonoid.toAddSemigroup.{max u_4 u_3} (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (AddCommMonoid.toAddMonoid.{max u_4 u_3} (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (LinearMap.addCommMonoid.{u_1, u_2, u_3, u_4} R R₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37 σ₁₂))))) (HSMul.hSMul.{u_5, max u_4 u_3, max u_4 u_3} S (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (instHSMul.{u_5, max u_4 u_3} S (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (LinearMap.instSMul.{u_1, u_2, u_5, u_3, u_4} R R₂ S M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37 σ₁₂ (DistribMulAction.toDistribSMul.{u_5, u_4} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.65 (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.68) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.72)) x._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.148 x._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.150) (HSMul.hSMul.{u_5, max u_4 u_3, max u_4 u_3} S (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (instHSMul.{u_5, max u_4 u_3} S (LinearMap.{u_1, u_2, u_3, u_4} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37) (LinearMap.instSMul.{u_1, u_2, u_5, u_3, u_4} R R₂ S M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37 σ₁₂ (DistribMulAction.toDistribSMul.{u_5, u_4} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.65 (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.68) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.72)) x._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.148 x._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.152))","typeFull":"∀ {R : Type u_1} {R₂ : Type u_2} {S : Type 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1511783732._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1511783732._hygCtx._hyg.41), Eq.{succ u_11} M₃ (DFunLike.coe.{max (succ u_11) (succ u_8), succ u_8, succ u_11} (LinearMap.{u_1, u_5, u_8, u_11} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1511783732._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1511783732._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1511783732._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1511783732._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1511783732._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1511783732._hygCtx._hyg.41) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₃) (LinearMap.instFunLike.{u_1, u_5, u_8, u_11} R S M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1511783732._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1511783732._hygCtx._hyg.18 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Type.{u_1}} {M₂ : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1234448945._hygCtx._hyg.79 : Subsingleton.{succ u_2} M₂], Subsingleton.{max (succ u_1) (succ u_2)} (M -> M₂)","typeFull":"∀ {M : Type u_1} {M₂ : Type u_2} [Subsingleton M₂], Subsingleton (M → M₂)","typeReadable":"∀ {M : Type u_1} {M₂ : Type u_2} [Subsingleton M₂], Subsingleton (M → M₂)","typeReferences":[["Subsingleton"]],"valueReferences":[["Pi","instSubsingleton"]]},{"isProp":true,"kind":"theorem","name":["IsLinearMap","isLinearMap_smul'"],"typeFallback":"forall {R : Type.{u_14}} {M : Type.{u_15}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4179659854._hygCtx._hyg.34 : Semiring.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4179659854._hygCtx._hyg.37 : AddCommMonoid.{u_15} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4179659854._hygCtx._hyg.40 : Module.{u_14, u_15} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4179659854._hygCtx._hyg.34 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[inst_1 : AddCommMonoid M] [inst_2 : Module R M] (a : M),\n  IsLinearMap R fun c => c • 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{R : Type.{u_1}} {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.156091895._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.156091895._hygCtx._hyg.18 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.156091895._hygCtx._hyg.21 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.156091895._hygCtx._hyg.24 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.156091895._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.156091895._hygCtx._hyg.18] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.156091895._hygCtx._hyg.28 : Module.{u_1, u_10} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.156091895._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.156091895._hygCtx._hyg.21] (f : M -> M₂), (IsLinearMap.{u_1, u_8, u_10} R M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.156091895._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.156091895._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.156091895._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.156091895._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.156091895._hygCtx._hyg.28 f) -> (LinearMap.{u_1, u_1, u_8, u_10} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.156091895._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.156091895._hygCtx._hyg.15 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.156091895._hygCtx._hyg.15)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.156091895._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.156091895._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.156091895._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.156091895._hygCtx._hyg.28)","typeFull":"{R : Type u_1} →\n  {M : Type u_8} →\n    {M₂ : Type u_10} →\n      [inst : Semiring R] →\n        [inst_1 : AddCommMonoid M] →\n          [inst_2 : AddCommMonoid M₂] →\n            [inst_3 : Module R M] → [inst_4 : Module R M₂] → (f : M → M₂) → IsLinearMap R f → M →ₗ[R] M₂","typeReadable":"{R : Type u_1} →\n  {M : Type u_8} →\n    {M₂ : Type u_10} →\n      [inst : Semiring R] →\n        [inst_1 : AddCommMonoid M] →\n          [inst_2 : AddCommMonoid M₂] →\n            [inst_3 : Module R M] → [inst_4 : Module R M₂] → (f : M → M₂) → IsLinearMap R f → M →ₗ[R] M₂","typeReferences":[["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Module"],["LinearMap"],["IsLinearMap"],["Semiring"]],"valueReferences":[["IsLinearMap","map_smul"],["Semiring","toNonAssocSemiring"],["AddCommMonoid","toAddCommSemigroup"],["RingHom","id"],["LinearMap","mk"],["AddHom","mk"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["IsLinearMap","map_add"]]},{"isProp":false,"kind":"definition","name":["LinearMap","addCommGroup"],"typeFallback":"forall {R₁ : Type.{u_2}} {R₂ : Type.{u_3}} {M : Type.{u_8}} {N₂ : Type.{u_12}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.15 : Semiring.{u_2} R₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.18 : Semiring.{u_3} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.24 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.33 : AddCommGroup.{u_12} N₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.39 : Module.{u_2, u_8} R₁ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.51 : Module.{u_3, u_12} R₂ N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.18 (AddCommGroup.toAddCommMonoid.{u_12} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.33)] {σ₁₂ : RingHom.{u_2, u_3} R₁ R₂ (Semiring.toNonAssocSemiring.{u_2} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.18)}, AddCommGroup.{max u_12 u_8} (LinearMap.{u_2, u_3, u_8, u_12} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.18 σ₁₂ M N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_12} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.33) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.51)","typeFull":"{R₁ : Type u_2} →\n  {R₂ : Type u_3} →\n    {M : Type u_8} →\n      {N₂ : Type u_12} →\n        [inst : Semiring R₁] →\n          [inst_1 : Semiring R₂] →\n            [inst_2 : AddCommMonoid M] →\n              [inst_3 : AddCommGroup N₂] →\n                [inst_4 : Module R₁ M] → [inst_5 : Module R₂ N₂] → {σ₁₂ : R₁ →+* R₂} → AddCommGroup (M →ₛₗ[σ₁₂] N₂)","typeReadable":"{R₁ : Type u_2} →\n  {R₂ : Type u_3} →\n    {M : Type u_8} →\n      {N₂ : Type u_12} →\n        [inst : Semiring R₁] →\n          [inst_1 : Semiring R₂] →\n            [inst_2 : AddCommMonoid M] →\n              [inst_3 : AddCommGroup N₂] →\n                [inst_4 : Module R₁ M] → [inst_5 : Module R₂ N₂] → {σ₁₂ : R₁ →+* R₂} → AddCommGroup (M →ₛₗ[σ₁₂] N₂)","typeReferences":[["RingHom"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["Module"],["AddCommGroup","toAddCommMonoid"],["AddCommGroup"],["LinearMap"],["Semiring"]],"valueReferences":[["LinearMap","addCommGroup","_proof_2"],["LinearMap","addCommGroup","_proof_7"],["LinearMap","addCommGroup","_proof_6"],["Int","instMonoid"],["LinearMap","instFunLike"],["AddCommGroup","toAddGroup"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["LinearMap","addCommGroup","_proof_1"],["LinearMap","instNeg"],["LinearMap","instAdd"],["AddGroup","toSubNegMonoid"],["LinearMap","instZero"],["Function","Injective","addCommGroup"],["LinearMap","addCommGroup","_proof_4"],["DistribMulAction","toDistribSMul"],["LinearMap"],["AddCommGroup","toIntModule"],["Int"],["LinearMap","addCommMonoid"],["Module","toDistribMulAction"],["AddMonoid","toNSMul"],["SubNegMonoid","toAddMonoid"],["Int","instSemiring"],["LinearMap","instSub"],["AddCommGroup","toAddCommMonoid"],["LinearMap","instSMul"],["LinearMap","addCommGroup","_proof_5"],["LinearMap","addCommGroup","_proof_8"],["LinearMap","addCommGroup","_proof_3"],["Pi","addCommGroup"]]},{"isProp":false,"kind":"definition","name":["Module","compHom","toLinearMap"],"typeFallback":"forall {R : Type.{u_14}} {S : Type.{u_15}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3803786946._hygCtx._hyg.17 : Semiring.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3803786946._hygCtx._hyg.20 : Semiring.{u_15} S] (g : RingHom.{u_14, u_15} R S (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3803786946._hygCtx._hyg.17) (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3803786946._hygCtx._hyg.20)), LinearMap.{u_14, u_14, u_14, u_15} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3803786946._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3803786946._hygCtx._hyg.17 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3803786946._hygCtx._hyg.17)) R S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_14} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3803786946._hygCtx._hyg.17))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_15} S (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3803786946._hygCtx._hyg.20))) (Semiring.toModule.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3803786946._hygCtx._hyg.17) (Module.compHom.{u_15, u_14, u_15} S R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3803786946._hygCtx._hyg.20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_15} S (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3803786946._hygCtx._hyg.20))) (Semiring.toModule.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3803786946._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3803786946._hygCtx._hyg.17 g)","typeFull":"{R : Type u_14} → {S : Type u_15} → [inst : Semiring R] → [inst_1 : Semiring S] → (g : R →+* S) → R →ₗ[R] S","typeReadable":"{R : Type u_14} → {S : Type u_15} → [inst : Semiring R] → [inst_1 : Semiring S] → (g : R →+* S) → R →ₗ[R] S","typeReferences":[["RingHom"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Semiring","toModule"],["LinearMap"],["Module","compHom"],["Semiring"]],"valueReferences":[["RingHom"],["LinearMap","mk"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["AddHom","mk"],["RingHom","instFunLike"],["DFunLike","coe"],["Module","compHom","toLinearMap","_proof_1"],["RingHom","toSemilinearMap","_proof_2"],["Semiring","toNonAssocSemiring"],["AddCommMonoid","toAddCommSemigroup"],["RingHom","id"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Semiring","toModule"],["Module","compHom"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","comp_smul"],"typeFallback":"forall {R : Type.{u_1}} {S : Type.{u_5}} {M : Type.{u_8}} {M₂ : Type.{u_10}} {M₃ : Type.{u_11}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.24 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.27 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.30 : AddCommMonoid.{u_11} M₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.33 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.65 : Monoid.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.68 : DistribMulAction.{u_5, u_10} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.65 (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.27)] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.89 : Module.{u_1, u_10} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.27] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.93 : Module.{u_1, u_11} R M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.30] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.97 : SMulCommClass.{u_1, u_5, u_10} R S M₂ (SMulZeroClass.toSMul.{u_1, u_10} R M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.27)))) (DistribSMul.toSMulZeroClass.{u_1, u_10} R M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.27)) (DistribMulAction.toDistribSMul.{u_1, u_10} R M₂ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15)) (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.27) (Module.toDistribMulAction.{u_1, u_10} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.89)))) (SMulZeroClass.toSMul.{u_5, u_10} S M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.27)))) (DistribSMul.toSMulZeroClass.{u_5, u_10} S M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.27)) (DistribMulAction.toDistribSMul.{u_5, u_10} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.65 (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.68)))] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.102 : DistribMulAction.{u_5, u_11} S M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.65 (AddCommMonoid.toAddMonoid.{u_11} M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.30)] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.106 : SMulCommClass.{u_1, u_5, u_11} R S M₃ (SMulZeroClass.toSMul.{u_1, u_11} R M₃ (AddZero.toZero.{u_11} M₃ (AddZeroClass.toAddZero.{u_11} M₃ (AddMonoid.toAddZeroClass.{u_11} M₃ (AddCommMonoid.toAddMonoid.{u_11} M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.30)))) (DistribSMul.toSMulZeroClass.{u_1, u_11} R M₃ (AddMonoid.toAddZeroClass.{u_11} M₃ (AddCommMonoid.toAddMonoid.{u_11} M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.30)) (DistribMulAction.toDistribSMul.{u_1, u_11} R M₃ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15)) (AddCommMonoid.toAddMonoid.{u_11} M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.30) (Module.toDistribMulAction.{u_1, u_11} R M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.93)))) (SMulZeroClass.toSMul.{u_5, u_11} S M₃ (AddZero.toZero.{u_11} M₃ (AddZeroClass.toAddZero.{u_11} M₃ (AddMonoid.toAddZeroClass.{u_11} M₃ (AddCommMonoid.toAddMonoid.{u_11} M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.30)))) (DistribSMul.toSMulZeroClass.{u_5, u_11} S M₃ (AddMonoid.toAddZeroClass.{u_11} M₃ (AddCommMonoid.toAddMonoid.{u_11} M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.30)) (DistribMulAction.toDistribSMul.{u_5, u_11} S M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.65 (AddCommMonoid.toAddMonoid.{u_11} M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.30) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.102)))] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.111 : LinearMap.CompatibleSMul.{u_11, u_10, u_5, u_1} M₃ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.27 S R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 (SMulZeroClass.toSMul.{u_5, u_11} S M₃ (AddZero.toZero.{u_11} M₃ (AddZeroClass.toAddZero.{u_11} M₃ (AddMonoid.toAddZeroClass.{u_11} M₃ (AddCommMonoid.toAddMonoid.{u_11} M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.30)))) (DistribSMul.toSMulZeroClass.{u_5, u_11} S M₃ (AddMonoid.toAddZeroClass.{u_11} M₃ (AddCommMonoid.toAddMonoid.{u_11} M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.30)) (DistribMulAction.toDistribSMul.{u_5, u_11} S M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.65 (AddCommMonoid.toAddMonoid.{u_11} M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.30) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.102))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.93 (SMulZeroClass.toSMul.{u_5, u_10} S M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.27)))) (DistribSMul.toSMulZeroClass.{u_5, u_10} S M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.27)) (DistribMulAction.toDistribSMul.{u_5, u_10} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.65 (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.68))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.89] (g : LinearMap.{u_1, u_1, u_11, u_10} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15)) M₃ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.93 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.89) (a : S) (f : LinearMap.{u_1, u_1, u_8, u_11} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15)) M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.93), Eq.{max (succ u_10) (succ u_8)} (LinearMap.{u_1, u_1, u_8, u_10} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.89) (LinearMap.comp.{u_1, u_1, u_1, u_8, u_11, u_10} R R R M M₃ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.93 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.89 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15)) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15)) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15)) (RingHomCompTriple.ids.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15))) g 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.93) (instHSMul.{u_5, max u_11 u_8} S (LinearMap.{u_1, u_1, u_8, u_11} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15)) M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.93) (LinearMap.instSMul.{u_1, u_1, u_5, u_8, u_11} R R S M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.93 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15)) (DistribMulAction.toDistribSMul.{u_5, u_11} S M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.65 (AddCommMonoid.toAddMonoid.{u_11} M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.30) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.102) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.106)) a f)) (HSMul.hSMul.{u_5, max u_10 u_8, max u_10 u_8} S (LinearMap.{u_1, u_1, u_8, u_10} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.89) (LinearMap.{u_1, u_1, u_8, u_10} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.89) (instHSMul.{u_5, max u_10 u_8} S (LinearMap.{u_1, u_1, u_8, u_10} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.89) (LinearMap.instSMul.{u_1, u_1, u_5, u_8, u_10} R R S M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.89 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15)) (DistribMulAction.toDistribSMul.{u_5, u_10} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.65 (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.68) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.97)) a (LinearMap.comp.{u_1, u_1, u_1, u_8, u_11, u_10} R R R M M₃ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.93 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.89 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15)) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15)) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15)) (RingHomCompTriple.ids.{u_1, u_1} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.525146200._hygCtx._hyg.15))) g f))","typeFull":"∀ {R : Type u_1} {S : Type u_5} {M : Type u_8} {M₂ : Type u_10} {M₃ : Type u_11} [inst : Semiring R]\n  [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M₂] [inst_3 : AddCommMonoid M₃] [inst_4 : Module R M]\n  [inst_5 : Monoid S] [inst_6 : DistribMulAction S M₂] [inst_7 : Module R M₂] [inst_8 : Module R M₃]\n  [inst_9 : SMulCommClass R S M₂] [inst_10 : DistribMulAction S M₃] [inst_11 : SMulCommClass R S M₃]\n  [LinearMap.CompatibleSMul M₃ M₂ S R] (g : M₃ →ₗ[R] M₂) (a : S) (f : M →ₗ[R] M₃), g ∘ₗ (a • f) = a • g ∘ₗ f","typeReadable":"∀ {R : Type u_1} {S : Type u_5} {M : Type u_8} {M₂ : Type u_10} {M₃ : Type u_11} [inst : Semiring R]\n  [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M₂] [inst_3 : AddCommMonoid M₃] [inst_4 : Module R M]\n  [inst_5 : Monoid S] [inst_6 : DistribMulAction S M₂] [inst_7 : Module R M₂] [inst_8 : Module R M₃]\n  [inst_9 : SMulCommClass R S M₂] [inst_10 : DistribMulAction S M₃] [inst_11 : SMulCommClass R S M₃]\n  [LinearMap.CompatibleSMul M₃ M₂ S R] (g : M₃ →ₗ[R] M₂) (a : S) (f : M →ₗ[R] M₃), g ∘ₗ (a • f) = a • g ∘ₗ f","typeReferences":[["LinearMap","CompatibleSMul"],["Module"],["RingHomCompTriple","ids"],["LinearMap","comp"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["LinearMap"],["SMulCommClass"],["Module","toDistribMulAction"],["HSMul","hSMul"],["Monoid"],["LinearMap","instSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["LinearMap","instFunLike"],["RingHomCompTriple","ids"],["DistribMulAction","toDistribSMul"],["LinearMap","comp"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["LinearMap"],["DFunLike","coe"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["LinearMap","ext"],["HSMul","hSMul"],["instHSMul"],["LinearMap","instSMul"],["LinearMap","map_smul_of_tower"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","neg_comp"],"typeFallback":"forall {R₁ : Type.{u_2}} {R₂ : Type.{u_3}} {R₃ : Type.{u_4}} {M : Type.{u_8}} {M₂ : Type.{u_10}} {N₃ : Type.{u_13}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.15 : Semiring.{u_2} R₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.18 : Semiring.{u_3} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.21 : Semiring.{u_4} R₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.24 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.27 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.36 : AddCommGroup.{u_13} N₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.39 : Module.{u_2, u_8} R₁ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.43 : Module.{u_3, u_10} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.27] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.55 : Module.{u_4, u_13} R₃ N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.21 (AddCommGroup.toAddCommMonoid.{u_13} N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.36)] {σ₁₂ : RingHom.{u_2, u_3} R₁ R₂ (Semiring.toNonAssocSemiring.{u_2} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.18)} {σ₂₃ : RingHom.{u_3, u_4} R₂ R₃ (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.18) (Semiring.toNonAssocSemiring.{u_4} R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.21)} {σ₁₃ : RingHom.{u_2, u_4} R₁ R₃ (Semiring.toNonAssocSemiring.{u_2} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_4} R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.21)} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.74 : RingHomCompTriple.{u_2, u_3, u_4} R₁ R₂ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.21 σ₁₂ σ₂₃ σ₁₃] (f : LinearMap.{u_2, u_3, u_8, u_10} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1019637513._hygCtx._hyg.24 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{R : Type.{u_1}} {S : Type.{u_5}} {M : Type.{u_8}} {M₃ : Type.{u_11}} {F : Type.{u_14}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168033._hygCtx._hyg.16 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168033._hygCtx._hyg.19 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168033._hygCtx._hyg.22 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168033._hygCtx._hyg.31 : AddCommMonoid.{u_11} M₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168033._hygCtx._hyg.34 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168033._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168033._hygCtx._hyg.22] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168033._hygCtx._hyg.42 : Module.{u_5, u_11} S M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168033._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168033._hygCtx._hyg.31] {σ : RingHom.{u_1, u_5} R S 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168033._hygCtx._hyg.52] {σ' : RingHom.{u_5, u_1} S R (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168033._hygCtx._hyg.19) (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168033._hygCtx._hyg.16)} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168033._hygCtx._hyg.68 : RingHomInvPair.{u_1, u_5} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168033._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168033._hygCtx._hyg.19 σ σ'] (c : S) (x : M), Eq.{succ u_11} M₃ (HSMul.hSMul.{u_5, u_11, u_11} S M₃ M₃ (instHSMul.{u_5, u_11} S M₃ (SMulZeroClass.toSMul.{u_5, u_11} S M₃ (AddZero.toZero.{u_11} M₃ (AddZeroClass.toAddZero.{u_11} M₃ (AddMonoid.toAddZeroClass.{u_11} M₃ (AddCommMonoid.toAddMonoid.{u_11} M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168033._hygCtx._hyg.31)))) (DistribSMul.toSMulZeroClass.{u_5, u_11} S M₃ 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168033._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168033._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168033._hygCtx._hyg.34))))) (DFunLike.coe.{max (succ u_1) (succ u_5), succ u_5, succ u_1} (RingHom.{u_5, u_1} S R (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168033._hygCtx._hyg.19) (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168033._hygCtx._hyg.16)) S (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : S) => R) (RingHom.instFunLike.{u_5, u_1} S R (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168033._hygCtx._hyg.19) (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3671168033._hygCtx._hyg.16)) σ' c) x))","typeFull":"∀ {R : Type u_1} {S : Type u_5} {M : Type u_8} {M₃ : Type u_11} {F : Type u_14} [inst : Semiring R]\n  [inst_1 : Semiring S] [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₃] [inst_4 : Module R M]\n  [inst_5 : Module S M₃] {σ : R →+* S} (f : F) [inst_6 : FunLike F M M₃] [SemilinearMapClass F σ M M₃] {σ' : S →+* R}\n  [RingHomInvPair σ σ'] (c : S) (x : M), c • f x = f (σ' c • x)","typeReadable":"∀ {R : Type u_1} {S : Type u_5} {M : Type u_8} {M₃ : Type u_11} {F : Type u_14} [inst : Semiring R]\n  [inst_1 : Semiring S] [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₃] [inst_4 : Module R M]\n  [inst_5 : Module S M₃] {σ : R →+* S} (f : F) [inst_6 : FunLike F M M₃] [SemilinearMapClass F σ M M₃] {σ' : S →+* R}\n  [RingHomInvPair σ σ'] (c : S) (x : M), c • f x = f (σ' c • x)","typeReferences":[["RingHom"],["FunLike"],["SemilinearMapClass"],["Module"],["DistribMulAction","toDistribSMul"],["RingHom","instFunLike"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["instHSMul"],["RingHomInvPair"],["Eq"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["RingHomInvPair","triples₂"],["RingHom"],["Eq","trans"],["RingHomCompTriple","comp_apply"],["RingHom","instFunLike"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["congrArg"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["instHSMul"],["congrFun'"],["Eq"],["DistribSMul","toSMulZeroClass"],["SemilinearMapClass","toMulActionSemiHomClass"],["True"],["outParam"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["eq_self"],["MulActionSemiHomClass","map_smulₛₗ"],["Module","toDistribMulAction"],["of_eq_true"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","mk","sizeOf_spec"],"typeFallback":"forall {R : Type.{u_14}} {S : Type.{u_15}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 : Semiring.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 : Semiring.{u_15} S] {σ : RingHom.{u_14, u_15} R S (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30) (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33)} {M : Type.{u_16}} {M₂ : Type.{u_17}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 : AddCommMonoid.{u_16} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 : AddCommMonoid.{u_17} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 : Module.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43] 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N₂)","typeReferences":[["RingHom"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["Module"],["AddCommGroup","toAddCommMonoid"],["Sub"],["AddCommGroup"],["LinearMap"],["Semiring"]],"valueReferences":[["LinearMap","instSub","_proof_1"],["Pi","instSub"],["LinearMap","instFunLike"],["AddCommGroup","toAddGroup"],["LinearMap","mk"],["AddHom","mk"],["LinearMap"],["DFunLike","coe"],["AddCommMonoid","toAddCommSemigroup"],["SubNegMonoid","toSub"],["Sub","mk"],["AddCommGroup","toAddCommMonoid"],["HSub","hSub"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["AddGroup","toSubNegMonoid"],["instHSub"],["LinearMap","instSub","_proof_2"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","isLinearMap_of_compatibleSMul"],"typeFallback":"forall (R : Type.{u_1}) {S : Type.{u_5}} {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2877613765._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2877613765._hygCtx._hyg.18 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[inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2877613765._hygCtx._hyg.75 : Module.{u_5, u_10} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2877613765._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2877613765._hygCtx._hyg.27] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2877613765._hygCtx._hyg.79 : LinearMap.CompatibleSMul.{u_8, u_10, u_1, u_5} M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2877613765._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2877613765._hygCtx._hyg.27 R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2877613765._hygCtx._hyg.18 (SMulZeroClass.toSMul.{u_1, u_8} R M (AddZero.toZero.{u_8} M (AddZeroClass.toAddZero.{u_8} M (AddMonoid.toAddZeroClass.{u_8} M (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2877613765._hygCtx._hyg.21)))) (DistribSMul.toSMulZeroClass.{u_1, u_8} R M (AddMonoid.toAddZeroClass.{u_8} M (AddCommMonoid.toAddMonoid.{u_8} M 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2877613765._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2877613765._hygCtx._hyg.18 (RingHom.id.{u_5} S (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2877613765._hygCtx._hyg.18)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2877613765._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2877613765._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2877613765._hygCtx._hyg.71 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2877613765._hygCtx._hyg.75) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) (LinearMap.instFunLike.{u_5, u_5, u_8, u_10} S S M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2877613765._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2877613765._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2877613765._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2877613765._hygCtx._hyg.27 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⇑f","typeReferences":[["LinearMap","CompatibleSMul"],["LinearMap","instFunLike"],["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["IsLinearMap"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["IsLinearMap","mk"],["LinearMap","instFunLike"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["SemilinearMapClass","toAddHomClass"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["LinearMap"],["DFunLike","coe"],["map_add"],["LinearMap","semilinearMapClass"],["Module","toDistribMulAction"],["Semiring","toNonAssocSemiring"],["AddCommMonoid","toAddCommSemigroup"],["RingHom","id"],["MonoidWithZero","toMonoid"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["LinearMap","map_smul_of_tower"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","IsScalarTower","compatibleSMul'"],"typeFallback":"forall 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{R₁ : Type.{u_2}} {R₂ : Type.{u_3}} {R₃ : Type.{u_4}} {M : Type.{u_8}} {M₂ : Type.{u_10}} {M₃ : Type.{u_11}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.15 : Semiring.{u_2} R₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.18 : Semiring.{u_3} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.21 : Semiring.{u_4} R₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.24 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.27 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.30 : AddCommMonoid.{u_11} M₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.39 : Module.{u_2, u_8} R₁ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.43 : Module.{u_3, u_10} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.27] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.47 : Module.{u_4, u_11} R₃ M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.30] {σ₁₂ : RingHom.{u_2, u_3} R₁ R₂ (Semiring.toNonAssocSemiring.{u_2} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.18)} {σ₂₃ : RingHom.{u_3, u_4} R₂ R₃ (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.18) (Semiring.toNonAssocSemiring.{u_4} R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.21)} {σ₁₃ : RingHom.{u_2, u_4} R₁ R₃ (Semiring.toNonAssocSemiring.{u_2} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_4} R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.21)} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.74 : RingHomCompTriple.{u_2, u_3, u_4} R₁ R₂ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.21 σ₁₂ σ₂₃ σ₁₃] (f : LinearMap.{u_2, u_3, u_8, u_10} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.43), Eq.{max (succ u_11) (succ u_8)} (LinearMap.{u_2, u_4, u_8, u_11} R₁ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.21 σ₁₃ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.47) (LinearMap.comp.{u_2, u_3, u_4, u_8, u_10, u_11} R₁ R₂ R₃ M M₂ M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.47 σ₁₂ σ₂₃ σ₁₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.74 (OfNat.ofNat.{max u_10 u_11} (LinearMap.{u_3, u_4, u_10, u_11} R₂ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.21 σ₂₃ M₂ M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.47) 0 (Zero.toOfNat0.{max u_10 u_11} (LinearMap.{u_3, u_4, u_10, u_11} R₂ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.21 σ₂₃ M₂ M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.47) (LinearMap.instZero.{u_3, u_4, u_10, u_11} R₂ R₃ M₂ M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.47 σ₂₃))) f) (OfNat.ofNat.{max u_11 u_8} (LinearMap.{u_2, u_4, u_8, u_11} R₁ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.21 σ₁₃ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.47) 0 (Zero.toOfNat0.{max u_11 u_8} (LinearMap.{u_2, u_4, u_8, u_11} R₁ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.21 σ₁₃ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.47) (LinearMap.instZero.{u_2, u_4, u_8, u_11} R₁ R₃ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4214758008._hygCtx._hyg.47 σ₁₃)))","typeFull":"∀ {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_8} {M₂ : Type u_10} {M₃ : Type u_11} [inst : Semiring R₁]\n  [inst_1 : Semiring R₂] [inst_2 : Semiring R₃] [inst_3 : AddCommMonoid M] [inst_4 : AddCommMonoid M₂]\n  [inst_5 : AddCommMonoid M₃] [inst_6 : Module R₁ M] [inst_7 : Module R₂ M₂] [inst_8 : Module R₃ M₃] {σ₁₂ : R₁ →+* R₂}\n  {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [inst_9 : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (f : M →ₛₗ[σ₁₂] M₂), 0 ∘ₛₗ f = 0","typeReadable":"∀ {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_8} {M₂ : Type u_10} {M₃ : Type u_11} [inst : Semiring R₁]\n  [inst_1 : Semiring R₂] [inst_2 : Semiring R₃] [inst_3 : AddCommMonoid M] [inst_4 : AddCommMonoid M₂]\n  [inst_5 : AddCommMonoid M₃] [inst_6 : Module R₁ M] [inst_7 : Module R₂ M₂] [inst_8 : Module R₃ M₃] {σ₁₂ : R₁ →+* R₂}\n  {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [inst_9 : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (f : M →ₛₗ[σ₁₂] M₂), 0 ∘ₛₗ f = 0","typeReferences":[["LinearMap","instZero"],["RingHom"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["Module"],["LinearMap","comp"],["Zero","toOfNat0"],["Eq"],["LinearMap"],["OfNat","ofNat"],["RingHomCompTriple"],["Semiring"]],"valueReferences":[["rfl"],["LinearMap","instZero"],["LinearMap","comp"],["Zero","toOfNat0"],["LinearMap"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","mulLeftRight_apply"],"typeFallback":"forall {R : Type.{u_14}} {A : Type.{u_15}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.17 : Semiring.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.20 : NonUnitalNonAssocSemiring.{u_15} A] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.23 : Module.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.20)] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.27 : SMulCommClass.{u_14, u_15, u_15} R A A (SMulZeroClass.toSMul.{u_14, u_15} R A (AddZero.toZero.{u_15} A (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_14, u_15} R A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.20))) (DistribMulAction.toDistribSMul.{u_14, u_15} R A (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.17)) (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.20)) (Module.toDistribMulAction.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.23)))) (SMulZeroClass.toSMul.{u_15, u_15} A A (AddZero.toZero.{u_15} A (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_15, u_15} A A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.20))) (instDistribSMul.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.20)))] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.32 : IsScalarTower.{u_14, u_15, u_15} R A A (SMulZeroClass.toSMul.{u_14, u_15} R A (AddZero.toZero.{u_15} A (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_14, u_15} R A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.20))) (DistribMulAction.toDistribSMul.{u_14, u_15} R A (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.17)) (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.20)) (Module.toDistribMulAction.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.23)))) (SMulZeroClass.toSMul.{u_15, u_15} A A (AddZero.toZero.{u_15} A (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_15, u_15} A A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.20))) (instDistribSMul.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.20))) (SMulZeroClass.toSMul.{u_14, u_15} R A (AddZero.toZero.{u_15} A (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_14, u_15} R A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.20))) (DistribMulAction.toDistribSMul.{u_14, u_15} R A (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.17)) (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.20)) (Module.toDistribMulAction.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.23))))] (a : A) (b : A) (x : A), Eq.{succ u_15} A (DFunLike.coe.{succ u_15, succ u_15, succ u_15} (LinearMap.{u_14, u_14, u_15, u_15} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.17 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.17)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.23) A (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : A) => A) (LinearMap.instFunLike.{u_14, u_14, u_15, u_15} R R A A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1386895219._hygCtx._hyg.23 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R 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[inst_1 : NonUnitalNonAssocSemiring A] [inst_2 : Module R A]\n  [inst_3 : SMulCommClass R A A] [inst_4 : IsScalarTower R A A] (a b x : A),\n  (LinearMap.mulLeftRight R (a, b)) x = a * x * b","typeReadable":"∀ {R : Type u_14} {A : Type u_15} [inst : Semiring R] [inst_1 : NonUnitalNonAssocSemiring A] [inst_2 : Module R A]\n  [inst_3 : SMulCommClass R A A] [inst_4 : IsScalarTower R A A] (a b x : A),\n  (LinearMap.mulLeftRight R (a, b)) x = a * x * b","typeReferences":[["LinearMap","instFunLike"],["Module"],["IsScalarTower"],["Prod","mk"],["HMul","hMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["instDistribSMul"],["DFunLike","coe"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["Eq"],["DistribSMul","toSMulZeroClass"],["LinearMap","mulLeftRight"],["NonUnitalNonAssocSemiring","toDistrib"],["Distrib","toMul"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["LinearMap"],["SMulCommClass"],["Module","toDistribMulAction"],["instHMul"],["AddZero","toZero"],["NonUnitalNonAssocSemiring"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["rfl"],["Semiring","toNonAssocSemiring"],["LinearMap","instFunLike"],["Prod","mk"],["RingHom","id"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["LinearMap"],["DFunLike","coe"],["LinearMap","mulLeftRight"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","addCommGroup","_proof_1"],"typeFallback":"forall {R₂ : Type.{u_1}} {N₂ : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.18 : Semiring.{u_1} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.33 : AddCommGroup.{u_2} N₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.51 : Module.{u_1, u_2} R₂ N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.18 (AddCommGroup.toAddCommMonoid.{u_2} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.33)], SMulCommClass.{u_1, 0, u_2} R₂ Int N₂ (SMulZeroClass.toSMul.{u_1, u_2} R₂ N₂ (AddZero.toZero.{u_2} N₂ (AddZeroClass.toAddZero.{u_2} N₂ (AddMonoid.toAddZeroClass.{u_2} N₂ (SubNegMonoid.toAddMonoid.{u_2} N₂ (AddGroup.toSubNegMonoid.{u_2} N₂ (AddCommGroup.toAddGroup.{u_2} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.33)))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R₂ N₂ (AddMonoid.toAddZeroClass.{u_2} N₂ (SubNegMonoid.toAddMonoid.{u_2} N₂ 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N₂","typeReferences":[["Module"],["AddCommGroup","toAddGroup"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["SMulZeroClass","toSMul"],["AddZeroClass","toAddZero"],["Int"],["SMulCommClass"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["MonoidWithZero","toMonoid"],["AddCommGroup","toAddCommMonoid"],["AddGroup","toSubNegMonoid"],["SubNegMonoid","toZSMul"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["AddCommGroup","toAddGroup"],["MonoidWithZero","toMonoid"],["AddCommGroup","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["AddGroup","int_smulCommClass'"],["Semiring","toMonoidWithZero"],["AddGroup","toSubNegMonoid"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","addCommGroup","_proof_2"],"typeFallback":"forall {R₁ : Type.{u_3}} {R₂ : Type.{u_4}} {M : Type.{u_2}} {N₂ : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.15 : Semiring.{u_3} R₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.18 : Semiring.{u_4} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.24 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.33 : AddCommGroup.{u_1} N₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.39 : Module.{u_3, u_2} R₁ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.51 : Module.{u_4, u_1} R₂ N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.18 (AddCommGroup.toAddCommMonoid.{u_1} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.33)] {σ₁₂ : RingHom.{u_3, u_4} R₁ R₂ (Semiring.toNonAssocSemiring.{u_3} R₁ 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M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53) -> (forall {R' : Type.{u_14}} {S' : Type.{u_15}} [inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 : Semiring.{u_14} R'] [inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 : Semiring.{u_15} S'] {σ' : RingHom.{u_14, u_15} R' S' (Semiring.toNonAssocSemiring.{u_14} R' inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30) (Semiring.toNonAssocSemiring.{u_15} S' inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33)} {M' : Type.{u_16}} {M₂' : Type.{u_17}} [inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 : AddCommMonoid.{u_16} M'] [inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 : AddCommMonoid.{u_17} M₂'] [inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 : Module.{u_14, u_16} R' M' inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43] [inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53 : Module.{u_15, u_17} S' M₂' inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46], (LinearMap.{u_14, u_15, u_16, u_17} R' S' inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 σ' M' M₂' inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53) -> Sort.{u}))","typeFull":"Sort u →\n  {R : Type u_14} →\n    {S : Type u_15} →\n      [inst : Semiring R] →\n        [inst_1 : Semiring S] ���\n          {σ : R →+* S} →\n            {M : Type u_16} →\n              {M₂ : Type u_17} →\n                [inst_2 : AddCommMonoid M] →\n                  [inst_3 : AddCommMonoid M₂] →\n                    [inst_4 : Module R M] →\n                      [inst_5 : Module S M₂] →\n                        (M →ₛₗ[σ] M₂) →\n                          {R' : Type u_14} →\n                            {S' : Type u_15} →\n                              [inst' : Semiring R'] →\n                                [inst'_1 : Semiring S'] →\n                                  {σ' : R' →+* S'} →\n                                    {M' : Type u_16} →\n                                      {M₂' : Type u_17} →\n                                        [inst'_2 : AddCommMonoid M'] →\n                                          [inst'_3 : AddCommMonoid M₂'] →\n                                            [inst'_4 : Module R' M'] →\n                                              [inst'_5 : Module S' M₂'] → (M' →ₛₗ[σ'] M₂') → Sort u","typeReadable":"Sort u →\n  {R : Type u_14} →\n    {S : Type u_15} →\n      [inst : Semiring R] →\n        [inst_1 : Semiring S] →\n          {σ : R →+* S} →\n            {M : Type u_16} →\n              {M₂ : Type u_17} →\n                [inst_2 : AddCommMonoid M] →\n                  [inst_3 : AddCommMonoid M₂] →\n                    [inst_4 : Module R M] →\n                      [inst_5 : Module S M₂] →\n                        (M →ₛₗ[σ] M₂) →\n                          {R' : Type u_14} →\n                            {S' : Type u_15} →\n                              [inst' : Semiring R'] →\n                                [inst'_1 : Semiring S'] →\n                                  {σ' : R' →+* S'} →\n                                    {M' : Type u_16} →\n                                      {M₂' : Type u_17} →\n                                        [inst'_2 : AddCommMonoid M'] →\n                                          [inst'_3 : AddCommMonoid M₂'] →\n                                            [inst'_4 : Module R' M'] →\n                                              [inst'_5 : Module S' M₂'] → (M' →ₛₗ[σ'] M₂') → Sort u","typeReferences":[["RingHom"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["Module"],["LinearMap"],["Semiring"]],"valueReferences":[["AddCommMonoid","toAddCommSemigroup"],["LinearMap","casesOn"],["HEq"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["AddHom"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","map_eq_zero_iff"],"typeFallback":"forall {R : Type.{u_1}} {S : Type.{u_5}} {M : Type.{u_8}} {M₃ : Type.{u_11}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4221540347._hygCtx._hyg.15 : 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2204729447._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2204729447._hygCtx._hyg.51 σ₁₂) f) x) (Neg.neg.{u_12} N₂ (NegZeroClass.toNeg.{u_12} N₂ (SubNegZeroMonoid.toNegZeroClass.{u_12} N₂ (SubtractionMonoid.toSubNegZeroMonoid.{u_12} N₂ (SubtractionCommMonoid.toSubtractionMonoid.{u_12} N₂ (AddCommGroup.toDivisionAddCommMonoid.{u_12} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2204729447._hygCtx._hyg.33))))) (DFunLike.coe.{max (succ u_12) (succ u_8), succ u_8, succ u_12} (LinearMap.{u_2, u_3, u_8, u_12} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2204729447._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2204729447._hygCtx._hyg.18 σ₁₂ M N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2204729447._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_12} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2204729447._hygCtx._hyg.33) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2204729447._hygCtx._hyg.39 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.51] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.61 : Module.{u_16, u_18} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.65 : FunLike.{succ u_14, succ u_17, succ u_18} F M M₂] [toAddHomClass : AddHomClass.{u_14, u_17, u_18} F M M₂ (AddCommMagma.toAdd.{u_17} M (AddCommSemigroup.toAddCommMagma.{u_17} M (AddCommMonoid.toAddCommSemigroup.{u_17} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.51))) (AddCommMagma.toAdd.{u_18} M₂ (AddCommSemigroup.toAddCommMagma.{u_18} M₂ (AddCommMonoid.toAddCommSemigroup.{u_18} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54))) 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[inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.18 : Semiring.{u_3} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.21 : Semiring.{u_4} R₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.24 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.33 : AddCommGroup.{u_12} N₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.36 : AddCommGroup.{u_13} N₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.39 : Module.{u_2, u_8} R₁ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.51 : Module.{u_3, u_12} R₂ N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.18 (AddCommGroup.toAddCommMonoid.{u_12} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.33)] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.55 : Module.{u_4, u_13} R₃ N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.21 (AddCommGroup.toAddCommMonoid.{u_13} N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.36)] {σ₁₂ : RingHom.{u_2, u_3} R₁ R₂ (Semiring.toNonAssocSemiring.{u_2} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.18)} {σ₂₃ : RingHom.{u_3, u_4} R₂ R₃ (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.18) (Semiring.toNonAssocSemiring.{u_4} R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.21)} {σ₁₃ : RingHom.{u_2, u_4} R₁ R₃ (Semiring.toNonAssocSemiring.{u_2} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_4} R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.21)} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.74 : RingHomCompTriple.{u_2, u_3, u_4} R₁ R₂ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.21 σ₁₂ σ₂₃ σ₁₃] (f : LinearMap.{u_2, u_3, u_8, u_12} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.18 σ₁₂ M N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_12} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.33) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.51) (g : LinearMap.{u_2, u_3, u_8, u_12} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.18 σ₁₂ M N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_12} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.33) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.51) (h : LinearMap.{u_3, u_4, u_12, u_13} R₂ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.21 σ₂₃ N₂ N₃ (AddCommGroup.toAddCommMonoid.{u_12} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.33) (AddCommGroup.toAddCommMonoid.{u_13} N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.36) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.51 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.55), Eq.{max (succ u_13) (succ u_8)} (LinearMap.{u_2, u_4, u_8, u_13} R₁ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.21 σ₁₃ M N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_13} N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.36) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.55) (LinearMap.comp.{u_2, u_3, u_4, u_8, u_12, u_13} R₁ R₂ R₃ M N₂ N₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1876214333._hygCtx._hyg.18 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: RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (f g : M →ₛₗ[σ₁₂] N₂)\n  (h : N₂ →ₛₗ[σ₂₃] N₃), h ∘ₛₗ (g - f) = h ∘ₛₗ g - h ∘ₛₗ f","typeReferences":[["RingHom"],["Module"],["LinearMap","comp"],["AddCommGroup"],["LinearMap"],["RingHomCompTriple"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["LinearMap","instSub"],["AddCommGroup","toAddCommMonoid"],["HSub","hSub"],["Eq"],["instHSub"],["Semiring"]],"valueReferences":[["LinearMap","map_sub"],["LinearMap","instFunLike"],["LinearMap","ext"],["LinearMap","instSub"],["HSub","hSub"],["AddCommGroup","toAddCommMonoid"],["LinearMap","comp"],["instHSub"],["LinearMap"],["DFunLike","coe"]]},{"isProp":true,"kind":"theorem","name":["DistribMulActionHom","toLinearMap_injective"],"typeFallback":"forall {R : Type.{u_1}} {S : Type.{u_5}} {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.15 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.18 : 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u_10} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.24)) S (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.31)) (RingHom.toMonoidHom.{u_1, u_5} R S (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.24) (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.31) σ) M (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.15) (Module.toDistribMulAction.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.27) M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.15) (Module.toDistribMulAction.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.27) M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.18) (Module.toDistribMulAction.{u_5, u_10} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.34)}, (Eq.{max (succ u_10) (succ u_8)} (LinearMap.{u_1, u_5, u_8, u_10} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.31 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.34) (SemilinearMapClass.semilinearMap.{u_1, u_5, u_8, u_10, max u_10 u_8} R S M M₂ (DistribMulActionHom.{u_1, u_5, u_8, u_10} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.24)) S (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.31)) (RingHom.toMonoidHom.{u_1, u_5} R S (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.24) (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325880._hygCtx._hyg.31) σ) M (AddCommMonoid.toAddMonoid.{u_8} 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{R₃ : Type.{u_5}} {M₁ : Type.{u_2}} {M₂ : Type.{u_3}} {M₃ : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.21 : Semiring.{u_6} R₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.24 : Semiring.{u_4} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.27 : Semiring.{u_5} R₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.33 : AddCommMonoid.{u_2} M₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.36 : AddCommMonoid.{u_3} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.39 : AddCommMonoid.{u_1} M₃] {module_M₁ : Module.{u_6, u_2} R₁ M₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.33} {module_M₂ : Module.{u_4, u_3} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.24 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.24 σ₁₂ M₁ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.36 module_M₁ module_M₂) M₁ (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M₁) => M₂) (LinearMap.instFunLike.{u_6, u_4, u_2, u_3} R₁ R₂ M₁ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1109371310._hygCtx._hyg.36 module_M₁ module_M₂ σ₁₂) g) x))","typeFull":"∀ {R₁ : Type u_6} {R₂ : Type u_4} {R₃ : Type u_5} {M₁ : Type u_2} {M₂ : Type u_3} {M₃ : Type u_1} [inst : Semiring R₁]\n  [inst_1 : Semiring R₂] [inst_2 : Semiring R₃] [inst_3 : AddCommMonoid M₁] [inst_4 : AddCommMonoid M₂]\n  [inst_5 : AddCommMonoid M₃] {module_M₁ : Module 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-> (HEq.{max (succ u_14) (succ u_15)} (RingHom.{u_14, u_15} R S (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30) (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33)) σ (RingHom.{u_14, u_15} R' S' (Semiring.toNonAssocSemiring.{u_14} R' inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30) (Semiring.toNonAssocSemiring.{u_15} S' inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33)) σ') -> (Eq.{succ (succ u_16)} Type.{u_16} M M') -> (Eq.{succ (succ u_17)} Type.{u_17} M₂ M₂') -> (HEq.{succ u_16} (AddCommMonoid.{u_16} M) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 (AddCommMonoid.{u_16} M') inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43) -> (HEq.{succ u_17} (AddCommMonoid.{u_17} M₂) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 (AddCommMonoid.{u_17} M₂') inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46) -> (HEq.{max (succ u_14) (succ u_16)} (Module.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 (Module.{u_14, u_16} R' M' inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43) inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49) -> (HEq.{max (succ u_15) (succ u_17)} (Module.{u_15, u_17} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53 (Module.{u_15, u_17} S' M₂' inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46) inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53) -> (HEq.{max (succ u_16) (succ u_17)} (LinearMap.{u_14, u_15, u_16, u_17} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53) t (LinearMap.{u_14, u_15, u_16, u_17} R' S' inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 σ' M' M₂' inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53) t') -> (LinearMap.noConfusionType.{u, u_14, u_15, u_16, u_17} P R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53 t R' S' inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 σ' M' M₂' inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 inst'._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53 t')","typeFull":"{P : Sort u} →\n  {R : Type u_14} →\n    {S : Type u_15} →\n      {inst : Semiring R} →\n        {inst_1 : Semiring S} →\n          {σ : R →+* S} →\n            {M : Type u_16} →\n              {M₂ : Type u_17} →\n                {inst_2 : AddCommMonoid M} →\n                  {inst_3 : AddCommMonoid M₂} →\n                    {inst_4 : Module R M} →\n                      {inst_5 : Module S M₂} →\n                        {t : M →ₛₗ[σ] M₂} →\n                          {R' : Type u_14} →\n                            {S' : Type u_15} →\n                              {inst' : Semiring R'} →\n                                {inst'_1 : Semiring S'} →\n                                  {σ' : R' →+* S'} →\n                                    {M' : Type u_16} →\n                                      {M₂' : Type u_17} →\n                                        {inst'_2 : AddCommMonoid M'} →\n                                          {inst'_3 : AddCommMonoid M₂'} →\n                                            {inst'_4 : Module R' M'} →\n                                              {inst'_5 : Module S' M₂'} →\n                                                {t' : M' →ₛₗ[σ'] M₂'} →\n                                                  R = R' →\n                                                    S = S' →\n                                                      inst ≍ inst' →\n                                                        inst_1 ≍ inst'_1 →\n                                                          σ ≍ σ' →\n                                                            M = M' →\n                                                              M₂ = M₂' →\n                                                                inst_2 ≍ inst'_2 →\n                                                                  inst_3 ≍ inst'_3 →\n                                                                    inst_4 ≍ inst'_4 →\n                                                                      inst_5 ≍ inst'_5 →\n                                                                        t ≍ t' → LinearMap.noConfusionType P t t'","typeReadable":"{P : Sort u} →\n  {R : Type u_14} →\n    {S : Type u_15} →\n      {inst : Semiring R} →\n        {inst_1 : Semiring S} →\n          {σ : R →+* S} →\n            {M : Type u_16} →\n              {M₂ : Type u_17} →\n                {inst_2 : AddCommMonoid M} →\n                  {inst_3 : AddCommMonoid M₂} →\n                    {inst_4 : Module R M} →\n                      {inst_5 : Module S M₂} →\n                        {t : M →ₛₗ[σ] M₂} →\n                          {R' : Type u_14} →\n                            {S' : Type u_15} →\n                              {inst' : Semiring R'} →\n                                {inst'_1 : Semiring S'} →\n                                  {σ' : R' →+* S'} →\n                                    {M' : Type u_16} →\n                                      {M₂' : Type u_17} →\n                                        {inst'_2 : AddCommMonoid M'} →\n                                          {inst'_3 : AddCommMonoid M₂'} →\n                                            {inst'_4 : Module R' M'} →\n                                              {inst'_5 : Module S' M₂'} →\n                                                {t' : M' →ₛₗ[σ'] M₂'} →\n                                                  R = R' →\n                                                    S = S' →\n                                                      inst ≍ inst' →\n                                                        inst_1 ≍ inst'_1 →\n                                                          σ ≍ σ' →\n                                                            M = M' →\n                                                              M₂ = M₂' →\n                                                                inst_2 ≍ inst'_2 →\n                                                                  inst_3 ≍ inst'_3 →\n                                                                    inst_4 ≍ inst'_4 →\n                                                                      inst_5 ≍ inst'_5 →\n                                                                        t ≍ t' → LinearMap.noConfusionType P t 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[inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.30 : AddCommMonoid.{u_1} M₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.33 : Module.{u_3, u_2} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.41 : Module.{u_4, u_1} S M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.30] {σ : RingHom.{u_3, u_4} R S (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_4} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.18)} (f : LinearMap.{u_3, u_4, u_2, u_1} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.41), Eq.{succ u_1} M₃ (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_2, succ u_1} (LinearMap.{u_3, u_4, u_2, u_1} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.41) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₃) (LinearMap.instFunLike.{u_3, u_4, u_2, u_1} R S M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.41 σ) f (OfNat.ofNat.{u_2} M 0 (Zero.toOfNat0.{u_2} M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.21))))))) (OfNat.ofNat.{u_1} M₃ 0 (Zero.toOfNat0.{u_1} M₃ (AddZero.toZero.{u_1} M₃ (AddZeroClass.toAddZero.{u_1} M₃ (AddMonoid.toAddZeroClass.{u_1} M₃ (AddCommMonoid.toAddMonoid.{u_1} M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.738115108._hygCtx._hyg.30))))))","typeFull":"∀ {R : Type u_3} {S : Type u_4} {M : Type u_2} {M₃ : Type u_1} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₃] [inst_4 : Module R M] [inst_5 : Module S M₃] {σ : R →+* S}\n  (f : M →ₛₗ[σ] M₃), f 0 = 0","typeReadable":"∀ {R : Type u_3} {S : Type u_4} {M : Type u_2} {M₃ : Type u_1} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₃] [inst_4 : Module R M] [inst_5 : Module S M₃] {σ : R →+* S}\n  (f : M →ₛₗ[σ] M₃), f 0 = 0","typeReferences":[["RingHom"],["LinearMap","instFunLike"],["Module"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["OfNat","ofNat"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["Zero","toOfNat0"],["Eq"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["RingHom"],["LinearMap","instFunLike"],["AddMonoidHomClass","toZeroHomClass"],["Semiring","toMonoidWithZero"],["RingHom","instFunLike"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["LinearMap","semilinearMapClass"],["DistribMulActionSemiHomClass","toAddMonoidHomClass"],["Module","toDistribMulAction"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["map_zero"],["AddZero","toZero"],["SemilinearMapClass","distribMulActionSemiHomClass"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","mk","injEq"],"typeFallback":"forall {R : Type.{u_14}} {S : Type.{u_15}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 : Semiring.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 : Semiring.{u_15} S] {σ : RingHom.{u_14, u_15} R S (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30) (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33)} {M : Type.{u_16}} {M₂ : Type.{u_17}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 : AddCommMonoid.{u_16} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 : AddCommMonoid.{u_17} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 : Module.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53 : Module.{u_15, u_17} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46] (toAddHom : AddHom.{u_16, u_17} M M₂ (AddCommMagma.toAdd.{u_16} M (AddCommSemigroup.toAddCommMagma.{u_16} M (AddCommMonoid.toAddCommSemigroup.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43))) (AddCommMagma.toAdd.{u_17} M₂ (AddCommSemigroup.toAddCommMagma.{u_17} M₂ (AddCommMonoid.toAddCommSemigroup.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46)))) (map_smul' : forall (m : R) (x : M), Eq.{succ u_17} M₂ (AddHom.toFun.{u_16, u_17} M M₂ (AddCommMagma.toAdd.{u_16} M (AddCommSemigroup.toAddCommMagma.{u_16} M (AddCommMonoid.toAddCommSemigroup.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43))) (AddCommMagma.toAdd.{u_17} M₂ 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u_3} {S : Type u_4} {M : Type u_2} {M₃ : Type u_1} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₃] [inst_4 : Module R M] [inst_5 : Module S M₃] {σ : R →+* S}\n  (f : M →ₛₗ[σ] M₃) (x y : M), f.toFun (x + y) = f.toFun x + f.toFun y","typeReadable":"∀ {R : Type u_3} {S : Type u_4} {M : Type u_2} {M₃ : Type u_1} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₃] [inst_4 : Module R M] [inst_5 : Module S M₃] {σ : R →+* S}\n  (f : M →ₛₗ[σ] M₃) (x y : M), f.toFun (x + y) = f.toFun x + f.toFun y","typeReferences":[["RingHom"],["instHAdd"],["Module"],["LinearMap","toAddHom"],["LinearMap"],["HAdd","hAdd"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["AddCommMonoid","toAddCommSemigroup"],["AddHom","toFun"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Eq"],["Semiring"]],"valueReferences":[["AddHom","map_add'"],["AddCommMonoid","toAddCommSemigroup"],["LinearMap","toAddHom"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"]]},{"isProp":true,"kind":"theorem","name":["SemilinearMapClass","semilinearMap","_proof_1"],"typeFallback":"forall {R : Type.{u_4}} {S : Type.{u_5}} {M : Type.{u_3}} {M₃ : Type.{u_1}} {F : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.16 : Semiring.{u_4} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.19 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.22 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.31 : AddCommMonoid.{u_1} M₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.34 : Module.{u_4, u_3} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.22] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.42 : Module.{u_5, u_1} S M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.31] {σ : RingHom.{u_4, u_5} R S (Semiring.toNonAssocSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.16) (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.19)} (f : F) [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.52 : FunLike.{succ u_2, succ u_3, succ u_1} F M M₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.57 : SemilinearMapClass.{u_2, u_4, u_5, u_3, u_1} F R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.19 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.42 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.52] (x : M) (y : M), Eq.{succ u_1} M₃ (DFunLike.coe.{succ u_2, succ u_3, succ u_1} F M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₃) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1133937772._hygCtx._hyg.52 f (HAdd.hAdd.{u_3, u_3, u_3} M M M (instHAdd.{u_3} M (AddCommMagma.toAdd.{u_3} M (AddCommSemigroup.toAddCommMagma.{u_3} M 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: Module S M₃] {σ : R →+* S}\n  (f : F) [inst_6 : FunLike F M M₃] [SemilinearMapClass F σ M M₃] (x y : M), f (x + y) = f x + f y","typeReadable":"∀ {R : Type u_4} {S : Type u_5} {M : Type u_3} {M₃ : Type u_1} {F : Type u_2} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₃] [inst_4 : Module R M] [inst_5 : Module S M₃] {σ : R →+* S}\n  (f : F) [inst_6 : FunLike F M M₃] [SemilinearMapClass F σ M M₃] (x y : M), f (x + y) = f x + f y","typeReferences":[["RingHom"],["FunLike"],["SemilinearMapClass"],["Module"],["instHAdd"],["DFunLike","coe"],["HAdd","hAdd"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["AddCommMonoid","toAddCommSemigroup"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Eq"],["Semiring"]],"valueReferences":[["map_add"],["AddCommMonoid","toAddCommSemigroup"],["SemilinearMapClass","toAddHomClass"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"]]},{"isProp":false,"kind":"definition","name":["SemilinearMapClass","casesOn"],"typeFallback":"forall {F : Type.{u_14}} {R : Type.{u_15}} {S : Type.{u_16}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.33 : Semiring.{u_15} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.36 : Semiring.{u_16} S] {σ : RingHom.{u_15, u_16} R S (Semiring.toNonAssocSemiring.{u_15} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.33) (Semiring.toNonAssocSemiring.{u_16} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.36)} {M : Type.{u_17}} {M₂ : Type.{u_18}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.51 : AddCommMonoid.{u_17} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54 : AddCommMonoid.{u_18} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.57 : Module.{u_15, u_17} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.51] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.61 : Module.{u_16, u_18} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.65 : FunLike.{succ u_14, succ u_17, succ u_18} F M M₂] {motive : (SemilinearMapClass.{u_14, u_15, u_16, u_17, u_18} F R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.36 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.51 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.57 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.61 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.65) -> Sort.{u}} (t : SemilinearMapClass.{u_14, u_15, u_16, u_17, u_18} F R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.36 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.51 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.57 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.61 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.65), (forall [toAddHomClass : AddHomClass.{u_14, u_17, u_18} F M M₂ (AddCommMagma.toAdd.{u_17} M (AddCommSemigroup.toAddCommMagma.{u_17} M (AddCommMonoid.toAddCommSemigroup.{u_17} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.51))) (AddCommMagma.toAdd.{u_18} M₂ (AddCommSemigroup.toAddCommMagma.{u_18} M₂ (AddCommMonoid.toAddCommSemigroup.{u_18} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.65] [toMulActionSemiHomClass : MulActionSemiHomClass.{u_14, u_15, u_16, u_17, u_18} F R S (DFunLike.coe.{max (succ u_15) (succ u_16), succ u_15, succ u_16} (outParam.{max (succ u_16) (succ u_15)} (RingHom.{u_15, u_16} R S (Semiring.toNonAssocSemiring.{u_15} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.33) (Semiring.toNonAssocSemiring.{u_16} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.36))) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => S) (RingHom.instFunLike.{u_15, u_16} R S (Semiring.toNonAssocSemiring.{u_15} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.33) (Semiring.toNonAssocSemiring.{u_16} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.36)) σ) M M₂ (SMulZeroClass.toSMul.{u_15, u_17} R M (AddZero.toZero.{u_17} M (AddZeroClass.toAddZero.{u_17} M (AddMonoid.toAddZeroClass.{u_17} M (AddCommMonoid.toAddMonoid.{u_17} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.51)))) (DistribSMul.toSMulZeroClass.{u_15, u_17} R M (AddMonoid.toAddZeroClass.{u_17} M (AddCommMonoid.toAddMonoid.{u_17} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.51)) (DistribMulAction.toDistribSMul.{u_15, u_17} R M (MonoidWithZero.toMonoid.{u_15} R 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[toMulActionSemiHomClass : MulActionSemiHomClass F (⇑σ) M M₂] → motive ⋯) →\n                                motive t","typeReadable":"{F : Type u_14} →\n  {R : Type u_15} →\n    {S : Type u_16} →\n      [inst : Semiring R] →\n        [inst_1 : Semiring S] →\n          {σ : R →+* S} →\n            {M : Type u_17} →\n              {M₂ : Type u_18} →\n                [inst_2 : AddCommMonoid M] →\n                  [inst_3 : AddCommMonoid M₂] →\n                    [inst_4 : Module R M] →\n                      [inst_5 : Module S M₂] →\n                        [inst_6 : FunLike F M M₂] →\n                          {motive : SemilinearMapClass F σ M M₂ → Sort u} →\n                            (t : SemilinearMapClass F σ M M₂) →\n                              ([toAddHomClass : AddHomClass F M M₂] →\n                                  [toMulActionSemiHomClass : MulActionSemiHomClass F (⇑σ) M M₂] → motive ⋯) →\n                                motive 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[inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.36 : Semiring.{u_16} S] {σ : RingHom.{u_15, u_16} R S (Semiring.toNonAssocSemiring.{u_15} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.33) (Semiring.toNonAssocSemiring.{u_16} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.36)} {M : Type.{u_17}} {M₂ : Type.{u_18}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.51 : AddCommMonoid.{u_17} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54 : AddCommMonoid.{u_18} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.57 : Module.{u_15, u_17} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.51] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.61 : Module.{u_16, u_18} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.65 : FunLike.{succ u_14, succ u_17, succ u_18} F M M₂] {motive : (SemilinearMapClass.{u_14, u_15, u_16, u_17, u_18} F R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.36 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.51 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.57 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.61 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.65) -> Sort.{u}}, (forall [toAddHomClass : AddHomClass.{u_14, u_17, u_18} F M M₂ (AddCommMagma.toAdd.{u_17} M (AddCommSemigroup.toAddCommMagma.{u_17} M (AddCommMonoid.toAddCommSemigroup.{u_17} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.51))) (AddCommMagma.toAdd.{u_18} M₂ (AddCommSemigroup.toAddCommMagma.{u_18} M₂ (AddCommMonoid.toAddCommSemigroup.{u_18} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.65] [toMulActionSemiHomClass : MulActionSemiHomClass.{u_14, u_15, u_16, u_17, u_18} F R S (DFunLike.coe.{max (succ u_15) (succ u_16), succ u_15, succ u_16} (outParam.{max (succ u_16) (succ u_15)} (RingHom.{u_15, u_16} R S (Semiring.toNonAssocSemiring.{u_15} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.33) (Semiring.toNonAssocSemiring.{u_16} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.36))) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => S) (RingHom.instFunLike.{u_15, u_16} R S (Semiring.toNonAssocSemiring.{u_15} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.33) (Semiring.toNonAssocSemiring.{u_16} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.36)) σ) M M₂ (SMulZeroClass.toSMul.{u_15, u_17} R M (AddZero.toZero.{u_17} M (AddZeroClass.toAddZero.{u_17} M (AddMonoid.toAddZeroClass.{u_17} M (AddCommMonoid.toAddMonoid.{u_17} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.51)))) (DistribSMul.toSMulZeroClass.{u_15, u_17} R M (AddMonoid.toAddZeroClass.{u_17} M (AddCommMonoid.toAddMonoid.{u_17} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.51)) (DistribMulAction.toDistribSMul.{u_15, u_17} R M (MonoidWithZero.toMonoid.{u_15} R (Semiring.toMonoidWithZero.{u_15} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.33)) (AddCommMonoid.toAddMonoid.{u_17} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.51) (Module.toDistribMulAction.{u_15, u_17} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.51 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.57)))) (SMulZeroClass.toSMul.{u_16, u_18} S M₂ (AddZero.toZero.{u_18} M₂ (AddZeroClass.toAddZero.{u_18} M₂ (AddMonoid.toAddZeroClass.{u_18} M₂ (AddCommMonoid.toAddMonoid.{u_18} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54)))) (DistribSMul.toSMulZeroClass.{u_16, u_18} S M₂ (AddMonoid.toAddZeroClass.{u_18} M₂ (AddCommMonoid.toAddMonoid.{u_18} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54)) (DistribMulAction.toDistribSMul.{u_16, u_18} S M₂ (MonoidWithZero.toMonoid.{u_16} S (Semiring.toMonoidWithZero.{u_16} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.36)) (AddCommMonoid.toAddMonoid.{u_18} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54) (Module.toDistribMulAction.{u_16, u_18} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.61)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.65], motive (SemilinearMapClass.mk.{u_14, u_15, u_16, u_17, u_18} F R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.36 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.51 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.57 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.61 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.65 toAddHomClass toMulActionSemiHomClass)) -> (forall (t : SemilinearMapClass.{u_14, u_15, u_16, u_17, u_18} F R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.36 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.51 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.57 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.61 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.65), motive t)","typeFull":"{F : Type u_14} →\n  {R : Type u_15} →\n    {S : Type u_16} →\n      [inst : Semiring R] →\n        [inst_1 : Semiring S] →\n          {σ : R →+* S} →\n            {M : Type u_17} →\n              {M₂ : Type u_18} →\n                [inst_2 : AddCommMonoid M] →\n                  [inst_3 : AddCommMonoid M₂] →\n                    [inst_4 : Module R M] →\n                      [inst_5 : Module S M₂] →\n                        [inst_6 : FunLike F M M₂] →\n                          {motive : SemilinearMapClass F σ M M₂ → Sort u} →\n                            ([toAddHomClass : AddHomClass F M M₂] →\n                                [toMulActionSemiHomClass : MulActionSemiHomClass F (⇑σ) M M₂] → motive ⋯) →\n                              (t : SemilinearMapClass F σ M M₂) → motive t","typeReadable":"{F : Type u_14} →\n  {R : Type u_15} →\n    {S : Type u_16} →\n      [inst : Semiring R] →\n        [inst_1 : Semiring S] →\n          {σ : R →+* S} →\n            {M : Type u_17} →\n              {M₂ : Type u_18} →\n                [inst_2 : AddCommMonoid M] →\n                  [inst_3 : AddCommMonoid M₂] →\n                    [inst_4 : Module R M] →\n                      [inst_5 : Module S M₂] →\n                        [inst_6 : FunLike F M M₂] →\n                          {motive : SemilinearMapClass F σ M M₂ → Sort u} →\n                            ([toAddHomClass : AddHomClass F M M₂] →\n                                [toMulActionSemiHomClass : MulActionSemiHomClass F (⇑σ) M M₂] → motive ⋯) →\n                              (t : SemilinearMapClass F σ M M₂) → motive t","typeReferences":[["RingHom"],["SemilinearMapClass"],["Module"],["RingHom","instFunLike"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["SemilinearMapClass","mk"],["DFunLike","coe"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["AddHomClass"],["DistribSMul","toSMulZeroClass"],["FunLike"],["outParam"],["MulActionSemiHomClass"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["LinearMap","copy_eq"],"typeFallback":"forall {R : Type.{u_1}} {S : Type.{u_5}} {M : Type.{u_8}} {M₃ : Type.{u_11}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.18 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.30 : AddCommMonoid.{u_11} M₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.33 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.41 : Module.{u_5, u_11} S M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.30] {σ : RingHom.{u_1, u_5} R S (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.18)} (f : LinearMap.{u_1, u_5, u_8, u_11} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.41) (f' : M -> M₃) (h : Eq.{max (succ u_11) (succ u_8)} (M -> M₃) f' (DFunLike.coe.{max (succ u_11) (succ u_8), succ u_8, succ u_11} (LinearMap.{u_1, u_5, u_8, u_11} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.41) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₃) (LinearMap.instFunLike.{u_1, u_5, u_8, u_11} R S M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.41 σ) f)), Eq.{max (succ u_11) (succ u_8)} (LinearMap.{u_1, u_5, u_8, u_11} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.41) (LinearMap.copy.{u_1, u_5, u_8, u_11} R S M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.4016340264._hygCtx._hyg.41 σ f f' h) f","typeFull":"∀ {R : Type u_1} {S : Type u_5} {M : Type u_8} {M₃ : Type u_11} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₃] [inst_4 : Module R M] [inst_5 : Module S M₃] {σ : R →+* S}\n  (f : M →ₛₗ[σ] M₃) (f' : M → M₃) (h : f' = ⇑f), f.copy f' h = f","typeReadable":"∀ {R : Type u_1} {S : Type u_5} {M : Type u_8} {M₃ : Type u_11} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₃] [inst_4 : Module R M] [inst_5 : Module S M₃] {σ : R →+* S}\n  (f : M →ₛₗ[σ] M₃) (f' : M → M₃) (h : f' = ⇑f), f.copy f' h = f","typeReferences":[["RingHom"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["LinearMap","instFunLike"],["Module"],["LinearMap","copy"],["Eq"],["LinearMap"],["DFunLike","coe"],["Semiring"]],"valueReferences":[["DFunLike","ext'"],["LinearMap","instFunLike"],["LinearMap","copy"],["LinearMap"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","restrictScalars_comp"],"typeFallback":"forall {R : Type.{u_14}} {S : Type.{u_15}} {M : Type.{u_16}} {N : Type.{u_17}} {P : Type.{u_18}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.20 : Semiring.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.23 : Semiring.{u_15} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.26 : AddCommMonoid.{u_16} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.29 : AddCommMonoid.{u_17} N] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.32 : Module.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.26] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.36 : Module.{u_14, u_17} R N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.40 : Module.{u_15, u_16} S M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.26] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.44 : Module.{u_15, u_17} S N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.48 : LinearMap.CompatibleSMul.{u_16, u_17, u_14, u_15} M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.29 R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.23 (SMulZeroClass.toSMul.{u_14, u_16} R M (AddZero.toZero.{u_16} M (AddZeroClass.toAddZero.{u_16} M (AddMonoid.toAddZeroClass.{u_16} M (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.26)))) (DistribSMul.toSMulZeroClass.{u_14, u_16} R M (AddMonoid.toAddZeroClass.{u_16} M (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.26)) (DistribMulAction.toDistribSMul.{u_14, u_16} R M (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.20)) (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.26) (Module.toDistribMulAction.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.32)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.40 (SMulZeroClass.toSMul.{u_14, u_17} R N (AddZero.toZero.{u_17} N (AddZeroClass.toAddZero.{u_17} N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.29)))) (DistribSMul.toSMulZeroClass.{u_14, u_17} R N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.29)) (DistribMulAction.toDistribSMul.{u_14, u_17} R N (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.20)) (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.29) (Module.toDistribMulAction.{u_14, u_17} R N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.36)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.44] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.72 : AddCommMonoid.{u_18} P] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.75 : Module.{u_15, u_18} S P inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.72] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.79 : Module.{u_14, u_18} R P inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.72] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.83 : LinearMap.CompatibleSMul.{u_17, u_18, u_14, u_15} N P inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.72 R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.23 (SMulZeroClass.toSMul.{u_14, u_17} R N (AddZero.toZero.{u_17} N (AddZeroClass.toAddZero.{u_17} N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.29)))) (DistribSMul.toSMulZeroClass.{u_14, u_17} R N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.29)) (DistribMulAction.toDistribSMul.{u_14, u_17} R N (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.20)) (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.29) (Module.toDistribMulAction.{u_14, u_17} R N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.36)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.44 (SMulZeroClass.toSMul.{u_14, u_18} R P (AddZero.toZero.{u_18} P (AddZeroClass.toAddZero.{u_18} P (AddMonoid.toAddZeroClass.{u_18} P (AddCommMonoid.toAddMonoid.{u_18} P inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.72)))) (DistribSMul.toSMulZeroClass.{u_14, u_18} R P (AddMonoid.toAddZeroClass.{u_18} P (AddCommMonoid.toAddMonoid.{u_18} P inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.72)) (DistribMulAction.toDistribSMul.{u_14, u_18} R P (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.20)) (AddCommMonoid.toAddMonoid.{u_18} P inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.72) (Module.toDistribMulAction.{u_14, u_18} R P inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.72 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.79)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.75] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.89 : LinearMap.CompatibleSMul.{u_16, u_18, u_14, u_15} M P inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.72 R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.23 (SMulZeroClass.toSMul.{u_14, u_16} R M (AddZero.toZero.{u_16} M (AddZeroClass.toAddZero.{u_16} M (AddMonoid.toAddZeroClass.{u_16} M (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.26)))) (DistribSMul.toSMulZeroClass.{u_14, u_16} R M (AddMonoid.toAddZeroClass.{u_16} M (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.26)) (DistribMulAction.toDistribSMul.{u_14, u_16} R M (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.20)) (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.26) (Module.toDistribMulAction.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.32)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.40 (SMulZeroClass.toSMul.{u_14, u_18} R P (AddZero.toZero.{u_18} P (AddZeroClass.toAddZero.{u_18} P (AddMonoid.toAddZeroClass.{u_18} P (AddCommMonoid.toAddMonoid.{u_18} P inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.72)))) (DistribSMul.toSMulZeroClass.{u_14, u_18} R P (AddMonoid.toAddZeroClass.{u_18} P (AddCommMonoid.toAddMonoid.{u_18} P inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.72)) (DistribMulAction.toDistribSMul.{u_14, u_18} R P (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.20)) (AddCommMonoid.toAddMonoid.{u_18} P inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.72) (Module.toDistribMulAction.{u_14, u_18} R P inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.72 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.79)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.75] (f : LinearMap.{u_15, u_15, u_17, u_18} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.23 (RingHom.id.{u_15} S (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.23)) N P inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.72 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.44 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.75) (g : LinearMap.{u_15, u_15, u_16, u_17} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.23 (RingHom.id.{u_15} S (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.23)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.44), Eq.{max (succ 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.79 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.75 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.89 (LinearMap.comp.{u_15, u_15, u_15, u_16, u_17, u_18} S S S M N P inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.72 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.351624844._hygCtx._hyg.44 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M]\n  [inst_5 : Module R N] [inst_6 : Module S M] [inst_7 : Module S N] [inst_8 : LinearMap.CompatibleSMul M N R S]\n  [inst_9 : AddCommMonoid P] [inst_10 : Module S P] [inst_11 : Module R P] [inst_12 : LinearMap.CompatibleSMul N P R S]\n  [inst_13 : LinearMap.CompatibleSMul M P R S] (f : N →ₗ[S] P) (g : M →ₗ[S] N), ↑R (f ∘ₗ g) = ↑R f ∘ₗ ↑R g","typeReadable":"∀ {R : Type u_14} {S : Type u_15} {M : Type u_16} {N : Type u_17} {P : Type u_18} [inst : Semiring R]\n  [inst_1 : Semiring S] [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid N] [inst_4 : Module R M]\n  [inst_5 : Module R N] [inst_6 : Module S M] [inst_7 : Module S N] [inst_8 : LinearMap.CompatibleSMul M N R S]\n  [inst_9 : AddCommMonoid P] [inst_10 : Module S P] [inst_11 : Module R P] [inst_12 : LinearMap.CompatibleSMul N P R S]\n  [inst_13 : LinearMap.CompatibleSMul M P R S] (f : N →ₗ[S] P) (g : M →ₗ[S] N), ↑R (f ∘ₗ g) = ↑R f ∘ₗ ↑R 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.65","typeFull":"∀ {F : Type u_14} {R : outParam (Type u_15)} {S : outParam (Type u_16)} {inst : Semiring R} {inst_1 : Semiring S}\n  {σ : outParam (R →+* S)} {M : outParam (Type u_17)} {M₂ : outParam (Type u_18)} {inst_2 : AddCommMonoid M}\n  {inst_3 : AddCommMonoid M₂} {inst_4 : Module R M} {inst_5 : Module S M₂} {inst_6 : FunLike F M M₂}\n  [self : SemilinearMapClass F σ M M₂], MulActionSemiHomClass F (⇑σ) M M₂","typeReadable":"∀ {F : Type u_14} {R : outParam (Type u_15)} {S : outParam (Type u_16)} {inst : Semiring R} {inst_1 : Semiring S}\n  {σ : outParam (R →+* S)} {M : outParam (Type u_17)} {M₂ : outParam (Type u_18)} {inst_2 : AddCommMonoid M}\n  {inst_3 : AddCommMonoid M₂} {inst_4 : Module R M} {inst_5 : Module S M₂} {inst_6 : FunLike F M M₂}\n  [self : SemilinearMapClass F σ M M₂], MulActionSemiHomClass F (⇑σ) M 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[inst._@.Mathlib.Algebra.Module.LinearMap.Defs.641934606._hygCtx._hyg.27 : Semiring.{u_4} R₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.641934606._hygCtx._hyg.33 : AddCommMonoid.{u_9} M₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.641934606._hygCtx._hyg.36 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.641934606._hygCtx._hyg.39 : AddCommMonoid.{u_11} M₃] {module_M₁ : Module.{u_2, u_9} R₁ M₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.641934606._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.641934606._hygCtx._hyg.33} {module_M₂ : Module.{u_3, u_10} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.641934606._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.641934606._hygCtx._hyg.36} {module_M₃ : Module.{u_4, u_11} R₃ M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.641934606._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.641934606._hygCtx._hyg.39} {σ₁₂ : RingHom.{u_2, u_3} R₁ R₂ 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→\n          [inst_1 : Semiring S] →\n            [inst_2 : AddCommMonoid M] →\n              [inst_3 : AddCommMonoid M₃] →\n                [inst_4 : Module R M] → [inst_5 : Module S M₃] → {σ : R →+* S} → FunLike (M →ₛₗ[σ] M₃) M M₃","typeReferences":[["FunLike"],["RingHom"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["Module"],["LinearMap"],["Semiring"]],"valueReferences":[["DFunLike","mk"],["AddCommMonoid","toAddCommSemigroup"],["LinearMap","toAddHom"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["AddHom","toFun"],["LinearMap","instFunLike","_proof_1"],["LinearMap"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","id'_apply"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_8}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.33 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.21] {σ : RingHom.{u_1, u_1} R R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.15)} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.55 : RingHomId.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.15 σ] (x : M), Eq.{succ u_8} M (DFunLike.coe.{succ u_8, succ u_8, succ u_8} (LinearMap.{u_1, u_1, u_8, u_8} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.15 σ M M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.33) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M) (LinearMap.instFunLike.{u_1, u_1, u_8, u_8} R R M M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.33 σ) (LinearMap.id'.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.33 σ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1094269995._hygCtx._hyg.55) x) x","typeFull":"∀ {R : Type u_1} {M : Type u_8} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {σ : R →+* R}\n  [inst_3 : RingHomId σ] (x : M), LinearMap.id' x = x","typeReadable":"∀ {R : Type u_1} {M : Type u_8} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {σ : R →+* R}\n  [inst_3 : RingHomId σ] (x : M), LinearMap.id' x = x","typeReferences":[["RingHom"],["LinearMap","id'"],["RingHomId"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["LinearMap","instFunLike"],["Module"],["LinearMap"],["Eq"],["DFunLike","coe"],["Semiring"]],"valueReferences":[["LinearMap","id'"],["LinearMap","instFunLike"],["Eq","refl"],["LinearMap"],["DFunLike","coe"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","smul_apply"],"typeFallback":"forall {R : Type.{u_1}} {R₂ : Type.{u_3}} {S : Type.{u_5}} {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.195159311._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.195159311._hygCtx._hyg.18 : Semiring.{u_3} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.195159311._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.195159311._hygCtx._hyg.24 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.195159311._hygCtx._hyg.27 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.195159311._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.195159311._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.195159311._hygCtx._hyg.31 : Module.{u_3, u_10} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.195159311._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.195159311._hygCtx._hyg.24] {σ₁₂ : RingHom.{u_1, u_3} R R₂ (Semiring.toNonAssocSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.51) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => N₂) (LinearMap.instFunLike.{u_1, u_2, u_3, u_4} R₁ R₂ M N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_4} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.33) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.51 σ₁₂) (HAdd.hAdd.{max u_4 u_3, max u_4 u_3, max u_4 u_3} (LinearMap.{u_1, u_2, u_3, u_4} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.18 σ₁₂ M N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_4} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.33) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.51) (LinearMap.{u_1, u_2, u_3, u_4} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.18 σ₁₂ M N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_4} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.33) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.51) (LinearMap.{u_1, u_2, u_3, u_4} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.18 σ₁₂ M N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_4} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.33) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.51) (instHAdd.{max u_4 u_3} (LinearMap.{u_1, u_2, u_3, u_4} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.18 σ₁₂ M N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_4} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.33) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.51) (LinearMap.instAdd.{u_1, u_2, u_3, u_4} R₁ R₂ 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[inst : Semiring R₁] [inst_1 : Semiring R₂]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommGroup N₂] [inst_4 : Module R₁ M] [inst_5 : Module R₂ N₂] {σ₁₂ : R₁ →+* R₂}\n  (x x_1 : M →ₛₗ[σ₁₂] N₂), ⇑(x + x_1) = ⇑(x + x_1)","typeReferences":[["RingHom"],["LinearMap","instFunLike"],["Module"],["instHAdd"],["AddCommGroup"],["LinearMap"],["DFunLike","coe"],["HAdd","hAdd"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["LinearMap","instAdd"],["AddCommGroup","toAddCommMonoid"],["Eq"],["Semiring"]],"valueReferences":[["HAdd","hAdd"],["rfl"],["LinearMap","instFunLike"],["instHAdd"],["LinearMap","instAdd"],["AddCommGroup","toAddCommMonoid"],["LinearMap"],["DFunLike","coe"]]},{"isProp":true,"kind":"definition","name":["SemilinearMapClass","mk","_flat_ctor"],"typeFallback":"forall {F : Type.{u_14}} {R : outParam.{succ (succ u_15)} Type.{u_15}} {S : outParam.{succ (succ u_16)} Type.{u_16}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.33 : Semiring.{u_15} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.36 : Semiring.{u_16} S] {σ : outParam.{max (succ u_16) (succ u_15)} (RingHom.{u_15, u_16} R S (Semiring.toNonAssocSemiring.{u_15} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.33) (Semiring.toNonAssocSemiring.{u_16} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.36))} {M : outParam.{succ (succ u_17)} Type.{u_17}} {M₂ : outParam.{succ (succ u_18)} Type.{u_18}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.51 : AddCommMonoid.{u_17} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54 : AddCommMonoid.{u_18} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.57 : Module.{u_15, u_17} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.51] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.61 : 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(AddZeroClass.toAddZero.{u_18} M₂ (AddMonoid.toAddZeroClass.{u_18} M₂ (AddCommMonoid.toAddMonoid.{u_18} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54)))) (DistribSMul.toSMulZeroClass.{u_16, u_18} S M₂ (AddMonoid.toAddZeroClass.{u_18} M₂ (AddCommMonoid.toAddMonoid.{u_18} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54)) (DistribMulAction.toDistribSMul.{u_16, u_18} S M₂ (MonoidWithZero.toMonoid.{u_16} S (Semiring.toMonoidWithZero.{u_16} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.36)) (AddCommMonoid.toAddMonoid.{u_18} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54) (Module.toDistribMulAction.{u_16, u_18} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.61))))) (DFunLike.coe.{max (succ u_15) (succ u_16), succ 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.36 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.51 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.54 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.57 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.61 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.988368597._hygCtx._hyg.65)","typeFull":"∀ {F : Type u_14} {R : outParam (Type u_15)} {S : outParam (Type u_16)} [inst : Semiring R] [inst_1 : Semiring S]\n  {σ : outParam (R →+* S)} {M : outParam (Type u_17)} {M₂ : outParam (Type u_18)} [inst_2 : AddCommMonoid M]\n  [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M] [inst_5 : Module S M₂] [inst_6 : FunLike F M M₂],\n  (∀ (f : F) (x y : M), f (x + y) = f x + f y) →\n    (∀ (f : F) (c : R) (x : M), f (c • x) = σ c • f x) → SemilinearMapClass F σ M M₂","typeReadable":"∀ {F : Type u_14} {R : outParam (Type u_15)} {S : outParam (Type u_16)} [inst : Semiring R] [inst_1 : Semiring S]\n  {σ : outParam (R →+* S)} {M : outParam (Type u_17)} {M₂ : outParam (Type u_18)} [inst_2 : AddCommMonoid M]\n  [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M] [inst_5 : Module S M₂] [inst_6 : FunLike F M M₂],\n  (∀ (f : F) (x y : M), f (x + y) = f x + f y) →\n    (∀ (f : F) (c : R) (x : M), f (c • x) = σ c • f x) → SemilinearMapClass F σ M 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{R : Type.{u_1}} {S : Type.{u_5}} {M : Type.{u_8}} {M₃ : Type.{u_11}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325879._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325879._hygCtx._hyg.18 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325879._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325879._hygCtx._hyg.30 : AddCommMonoid.{u_11} M₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325879._hygCtx._hyg.33 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325879._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325879._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325879._hygCtx._hyg.41 : Module.{u_5, u_11} S M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325879._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325879._hygCtx._hyg.30] {σ : RingHom.{u_1, u_5} R S (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325879._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325879._hygCtx._hyg.18)} {F : Type.{u_14}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325879._hygCtx._hyg.51 : FunLike.{succ u_14, succ u_8, succ u_11} F M M₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325879._hygCtx._hyg.56 : SemilinearMapClass.{u_14, u_1, u_5, u_8, u_11} F R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325879._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325879._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325879._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325879._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325879._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2358325879._hygCtx._hyg.41 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.41) (f' : M -> M₃), (Eq.{max (succ u_1) (succ u_2)} (M -> M₃) f' (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_2, succ u_1} (LinearMap.{u_3, u_4, u_2, u_1} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.41) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₃) (LinearMap.instFunLike.{u_3, u_4, u_2, u_1} R S M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2619604494._hygCtx._hyg.15 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Prop","typeReadable":"(F : Type u_14) →\n  (R : outParam (Type u_15)) →\n    (M : Type u_16) →\n      (M₂ : Type u_17) →\n        [inst : Semiring R] →\n          [inst_1 : AddCommMonoid M] →\n            [inst_2 : AddCommMonoid M₂] → [Module R M] → [Module R M₂] → [FunLike F M M₂] → Prop","typeReferences":[["FunLike"],["AddCommMonoid"],["Module"],["outParam"],["Semiring"]],"valueReferences":[["Semiring","toNonAssocSemiring"],["SemilinearMapClass"],["RingHom","id"]]},{"isProp":false,"kind":"definition","name":["LinearMap","instNeg"],"typeFallback":"forall {R₁ : Type.{u_2}} {R₂ : Type.{u_3}} {M : Type.{u_8}} {N₂ : Type.{u_12}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.15 : Semiring.{u_2} R₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.18 : Semiring.{u_3} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.24 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.33 : AddCommGroup.{u_12} N₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.39 : Module.{u_2, u_8} R₁ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.51 : Module.{u_3, u_12} R₂ N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.18 (AddCommGroup.toAddCommMonoid.{u_12} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.33)] {σ₁₂ : RingHom.{u_2, u_3} R₁ R₂ (Semiring.toNonAssocSemiring.{u_2} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.18)}, Neg.{max u_12 u_8} (LinearMap.{u_2, u_3, u_8, u_12} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.18 σ₁₂ M N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_12} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.33) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1827621731._hygCtx._hyg.51)","typeFull":"{R₁ : Type u_2} →\n  {R₂ : Type u_3} →\n    {M : Type u_8} →\n      {N₂ : Type u_12} →\n        [inst : Semiring R₁] →\n          [inst_1 : Semiring R₂] →\n            [inst_2 : AddCommMonoid M] →\n              [inst_3 : AddCommGroup N₂] →\n                [inst_4 : Module R₁ M] → [inst_5 : Module R₂ N₂] → {σ₁₂ : R₁ →+* R₂} → Neg (M →ₛₗ[σ₁₂] N₂)","typeReadable":"{R₁ : Type u_2} →\n  {R₂ : Type u_3} →\n    {M : Type u_8} →\n      {N₂ : Type u_12} →\n        [inst : Semiring R₁] →\n          [inst_1 : Semiring R₂] →\n            [inst_2 : AddCommMonoid M] →\n              [inst_3 : AddCommGroup N₂] →\n                [inst_4 : Module R₁ M] → [inst_5 : Module R₂ N₂] → {σ₁₂ : R₁ →+* R₂} → Neg (M →ₛₗ[σ₁₂] N₂)","typeReferences":[["RingHom"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["Module"],["AddCommGroup","toAddCommMonoid"],["Neg"],["AddCommGroup"],["LinearMap"],["Semiring"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Pi","instNeg"],["LinearMap","instFunLike"],["LinearMap","instNeg","_proof_1"],["Neg","neg"],["LinearMap","mk"],["AddHom","mk"],["LinearMap","instNeg","_proof_2"],["SubtractionCommMonoid","toSubtractionMonoid"],["LinearMap"],["DFunLike","coe"],["SubNegZeroMonoid","toNegZeroClass"],["NegZeroClass","toNeg"],["AddCommGroup","toDivisionAddCommMonoid"],["AddCommMonoid","toAddCommSemigroup"],["AddCommGroup","toAddCommMonoid"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Neg","mk"]]},{"isProp":false,"kind":"definition","name":["AddMonoidHom","toNatLinearMap"],"typeFallback":"forall {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3419663415._hygCtx._hyg.15 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3419663415._hygCtx._hyg.18 : AddCommMonoid.{u_10} M₂], (AddMonoidHom.{u_8, u_10} M M₂ (AddZeroClass.toAddZero.{u_8} M (AddMonoid.toAddZeroClass.{u_8} M (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3419663415._hygCtx._hyg.15))) (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3419663415._hygCtx._hyg.18)))) -> (LinearMap.{0, 0, u_8, u_10} Nat Nat Nat.instSemiring Nat.instSemiring (RingHom.id.{0} Nat Nat.instNonAssocSemiring) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3419663415._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3419663415._hygCtx._hyg.18 (AddCommMonoid.toNatModule.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3419663415._hygCtx._hyg.15) (AddCommMonoid.toNatModule.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3419663415._hygCtx._hyg.18))","typeFull":"{M : Type u_8} → {M₂ : Type u_10} → [inst : AddCommMonoid M] → [inst_1 : AddCommMonoid M₂] → (M →+ M₂) → M →ₗ[ℕ] M₂","typeReadable":"{M : Type u_8} → {M₂ : Type u_10} → [inst : AddCommMonoid M] → [inst_1 : AddCommMonoid M₂] → (M →+ M₂) → M →ₗ[ℕ] M₂","typeReferences":[["Nat"],["AddCommMonoid"],["Nat","instNonAssocSemiring"],["Nat","instSemiring"],["RingHom","id"],["AddMonoidHom"],["AddCommMonoid","toNatModule"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["LinearMap","mk"],["AddHom","mk"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["AddMonoidHom","toNatLinearMap","_proof_1"],["Nat"],["AddMonoidHom","toNatLinearMap","_proof_2"],["Nat","instSemiring"],["Nat","instNonAssocSemiring"],["AddCommMonoid","toAddCommSemigroup"],["RingHom","id"],["AddMonoidHom"],["AddMonoidHom","instFunLike"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["AddCommMonoid","toNatModule"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"definition","name":["LinearMap","restrictScalarsₗ"],"typeFallback":"forall (R : Type.{u_14}) (S : Type.{u_15}) (M : Type.{u_16}) (N : Type.{u_17}) [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 : Semiring.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 : Semiring.{u_15} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26 : AddCommMonoid.{u_16} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 : AddCommMonoid.{u_17} N] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.32 : Module.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.36 : Module.{u_14, u_17} R N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.40 : Module.{u_15, u_16} S M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.44 : Module.{u_15, u_17} S N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.48 : LinearMap.CompatibleSMul.{u_16, u_17, u_14, u_15} M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 (SMulZeroClass.toSMul.{u_14, u_16} R M (AddZero.toZero.{u_16} M (AddZeroClass.toAddZero.{u_16} M (AddMonoid.toAddZeroClass.{u_16} M (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26)))) (DistribSMul.toSMulZeroClass.{u_14, u_16} R M (AddMonoid.toAddZeroClass.{u_16} M (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26)) (DistribMulAction.toDistribSMul.{u_14, u_16} R M (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20)) (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26) (Module.toDistribMulAction.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.32)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.40 (SMulZeroClass.toSMul.{u_14, u_17} R N (AddZero.toZero.{u_17} N (AddZeroClass.toAddZero.{u_17} N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29)))) (DistribSMul.toSMulZeroClass.{u_14, u_17} R N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29)) (DistribMulAction.toDistribSMul.{u_14, u_17} R N (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20)) (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29) (Module.toDistribMulAction.{u_14, u_17} R N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.36)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.44] (R₁ : Type.{u_19}) [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.55 : Semiring.{u_19} R₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.58 : Module.{u_19, u_17} R₁ N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.55 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.62 : SMulCommClass.{u_15, u_19, u_17} S R₁ N (SMulZeroClass.toSMul.{u_15, u_17} S N (AddZero.toZero.{u_17} N (AddZeroClass.toAddZero.{u_17} N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29)))) (DistribSMul.toSMulZeroClass.{u_15, u_17} S N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29)) (DistribMulAction.toDistribSMul.{u_15, u_17} S N (MonoidWithZero.toMonoid.{u_15} S (Semiring.toMonoidWithZero.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23)) (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29) (Module.toDistribMulAction.{u_15, u_17} S N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.44)))) (SMulZeroClass.toSMul.{u_19, u_17} R₁ N (AddZero.toZero.{u_17} N (AddZeroClass.toAddZero.{u_17} N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29)))) (DistribSMul.toSMulZeroClass.{u_19, u_17} R₁ N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29)) (DistribMulAction.toDistribSMul.{u_19, u_17} R₁ N (MonoidWithZero.toMonoid.{u_19} R₁ (Semiring.toMonoidWithZero.{u_19} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.55)) (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29) (Module.toDistribMulAction.{u_19, u_17} R₁ N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.55 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.58))))] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.67 : SMulCommClass.{u_14, u_19, u_17} R R₁ N (SMulZeroClass.toSMul.{u_14, u_17} R N (AddZero.toZero.{u_17} N (AddZeroClass.toAddZero.{u_17} N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29)))) (DistribSMul.toSMulZeroClass.{u_14, u_17} R N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29)) (DistribMulAction.toDistribSMul.{u_14, u_17} R N (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20)) (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29) (Module.toDistribMulAction.{u_14, u_17} R N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.36)))) (SMulZeroClass.toSMul.{u_19, u_17} R₁ N (AddZero.toZero.{u_17} N (AddZeroClass.toAddZero.{u_17} N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29)))) (DistribSMul.toSMulZeroClass.{u_19, u_17} R₁ N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29)) (DistribMulAction.toDistribSMul.{u_19, u_17} R₁ N (MonoidWithZero.toMonoid.{u_19} R₁ (Semiring.toMonoidWithZero.{u_19} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.55)) (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29) (Module.toDistribMulAction.{u_19, u_17} R₁ N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.55 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.58))))], LinearMap.{u_19, u_19, max u_17 u_16, max u_17 u_16} R₁ R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.55 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.55 (RingHom.id.{u_19} R₁ (Semiring.toNonAssocSemiring.{u_19} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.55)) (LinearMap.{u_15, u_15, u_16, u_17} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 (RingHom.id.{u_15} S (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.44) (LinearMap.{u_14, u_14, u_16, u_17} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.20)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.36) (LinearMap.addCommMonoid.{u_15, u_15, u_16, u_17} S S M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2709294924._hygCtx._hyg.44 (RingHom.id.{u_15} S (Semiring.toNonAssocSemiring.{u_15} S 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[inst_12 : SMulCommClass R R₁ N] → (M →ₗ[S] N) →ₗ[R₁] M →ₗ[R] 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{R : Type.{u_1}} {M₁ : Type.{u_9}} {M₂ : Type.{u_10}} {F : Type.{u_14}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1907261298._hygCtx._hyg.16 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1907261298._hygCtx._hyg.19 : AddCommMonoid.{u_9} M₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1907261298._hygCtx._hyg.22 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1907261298._hygCtx._hyg.25 : Module.{u_1, u_9} R M₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1907261298._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1907261298._hygCtx._hyg.19] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1907261298._hygCtx._hyg.29 : Module.{u_1, u_10} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1907261298._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1907261298._hygCtx._hyg.22] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1907261298._hygCtx._hyg.34 : FunLike.{succ u_14, succ u_9, succ u_10} F M₁ M₂] 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          [inst_5 : FunLike F M₁ M₂] → [LinearMapClass F R M₁ M₂] → CoeHead F (M₁ →ₗ[R] M₂)","typeReferences":[["FunLike"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Module"],["CoeHead"],["LinearMap"],["LinearMapClass"],["Semiring"]],"valueReferences":[["Semiring","toNonAssocSemiring"],["CoeHead","mk"],["RingHom","id"],["SemilinearMapClass","semilinearMap"],["LinearMap"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","coe_restrictScalars"],"typeFallback":"forall (R : Type.{u_1}) {S : Type.{u_5}} {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3774992966._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3774992966._hygCtx._hyg.18 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3774992966._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3774992966._hygCtx._hyg.27 : AddCommMonoid.{u_10} M₂] 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3774992966._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3774992966._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3774992966._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3774992966._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3774992966._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3774992966._hygCtx._hyg.37 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3774992966._hygCtx._hyg.15))) (LinearMap.restrictScalars.{u_1, u_5, u_8, u_10} R S M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3774992966._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3774992966._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3774992966._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3774992966._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3774992966._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3774992966._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3774992966._hygCtx._hyg.71 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3774992966._hygCtx._hyg.75 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3774992966._hygCtx._hyg.79 f)) (DFunLike.coe.{max (succ u_10) (succ u_8), succ u_8, succ u_10} (LinearMap.{u_5, u_5, u_8, u_10} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3774992966._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3774992966._hygCtx._hyg.18 (RingHom.id.{u_5} S (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3774992966._hygCtx._hyg.18)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3774992966._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3774992966._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3774992966._hygCtx._hyg.71 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3774992966._hygCtx._hyg.75) M (fun 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LinearMap.CompatibleSMul M M₂ R S] (f : M →ₗ[S] M₂),\n  ⇑(↑R f) = ⇑f","typeReadable":"∀ (R : Type u_1) {S : Type u_5} {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M] [inst_5 : Module R M₂]\n  [inst_6 : Module S M] [inst_7 : Module S M₂] [inst_8 : LinearMap.CompatibleSMul M M₂ R S] (f : M →ₗ[S] M₂),\n  ⇑(↑R f) = ⇑f","typeReferences":[["LinearMap","CompatibleSMul"],["LinearMap","instFunLike"],["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["LinearMap","restrictScalars"],["Eq"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["rfl"],["Semiring","toNonAssocSemiring"],["LinearMap","instFunLike"],["RingHom","id"],["LinearMap","restrictScalars"],["LinearMap"],["DFunLike","coe"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","mulLeftRight","_proof_1"],"typeFallback":"forall (R : Type.{u_1}) [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.17 : Semiring.{u_1} R], RingHomCompTriple.{u_1, u_1, u_1} R R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.17 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.17)) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.17)) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2477694200._hygCtx._hyg.17))","typeFull":"∀ (R : Type u_1) [inst : Semiring R], RingHomCompTriple (RingHom.id R) (RingHom.id R) (RingHom.id R)","typeReadable":"∀ (R : Type u_1) [inst : Semiring R], RingHomCompTriple (RingHom.id R) (RingHom.id R) (RingHom.id R)","typeReferences":[["Semiring","toNonAssocSemiring"],["RingHom","id"],["RingHomCompTriple"],["Semiring"]],"valueReferences":[["Semiring","toNonAssocSemiring"],["RingHom","id"],["RingHomCompTriple","ids"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","mulRight_zero_eq_zero"],"typeFallback":"forall (R : Type.{u_14}) (A : Type.{u_15}) [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.17 : Semiring.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.20 : NonUnitalNonAssocSemiring.{u_15} A] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.23 : Module.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.20)] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.27 : IsScalarTower.{u_14, u_15, u_15} R A A (SMulZeroClass.toSMul.{u_14, u_15} R A (AddZero.toZero.{u_15} A (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_14, u_15} R A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.20))) (DistribMulAction.toDistribSMul.{u_14, u_15} R A (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.17)) (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.20)) (Module.toDistribMulAction.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.23)))) (SMulZeroClass.toSMul.{u_15, u_15} A A (AddZero.toZero.{u_15} A (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_15, u_15} A A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.20))) (instDistribSMul.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.20))) (SMulZeroClass.toSMul.{u_14, u_15} R A (AddZero.toZero.{u_15} A (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_14, u_15} R A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.20))) (DistribMulAction.toDistribSMul.{u_14, u_15} R A (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.17)) (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.20)) (Module.toDistribMulAction.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.23))))], Eq.{succ u_15} (LinearMap.{u_14, u_14, u_15, u_15} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.17 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.17)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.23) (LinearMap.mulRight.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.27 (OfNat.ofNat.{u_15} A 0 (Zero.toOfNat0.{u_15} A (MulZeroClass.toZero.{u_15} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.20))))) (OfNat.ofNat.{u_15} (LinearMap.{u_14, u_14, u_15, u_15} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.17 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.17)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.23) 0 (Zero.toOfNat0.{u_15} (LinearMap.{u_14, u_14, u_15, u_15} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.17 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.17)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.23) (LinearMap.instZero.{u_14, u_14, u_15, u_15} R R A A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.23 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1423981863._hygCtx._hyg.17)))))","typeFull":"∀ (R : Type u_14) (A : Type u_15) [inst : Semiring R] [inst_1 : NonUnitalNonAssocSemiring A] [inst_2 : Module R A]\n  [inst_3 : IsScalarTower R A A], LinearMap.mulRight R 0 = 0","typeReadable":"∀ (R : Type u_14) (A : Type u_15) [inst : Semiring R] [inst_1 : NonUnitalNonAssocSemiring A] [inst_2 : Module R A]\n  [inst_3 : IsScalarTower R A A], LinearMap.mulRight R 0 = 0","typeReferences":[["IsScalarTower"],["Module"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["instDistribSMul"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["Zero","toOfNat0"],["Eq"],["DistribSMul","toSMulZeroClass"],["LinearMap","instZero"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["LinearMap"],["OfNat","ofNat"],["Module","toDistribMulAction"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["LinearMap","mulRight"],["AddZero","toZero"],["NonUnitalNonAssocSemiring"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["LinearMap","instZero"],["Semiring","toNonAssocSemiring"],["MulZeroClass","toZero"],["RingHom","id"],["LinearMap","ext"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Zero","toOfNat0"],["LinearMap","mulRight"],["MulZeroClass","mul_zero"],["LinearMap"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","restrictScalars_smul"],"typeFallback":"forall {R : Type.{u_14}} {S : Type.{u_15}} {M : Type.{u_16}} {N : Type.{u_17}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.20 : Semiring.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.23 : Semiring.{u_15} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.26 : AddCommMonoid.{u_16} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29 : AddCommMonoid.{u_17} N] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.32 : Module.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.26] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.36 : Module.{u_14, u_17} R N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.40 : Module.{u_15, u_16} S M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.26] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.44 : Module.{u_15, u_17} S N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.48 : LinearMap.CompatibleSMul.{u_16, u_17, u_14, u_15} M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29 R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.23 (SMulZeroClass.toSMul.{u_14, u_16} R M (AddZero.toZero.{u_16} M (AddZeroClass.toAddZero.{u_16} M (AddMonoid.toAddZeroClass.{u_16} M (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.26)))) (DistribSMul.toSMulZeroClass.{u_14, u_16} R M (AddMonoid.toAddZeroClass.{u_16} M (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.26)) (DistribMulAction.toDistribSMul.{u_14, u_16} R M (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.20)) (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.26) (Module.toDistribMulAction.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.32)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.40 (SMulZeroClass.toSMul.{u_14, u_17} R N (AddZero.toZero.{u_17} N (AddZeroClass.toAddZero.{u_17} N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29)))) (DistribSMul.toSMulZeroClass.{u_14, u_17} R N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29)) (DistribMulAction.toDistribSMul.{u_14, u_17} R N (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.20)) (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29) (Module.toDistribMulAction.{u_14, u_17} R N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.36)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.44] {R₁ : Type.{u_19}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.55 : Semiring.{u_19} R₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.58 : Module.{u_19, u_17} R₁ N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.55 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.62 : SMulCommClass.{u_15, u_19, u_17} S R₁ N (SMulZeroClass.toSMul.{u_15, u_17} S N (AddZero.toZero.{u_17} N (AddZeroClass.toAddZero.{u_17} N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29)))) (DistribSMul.toSMulZeroClass.{u_15, u_17} S N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29)) (DistribMulAction.toDistribSMul.{u_15, u_17} S N (MonoidWithZero.toMonoid.{u_15} S (Semiring.toMonoidWithZero.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.23)) (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29) (Module.toDistribMulAction.{u_15, u_17} S N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.44)))) (SMulZeroClass.toSMul.{u_19, u_17} R₁ N (AddZero.toZero.{u_17} N (AddZeroClass.toAddZero.{u_17} N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29)))) (DistribSMul.toSMulZeroClass.{u_19, u_17} R₁ N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29)) (DistribMulAction.toDistribSMul.{u_19, u_17} R₁ N (MonoidWithZero.toMonoid.{u_19} R₁ (Semiring.toMonoidWithZero.{u_19} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.55)) (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29) (Module.toDistribMulAction.{u_19, u_17} R₁ N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.55 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.58))))] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.67 : SMulCommClass.{u_14, u_19, u_17} R R₁ N (SMulZeroClass.toSMul.{u_14, u_17} R N (AddZero.toZero.{u_17} N (AddZeroClass.toAddZero.{u_17} N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29)))) (DistribSMul.toSMulZeroClass.{u_14, u_17} R N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29)) (DistribMulAction.toDistribSMul.{u_14, u_17} R N (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.20)) (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29) (Module.toDistribMulAction.{u_14, u_17} R N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.36)))) (SMulZeroClass.toSMul.{u_19, u_17} R₁ N (AddZero.toZero.{u_17} N (AddZeroClass.toAddZero.{u_17} N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29)))) (DistribSMul.toSMulZeroClass.{u_19, u_17} R₁ N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29)) (DistribMulAction.toDistribSMul.{u_19, u_17} R₁ N (MonoidWithZero.toMonoid.{u_19} R₁ (Semiring.toMonoidWithZero.{u_19} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.55)) (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29) (Module.toDistribMulAction.{u_19, u_17} R₁ N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.55 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.58))))] (c : R₁) (f : LinearMap.{u_15, u_15, u_16, u_17} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.23 (RingHom.id.{u_15} S (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.23)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.44), Eq.{max (succ u_16) (succ u_17)} (LinearMap.{u_14, u_14, u_16, u_17} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.20 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.20)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.36) (LinearMap.restrictScalars.{u_14, u_15, u_16, u_17} R S M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.44 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.23)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.44) (instHSMul.{u_19, max u_16 u_17} R₁ (LinearMap.{u_15, u_15, u_16, u_17} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.23 (RingHom.id.{u_15} S (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.23)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.44) (LinearMap.instSMul.{u_15, u_15, u_19, u_16, u_17} S S R₁ M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.44 (RingHom.id.{u_15} S (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.23)) (DistribMulAction.toDistribSMul.{u_19, u_17} R₁ N (MonoidWithZero.toMonoid.{u_19} R₁ (Semiring.toMonoidWithZero.{u_19} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.55)) (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29) (Module.toDistribMulAction.{u_19, u_17} R₁ N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.55 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.58)) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.62)) c f)) (HSMul.hSMul.{u_19, max u_16 u_17, max u_16 u_17} R₁ (LinearMap.{u_14, u_14, u_16, u_17} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.20 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.20)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.36) (LinearMap.{u_14, u_14, u_16, u_17} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.20 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.20)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.36) (instHSMul.{u_19, max u_16 u_17} R₁ (LinearMap.{u_14, u_14, u_16, u_17} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.20 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.20)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.36) (LinearMap.instSMul.{u_14, u_14, u_19, u_16, u_17} R R R₁ M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.36 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_19, u_17} R₁ N (MonoidWithZero.toMonoid.{u_19} R₁ (Semiring.toMonoidWithZero.{u_19} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.55)) (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29) (Module.toDistribMulAction.{u_19, u_17} R₁ N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.55 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.58)) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.67)) c (LinearMap.restrictScalars.{u_14, u_15, u_16, u_17} R S M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.44 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2424490603._hygCtx._hyg.48 f))","typeFull":"∀ {R : Type u_14} {S : Type u_15} {M : Type u_16} {N : Type u_17} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid N] [inst_4 : Module R M] [inst_5 : Module R N]\n  [inst_6 : Module S M] [inst_7 : Module S N] [inst_8 : LinearMap.CompatibleSMul M N R S] {R₁ : Type u_19}\n  [inst_9 : Semiring R₁] [inst_10 : Module R₁ N] [inst_11 : SMulCommClass S R₁ N] [inst_12 : SMulCommClass R R₁ N]\n  (c : R₁) (f : M →ₗ[S] N), ↑R (c • f) = c • ↑R f","typeReadable":"∀ {R : Type u_14} {S : Type u_15} {M : Type u_16} {N : Type u_17} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid N] [inst_4 : Module R M] [inst_5 : Module R N]\n  [inst_6 : Module S M] [inst_7 : Module S N] [inst_8 : LinearMap.CompatibleSMul M N R S] {R₁ : Type u_19}\n  [inst_9 : Semiring R₁] [inst_10 : Module R₁ N] [inst_11 : SMulCommClass S R₁ N] [inst_12 : SMulCommClass R R₁ N]\n  (c : R₁) (f : M →ₗ[S] N), ↑R (c • f) = c • ↑R 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{R₁ : Type.{u_1}} {R₂ : Type.{u_2}} {M : Type.{u_3}} {M₂ : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.15 : Semiring.{u_1} R₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.18 : Semiring.{u_2} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.24 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.27 : AddCommMonoid.{u_4} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.39 : Module.{u_1, u_3} R₁ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.43 : Module.{u_2, u_4} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.27] {σ₁₂ : RingHom.{u_1, u_2} R₁ R₂ 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.43), Eq.{succ (max u_4 u_3)} (AddMonoidHom.{u_3, u_4} M M₂ (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.24))) (AddZeroClass.toAddZero.{u_4} M₂ (AddMonoid.toAddZeroClass.{u_4} M₂ (AddCommMonoid.toAddMonoid.{u_4} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.27)))) (LinearMap.toAddMonoidHom.{u_1, u_2, u_3, u_4} R₁ R₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.43 σ₁₂ (HAdd.hAdd.{max u_4 u_3, max u_4 u_3, max u_4 u_3} (LinearMap.{u_1, u_2, u_3, u_4} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.43) (LinearMap.{u_1, u_2, u_3, u_4} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.512997599._hygCtx._hyg.24 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Semiring.{u_14} R} {inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 : Semiring.{u_15} S} {σ : RingHom.{u_14, u_15} R S (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30) (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33)} {M : Type.{u_16}} {M₂ : Type.{u_17}} {inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 : AddCommMonoid.{u_16} M} {inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 : AddCommMonoid.{u_17} M₂} {inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 : Module.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43} {inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53 : Module.{u_15, u_17} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46} {P : Sort.{u}} {toAddHom : AddHom.{u_16, u_17} M M₂ (AddCommMagma.toAdd.{u_16} M (AddCommSemigroup.toAddCommMagma.{u_16} M (AddCommMonoid.toAddCommSemigroup.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43))) (AddCommMagma.toAdd.{u_17} M₂ (AddCommSemigroup.toAddCommMagma.{u_17} M₂ (AddCommMonoid.toAddCommSemigroup.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46)))} {map_smul' : forall (m : R) (x : M), Eq.{succ u_17} M₂ (AddHom.toFun.{u_16, u_17} M M₂ (AddCommMagma.toAdd.{u_16} M (AddCommSemigroup.toAddCommMagma.{u_16} M (AddCommMonoid.toAddCommSemigroup.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43))) (AddCommMagma.toAdd.{u_17} M₂ (AddCommSemigroup.toAddCommMagma.{u_17} M₂ (AddCommMonoid.toAddCommSemigroup.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46))) toAddHom (HSMul.hSMul.{u_14, u_16, u_16} R M M (instHSMul.{u_14, u_16} R M (SMulZeroClass.toSMul.{u_14, u_16} R M (AddZero.toZero.{u_16} M (AddZeroClass.toAddZero.{u_16} M (AddMonoid.toAddZeroClass.{u_16} M (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43)))) (DistribSMul.toSMulZeroClass.{u_14, u_16} R M (AddMonoid.toAddZeroClass.{u_16} M (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43)) (DistribMulAction.toDistribSMul.{u_14, u_16} R M (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30)) (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43) (Module.toDistribMulAction.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49))))) m x)) (HSMul.hSMul.{u_15, u_17, u_17} S M₂ M₂ (instHSMul.{u_15, u_17} S M₂ (SMulZeroClass.toSMul.{u_15, u_17} S M₂ (AddZero.toZero.{u_17} M₂ (AddZeroClass.toAddZero.{u_17} M₂ (AddMonoid.toAddZeroClass.{u_17} M₂ (AddCommMonoid.toAddMonoid.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46)))) (DistribSMul.toSMulZeroClass.{u_15, u_17} S M₂ (AddMonoid.toAddZeroClass.{u_17} M₂ (AddCommMonoid.toAddMonoid.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46)) (DistribMulAction.toDistribSMul.{u_15, u_17} S M₂ (MonoidWithZero.toMonoid.{u_15} S (Semiring.toMonoidWithZero.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33)) (AddCommMonoid.toAddMonoid.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46) (Module.toDistribMulAction.{u_15, u_17} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53))))) (DFunLike.coe.{max (succ u_14) (succ u_15), succ u_14, succ u_15} (RingHom.{u_14, u_15} R S (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30) (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33)) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => S) (RingHom.instFunLike.{u_14, u_15} R S (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30) (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33)) σ m) (AddHom.toFun.{u_16, u_17} M M₂ (AddCommMagma.toAdd.{u_16} M (AddCommSemigroup.toAddCommMagma.{u_16} M (AddCommMonoid.toAddCommSemigroup.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43))) (AddCommMagma.toAdd.{u_17} M₂ (AddCommSemigroup.toAddCommMagma.{u_17} M₂ (AddCommMonoid.toAddCommSemigroup.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46))) toAddHom x))} {toAddHom' : AddHom.{u_16, u_17} M M₂ (AddCommMagma.toAdd.{u_16} M (AddCommSemigroup.toAddCommMagma.{u_16} M (AddCommMonoid.toAddCommSemigroup.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43))) (AddCommMagma.toAdd.{u_17} M₂ (AddCommSemigroup.toAddCommMagma.{u_17} M₂ (AddCommMonoid.toAddCommSemigroup.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46)))} {map_smul'' : forall (m : R) (x : M), Eq.{succ u_17} M₂ (AddHom.toFun.{u_16, u_17} M M₂ (AddCommMagma.toAdd.{u_16} M (AddCommSemigroup.toAddCommMagma.{u_16} M (AddCommMonoid.toAddCommSemigroup.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43))) (AddCommMagma.toAdd.{u_17} M₂ (AddCommSemigroup.toAddCommMagma.{u_17} M₂ (AddCommMonoid.toAddCommSemigroup.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46))) toAddHom' (HSMul.hSMul.{u_14, u_16, u_16} R M M (instHSMul.{u_14, u_16} R M (SMulZeroClass.toSMul.{u_14, u_16} R M (AddZero.toZero.{u_16} M (AddZeroClass.toAddZero.{u_16} M (AddMonoid.toAddZeroClass.{u_16} M (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43)))) (DistribSMul.toSMulZeroClass.{u_14, u_16} R M (AddMonoid.toAddZeroClass.{u_16} M (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43)) (DistribMulAction.toDistribSMul.{u_14, u_16} R M (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30)) (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43) (Module.toDistribMulAction.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49))))) m x)) (HSMul.hSMul.{u_15, u_17, u_17} S M₂ M₂ (instHSMul.{u_15, u_17} S M₂ (SMulZeroClass.toSMul.{u_15, u_17} S M₂ (AddZero.toZero.{u_17} M₂ (AddZeroClass.toAddZero.{u_17} M₂ (AddMonoid.toAddZeroClass.{u_17} M₂ (AddCommMonoid.toAddMonoid.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46)))) (DistribSMul.toSMulZeroClass.{u_15, u_17} S M₂ (AddMonoid.toAddZeroClass.{u_17} M₂ (AddCommMonoid.toAddMonoid.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46)) (DistribMulAction.toDistribSMul.{u_15, u_17} S M₂ (MonoidWithZero.toMonoid.{u_15} S (Semiring.toMonoidWithZero.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33)) (AddCommMonoid.toAddMonoid.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46) (Module.toDistribMulAction.{u_15, u_17} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53))))) (DFunLike.coe.{max (succ u_14) (succ u_15), succ u_14, succ u_15} (RingHom.{u_14, u_15} R S (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30) (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33)) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => S) (RingHom.instFunLike.{u_14, u_15} R S (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30) (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33)) σ m) (AddHom.toFun.{u_16, u_17} M M₂ (AddCommMagma.toAdd.{u_16} M (AddCommSemigroup.toAddCommMagma.{u_16} M (AddCommMonoid.toAddCommSemigroup.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43))) (AddCommMagma.toAdd.{u_17} M₂ (AddCommSemigroup.toAddCommMagma.{u_17} M₂ (AddCommMonoid.toAddCommSemigroup.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46))) toAddHom' x))}, (Eq.{max (succ u_16) (succ u_17)} (LinearMap.{u_14, u_15, u_16, u_17} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53) (LinearMap.mk.{u_14, u_15, u_16, u_17} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53 toAddHom map_smul') (LinearMap.mk.{u_14, u_15, u_16, u_17} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53 toAddHom' map_smul'')) -> ((HEq.{max (succ u_16) (succ u_17)} (AddHom.{u_16, u_17} M M₂ (AddCommMagma.toAdd.{u_16} M (AddCommSemigroup.toAddCommMagma.{u_16} M (AddCommMonoid.toAddCommSemigroup.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43))) (AddCommMagma.toAdd.{u_17} M₂ (AddCommSemigroup.toAddCommMagma.{u_17} M₂ (AddCommMonoid.toAddCommSemigroup.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46)))) toAddHom (AddHom.{u_16, u_17} M M₂ (AddCommMagma.toAdd.{u_16} M (AddCommSemigroup.toAddCommMagma.{u_16} M (AddCommMonoid.toAddCommSemigroup.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43))) (AddCommMagma.toAdd.{u_17} M₂ (AddCommSemigroup.toAddCommMagma.{u_17} M₂ (AddCommMonoid.toAddCommSemigroup.{u_17} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46)))) toAddHom') -> P) -> P","typeFull":"{R : Type u_14} →\n  {S : Type u_15} →\n    {inst : Semiring R} →\n      {inst_1 : Semiring S} →\n        {σ : R →+* S} →\n          {M : Type u_16} →\n            {M₂ : Type u_17} →\n              {inst_2 : AddCommMonoid M} →\n                {inst_3 : AddCommMonoid M₂} →\n                  {inst_4 : Module R M} →\n                    {inst_5 : Module S M₂} →\n                      {P : Sort u} →\n                        {toAddHom : M →ₙ+ M₂} →\n                          {map_smul' : ∀ (m : R) (x : M), toAddHom.toFun (m • x) = σ m • toAddHom.toFun x} →\n                            {toAddHom' : M →ₙ+ M₂} →\n                              {map_smul'' : ∀ (m : R) (x : M), toAddHom'.toFun (m • x) = σ m • toAddHom'.toFun x} →\n                                { toAddHom := toAddHom, map_smul' := map_smul' } =\n                                    { toAddHom := toAddHom', map_smul' := map_smul'' } →\n                                  (toAddHom ≍ toAddHom' → P) → P","typeReadable":"{R : Type u_14} →\n  {S : Type u_15} →\n    {inst : Semiring R} →\n      {inst_1 : Semiring S} →\n        {σ : R →+* S} →\n          {M : Type u_16} →\n            {M₂ : Type u_17} →\n              {inst_2 : AddCommMonoid M} →\n                {inst_3 : AddCommMonoid M₂} →\n                  {inst_4 : Module R M} →\n                    {inst_5 : Module S M₂} →\n                      {P : Sort u} →\n                        {toAddHom : M →ₙ+ M₂} →\n                          {map_smul' : ∀ (m : R) (x : M), toAddHom.toFun (m • x) = σ m • toAddHom.toFun x} →\n                            {toAddHom' : M →ₙ+ M₂} →\n                              {map_smul'' : ∀ (m : R) (x : M), toAddHom'.toFun (m • x) = σ m • toAddHom'.toFun x} →\n                                { toAddHom := toAddHom, map_smul' := map_smul' } =\n                                    { toAddHom := toAddHom', map_smul' := map_smul'' } →\n                                  (toAddHom ≍ toAddHom' → P) → P","typeReferences":[["RingHom"],["Module"],["LinearMap","mk"],["RingHom","instFunLike"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddHom","toFun"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Eq"],["DistribSMul","toSMulZeroClass"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["LinearMap"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["HSMul","hSMul"],["HEq"],["AddHom"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["RingHom"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["HEq","refl"],["Module"],["Eq","refl"],["LinearMap","mk"],["id"],["heq_of_eq"],["LinearMap"],["Semiring"],["LinearMap","noConfusion"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","comp_add"],"typeFallback":"forall {R₁ : Type.{u_2}} {R₂ : Type.{u_3}} {R₃ : Type.{u_4}} {M : Type.{u_8}} {M₂ : Type.{u_10}} {M₃ : Type.{u_11}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.15 : Semiring.{u_2} R₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.18 : Semiring.{u_3} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.21 : Semiring.{u_4} R₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.24 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.27 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.30 : AddCommMonoid.{u_11} M₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.39 : Module.{u_2, u_8} R₁ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.43 : Module.{u_3, u_10} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.27] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.47 : Module.{u_4, u_11} R₃ M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.30] {σ₁₂ : RingHom.{u_2, u_3} R₁ R₂ (Semiring.toNonAssocSemiring.{u_2} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.18)} {σ₂₃ : RingHom.{u_3, u_4} R₂ R₃ (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.18) (Semiring.toNonAssocSemiring.{u_4} R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.21)} {σ₁₃ : RingHom.{u_2, u_4} R₁ R₃ (Semiring.toNonAssocSemiring.{u_2} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_4} R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.21)} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.74 : RingHomCompTriple.{u_2, u_3, u_4} R₁ R₂ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.21 σ₁₂ σ₂₃ σ₁₃] (f : LinearMap.{u_2, u_3, u_8, u_10} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.39 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(HAdd.hAdd.{max u_11 u_8, max u_11 u_8, max u_11 u_8} (LinearMap.{u_2, u_4, u_8, u_11} R₁ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.21 σ₁₃ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.47) (LinearMap.{u_2, u_4, u_8, u_11} R₁ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.21 σ₁₃ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.47) (LinearMap.{u_2, u_4, u_8, u_11} R₁ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.21 σ₁₃ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.47) (instHAdd.{max u_11 u_8} (LinearMap.{u_2, u_4, u_8, u_11} R₁ R₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.21 σ₁₃ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.47) (LinearMap.instAdd.{u_2, u_4, u_8, u_11} R₁ R₃ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.47 σ₁₃)) (LinearMap.comp.{u_2, u_3, u_4, u_8, u_10, u_11} R₁ R₂ R₃ M M₂ M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.47 σ₁₂ σ₂₃ σ₁₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.74 h f) (LinearMap.comp.{u_2, u_3, u_4, u_8, u_10, u_11} R₁ R₂ R₃ M M₂ M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.47 σ₁₂ σ₂₃ σ₁₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1609400224._hygCtx._hyg.74 h g))","typeFull":"∀ {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_8} {M₂ : Type u_10} {M₃ : Type u_11} [inst : Semiring R₁]\n  [inst_1 : Semiring R₂] [inst_2 : Semiring R₃] [inst_3 : AddCommMonoid M] [inst_4 : AddCommMonoid M₂]\n  [inst_5 : AddCommMonoid M₃] [inst_6 : Module R₁ M] [inst_7 : Module R₂ M₂] [inst_8 : Module R₃ M₃] {σ₁₂ : R₁ →+* R₂}\n  {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [inst_9 : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (f g : M →ₛₗ[σ₁₂] M₂)\n  (h : M₂ →ₛₗ[σ₂₃] M₃), h ∘ₛₗ (f + g) = h ∘ₛₗ f + h ∘ₛₗ g","typeReadable":"∀ {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_8} {M₂ : Type u_10} {M₃ : Type u_11} [inst : Semiring R₁]\n  [inst_1 : Semiring R₂] [inst_2 : Semiring R₃] [inst_3 : AddCommMonoid M] [inst_4 : AddCommMonoid M₂]\n  [inst_5 : AddCommMonoid M₃] [inst_6 : Module R₁ M] [inst_7 : Module R₂ M₂] [inst_8 : Module R₃ M₃] {σ₁₂ : R₁ →+* R₂}\n  {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [inst_9 : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (f g : M →ₛₗ[σ₁₂] M₂)\n  (h : M₂ →ₛₗ[σ₂₃] M₃), h ∘ₛₗ (f + g) = h ∘ₛₗ f + h ∘ₛₗ g","typeReferences":[["HAdd","hAdd"],["RingHom"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["instHAdd"],["Module"],["LinearMap","instAdd"],["LinearMap","comp"],["Eq"],["LinearMap"],["RingHomCompTriple"],["Semiring"]],"valueReferences":[["HAdd","hAdd"],["LinearMap","map_add"],["LinearMap","instFunLike"],["instHAdd"],["LinearMap","ext"],["LinearMap","instAdd"],["LinearMap","comp"],["LinearMap"],["DFunLike","coe"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","restrictScalars_inj"],"typeFallback":"forall (R : Type.{u_1}) {S : Type.{u_5}} {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.18 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.27 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.33 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.37 : Module.{u_1, u_10} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.27] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.71 : Module.{u_5, u_8} S M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.75 : Module.{u_5, u_10} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.27] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.79 : LinearMap.CompatibleSMul.{u_8, u_10, u_1, u_5} M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.27 R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.18 (SMulZeroClass.toSMul.{u_1, u_8} R M (AddZero.toZero.{u_8} M (AddZeroClass.toAddZero.{u_8} M (AddMonoid.toAddZeroClass.{u_8} M (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.21)))) (DistribSMul.toSMulZeroClass.{u_1, u_8} R M (AddMonoid.toAddZeroClass.{u_8} M (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.21)) (DistribMulAction.toDistribSMul.{u_1, u_8} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.15)) (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.21) (Module.toDistribMulAction.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.33)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.71 (SMulZeroClass.toSMul.{u_1, u_10} R M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.27)))) (DistribSMul.toSMulZeroClass.{u_1, u_10} R M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.27)) (DistribMulAction.toDistribSMul.{u_1, u_10} R M₂ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.15)) (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.27) (Module.toDistribMulAction.{u_1, u_10} R M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.37)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.75] (fₗ : LinearMap.{u_5, u_5, u_8, u_10} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2573026654._hygCtx._hyg.18 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.18 σ₁₂ M N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_4} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.33) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.51) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => N₂) (LinearMap.instFunLike.{u_1, u_2, u_3, u_4} R₁ R₂ M N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.24 (AddCommGroup.toAddCommMonoid.{u_4} N₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.33) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.39 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.51 σ₁₂) x._@.Mathlib.Algebra.Module.LinearMap.Defs.405133445._hygCtx._hyg.104))","typeFull":"∀ {R₁ : Type u_1} {R₂ : Type u_2} {M : Type u_3} {N₂ : Type u_4} [inst : Semiring R₁] [inst_1 : Semiring R₂]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommGroup N₂] [inst_4 : Module R₁ M] [inst_5 : Module R₂ N₂] {σ₁₂ : R₁ →+* R₂}\n  (x : M →ₛₗ[σ₁₂] N₂), ⇑(-x) = ⇑(-x)","typeReadable":"∀ {R₁ : Type u_1} {R₂ : Type u_2} {M : Type u_3} {N₂ : Type u_4} [inst : Semiring R₁] [inst_1 : Semiring R₂]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommGroup N₂] [inst_4 : Module R₁ M] [inst_5 : Module R₂ N₂] {σ₁₂ : R₁ →+* R₂}\n  (x : M →ₛₗ[σ₁₂] N₂), ⇑(-x) = ⇑(-x)","typeReferences":[["RingHom"],["LinearMap","instFunLike"],["Module"],["Neg","neg"],["AddCommGroup"],["LinearMap"],["DFunLike","coe"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["LinearMap","instNeg"],["AddCommGroup","toAddCommMonoid"],["Eq"],["Semiring"]],"valueReferences":[["rfl"],["LinearMap","instNeg"],["LinearMap","instFunLike"],["Neg","neg"],["AddCommGroup","toAddCommMonoid"],["LinearMap"],["DFunLike","coe"]]},{"isProp":false,"kind":"definition","name":["LinearMap","instZero"],"typeFallback":"forall {R₁ : Type.{u_2}} {R₂ : Type.{u_3}} {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.15 : Semiring.{u_2} R₁] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.18 : Semiring.{u_3} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.24 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.27 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.39 : Module.{u_2, u_8} R₁ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.43 : Module.{u_3, u_10} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.27] {σ₁₂ : RingHom.{u_2, u_3} R₁ R₂ (Semiring.toNonAssocSemiring.{u_2} R₁ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.18)}, Zero.{max u_10 u_8} (LinearMap.{u_2, u_3, u_8, u_10} R₁ R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2736031858._hygCtx._hyg.43)","typeFull":"{R₁ : Type u_2} →\n  {R₂ : Type u_3} →\n    {M : Type u_8} →\n      {M₂ : Type u_10} →\n        [inst : Semiring R₁] →\n          [inst_1 : Semiring R₂] →\n            [inst_2 : AddCommMonoid M] →\n              [inst_3 : AddCommMonoid M₂] →\n                [inst_4 : Module R₁ M] → [inst_5 : Module R₂ M₂] → {σ₁₂ : R₁ →+* R₂} → Zero (M →ₛₗ[σ₁₂] M₂)","typeReadable":"{R₁ : Type u_2} →\n  {R₂ : Type u_3} →\n    {M : Type u_8} →\n      {M₂ : Type u_10} →\n        [inst : Semiring R₁] →\n          [inst_1 : Semiring R₂] →\n            [inst_2 : AddCommMonoid M] →\n              [inst_3 : AddCommMonoid M₂] →\n                [inst_4 : Module R₁ M] → [inst_5 : Module R₂ M₂] → {σ₁₂ : R₁ →+* R₂} → Zero (M →ₛₗ[σ₁₂] M₂)","typeReferences":[["RingHom"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["Module"],["Zero"],["LinearMap"],["Semiring"]],"valueReferences":[["LinearMap","instZero","_proof_1"],["LinearMap","mk"],["AddHom","mk"],["LinearMap","instZero","_proof_2"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Zero","mk"],["AddCommMonoid","toAddCommSemigroup"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Zero","toOfNat0"],["AddZero","toZero"],["Pi","instZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","restrictScalars_id"],"typeFallback":"forall (R : Type.{u_1}) {S : Type.{u_5}} {M : Type.{u_8}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.18 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.33 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.71 : Module.{u_5, u_8} S M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.85 : LinearMap.CompatibleSMul.{u_8, u_8, u_1, u_5} M M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.21 R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.18 (SMulZeroClass.toSMul.{u_1, u_8} R M (AddZero.toZero.{u_8} M (AddZeroClass.toAddZero.{u_8} M (AddMonoid.toAddZeroClass.{u_8} M (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.21)))) (DistribSMul.toSMulZeroClass.{u_1, u_8} R M (AddMonoid.toAddZeroClass.{u_8} M (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.21)) (DistribMulAction.toDistribSMul.{u_1, u_8} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.15)) (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.21) (Module.toDistribMulAction.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.33)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.71 (SMulZeroClass.toSMul.{u_1, u_8} R M (AddZero.toZero.{u_8} M (AddZeroClass.toAddZero.{u_8} M (AddMonoid.toAddZeroClass.{u_8} M (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.21)))) (DistribSMul.toSMulZeroClass.{u_1, u_8} R M (AddMonoid.toAddZeroClass.{u_8} M (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.21)) (DistribMulAction.toDistribSMul.{u_1, u_8} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.15)) (AddCommMonoid.toAddMonoid.{u_8} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.21) (Module.toDistribMulAction.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.33)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.71], Eq.{succ u_8} (LinearMap.{u_1, u_1, u_8, u_8} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.15 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.15)) M M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.33) (LinearMap.restrictScalars.{u_1, u_5, u_8, u_8} R S M M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.71 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.71 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.85 (LinearMap.id.{u_5, u_8} S M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.71)) (LinearMap.id.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3626818416._hygCtx._hyg.33)","typeFull":"∀ (R : Type u_1) {S : Type u_5} {M : Type u_8} [inst : Semiring R] [inst_1 : Semiring S] [inst_2 : AddCommMonoid M]\n  [inst_3 : Module R M] [inst_4 : Module S M] [inst_5 : LinearMap.CompatibleSMul M M R S],\n  ↑R LinearMap.id = LinearMap.id","typeReadable":"∀ (R : Type u_1) {S : Type u_5} {M : Type u_8} [inst : Semiring R] [inst_1 : Semiring S] [inst_2 : AddCommMonoid M]\n  [inst_3 : Module R M] [inst_4 : Module S M] [inst_5 : LinearMap.CompatibleSMul M M R S],\n  ↑R LinearMap.id = LinearMap.id","typeReferences":[["LinearMap","CompatibleSMul"],["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["LinearMap","id"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["LinearMap","restrictScalars"],["Eq"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["rfl"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["LinearMap","restrictScalars"],["LinearMap"],["LinearMap","id"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","restrictScalars_trans"],"typeFallback":"forall {R : Type.{u_14}} {S : Type.{u_15}} {M : Type.{u_16}} {N : Type.{u_17}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.20 : Semiring.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.23 : Semiring.{u_15} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.26 : AddCommMonoid.{u_16} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.29 : AddCommMonoid.{u_17} N] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.32 : Module.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.26] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.36 : Module.{u_14, u_17} R N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.40 : Module.{u_15, u_16} S M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.26] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.44 : Module.{u_15, u_17} S N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.48 : LinearMap.CompatibleSMul.{u_16, u_17, u_14, u_15} M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.29 R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.23 (SMulZeroClass.toSMul.{u_14, u_16} R M (AddZero.toZero.{u_16} M (AddZeroClass.toAddZero.{u_16} M (AddMonoid.toAddZeroClass.{u_16} M (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.26)))) (DistribSMul.toSMulZeroClass.{u_14, u_16} R M (AddMonoid.toAddZeroClass.{u_16} M (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.26)) (DistribMulAction.toDistribSMul.{u_14, u_16} R M (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.20)) (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.26) (Module.toDistribMulAction.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.32)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.40 (SMulZeroClass.toSMul.{u_14, u_17} R N (AddZero.toZero.{u_17} N (AddZeroClass.toAddZero.{u_17} N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.29)))) (DistribSMul.toSMulZeroClass.{u_14, u_17} R N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.29)) (DistribMulAction.toDistribSMul.{u_14, u_17} R N (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.20)) (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.29) (Module.toDistribMulAction.{u_14, u_17} R N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.36)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.44] {T : Type.{u_20}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.73 : Semiring.{u_20} T] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.76 : Module.{u_20, u_16} T M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.26] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.80 : Module.{u_20, u_17} T N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.73 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.84 : LinearMap.CompatibleSMul.{u_16, u_17, u_15, u_20} M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.29 S T inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.73 (SMulZeroClass.toSMul.{u_15, u_16} S M (AddZero.toZero.{u_16} M (AddZeroClass.toAddZero.{u_16} M (AddMonoid.toAddZeroClass.{u_16} M 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.80] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.90 : LinearMap.CompatibleSMul.{u_16, u_17, u_14, u_20} M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.29 R T inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.73 (SMulZeroClass.toSMul.{u_14, u_16} R M (AddZero.toZero.{u_16} M (AddZeroClass.toAddZero.{u_16} M (AddMonoid.toAddZeroClass.{u_16} M (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.26)))) (DistribSMul.toSMulZeroClass.{u_14, u_16} R M (AddMonoid.toAddZeroClass.{u_16} M (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.26)) (DistribMulAction.toDistribSMul.{u_14, u_16} R M (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.76 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.80), Eq.{max (succ u_16) (succ u_17)} (LinearMap.{u_14, u_14, u_16, u_17} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.20 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.20)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2912025229._hygCtx._hyg.36) (LinearMap.restrictScalars.{u_14, u_15, u_16, u_17} R S 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.31)))) (SMulZeroClass.toSMul.{u_5, u_10} S M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.24)))) (DistribSMul.toSMulZeroClass.{u_5, u_10} S M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.24)) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.40))] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.49 : DistribSMul.{u_7, u_10} T M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.24))] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.53 : SMulCommClass.{u_3, u_7, u_10} R₂ T M₂ (SMulZeroClass.toSMul.{u_3, u_10} R₂ M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.24)))) (DistribSMul.toSMulZeroClass.{u_3, u_10} R₂ M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.24)) (DistribMulAction.toDistribSMul.{u_3, u_10} R₂ M₂ (MonoidWithZero.toMonoid.{u_3} R₂ (Semiring.toMonoidWithZero.{u_3} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.18)) (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.24) (Module.toDistribMulAction.{u_3, u_10} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.31)))) (SMulZeroClass.toSMul.{u_7, u_10} T M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.24)))) (DistribSMul.toSMulZeroClass.{u_7, u_10} T M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.24)) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.49))] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.58 : SMul.{u_5, u_7} S T] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.62 : IsScalarTower.{u_5, u_7, u_10} S T M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.58 (SMulZeroClass.toSMul.{u_7, u_10} T M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.24)))) (DistribSMul.toSMulZeroClass.{u_7, u_10} T M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.24)) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.49)) (SMulZeroClass.toSMul.{u_5, u_10} S M₂ (AddZero.toZero.{u_10} M₂ (AddZeroClass.toAddZero.{u_10} M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.24)))) (DistribSMul.toSMulZeroClass.{u_5, u_10} S M₂ (AddMonoid.toAddZeroClass.{u_10} M₂ (AddCommMonoid.toAddMonoid.{u_10} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.24)) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.40))], IsScalarTower.{u_5, u_7, max u_10 u_8} S T (LinearMap.{u_1, u_3, u_8, u_10} R R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.18 σ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.31) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.58 (LinearMap.instSMul.{u_1, u_3, u_7, u_8, u_10} R R₂ T M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.31 σ₁₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.49 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.53) (LinearMap.instSMul.{u_1, u_3, u_5, u_8, u_10} R R₂ S M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.31 σ₁₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.772748180._hygCtx._hyg.44)","typeFull":"∀ {R : Type u_1} {R₂ : Type u_3} {S : Type u_5} {T : Type u_7} {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R]\n  [inst_1 : Semiring R₂] [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M]\n  [inst_5 : Module R₂ M₂] {σ₁₂ : R →+* R₂} [inst_6 : DistribSMul S M₂] [inst_7 : SMulCommClass R₂ S M₂]\n  [inst_8 : DistribSMul T M₂] [inst_9 : SMulCommClass R₂ T M₂] [inst_10 : SMul S T] [IsScalarTower S T M₂],\n  IsScalarTower S T (M →ₛₗ[σ₁₂] M₂)","typeReadable":"∀ {R : Type u_1} {R₂ : Type u_3} {S : Type u_5} {T : Type u_7} {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R]\n  [inst_1 : Semiring R₂] [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M]\n  [inst_5 : Module R₂ M₂] {σ₁₂ : R →+* R₂} [inst_6 : DistribSMul S M₂] [inst_7 : SMulCommClass R₂ S M₂]\n  [inst_8 : DistribSMul T M₂] [inst_9 : SMulCommClass R₂ T M₂] [inst_10 : SMul S T] [IsScalarTower S T M₂],\n  IsScalarTower S T (M →ₛₗ[σ₁₂] 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[inst_1 : Semiring R₂]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommGroup N₂] [inst_4 : Module R₁ M] [inst_5 : Module R₂ N₂] {σ₁₂ : R₁ →+* R₂}\n  (x : M →ₛₗ[σ₁₂] N₂) (x_1 : ℤ), ⇑(x_1 • x) = ⇑(x_1 • 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{R : Type.{u_1}} {S : Type.{u_5}} {M : Type.{u_8}} {M₃ : Type.{u_11}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3425304009._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3425304009._hygCtx._hyg.18 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3425304009._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3425304009._hygCtx._hyg.30 : AddCommMonoid.{u_11} M₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3425304009._hygCtx._hyg.33 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3425304009._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3425304009._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3425304009._hygCtx._hyg.41 : Module.{u_5, u_11} S M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3425304009._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3425304009._hygCtx._hyg.30] {σ : RingHom.{u_1, u_5} R S 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AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.96 : AddCommMonoid.{u_10} M₂] {R : Type.{u_14}} {S : Type.{u_15}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.142 : Semiring.{u_15} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.145 : SMul.{u_14, u_8} R M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.149 : Module.{u_15, u_8} S M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.142 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.90] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.153 : SMul.{u_14, u_10} R M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.157 : Module.{u_15, u_10} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.142 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.96] {motive : (LinearMap.CompatibleSMul.{u_8, u_10, u_14, u_15} M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.90 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.96 R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.142 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.145 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.149 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.153 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.157) -> Sort.{u}}, (forall (map_smul : forall (fₗ : LinearMap.{u_15, u_15, u_8, u_10} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.142 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.142 (RingHom.id.{u_15} S (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.142)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.90 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.96 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.149 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.157) (c : R) (x : M), Eq.{succ u_10} M₂ (DFunLike.coe.{max (succ u_10) (succ u_8), succ u_8, succ u_10} (LinearMap.{u_15, u_15, u_8, u_10} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.142 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.142 (RingHom.id.{u_15} S (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.142)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.90 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.96 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.149 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.157) M (fun 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(DFunLike.coe.{max (succ u_10) (succ u_8), succ u_8, succ u_10} (LinearMap.{u_15, u_15, u_8, u_10} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.142 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.142 (RingHom.id.{u_15} S (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.142)) M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.90 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.96 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.149 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.157) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) (LinearMap.instFunLike.{u_15, u_15, u_8, u_10} S S M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.142 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2006690188._hygCtx._hyg.142 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46))) toAddHom x))) -> (LinearMap.{u_14, u_15, u_16, u_17} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.33 σ M M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.49 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.2302359325._hygCtx._hyg.53)","typeFull":"{R : Type u_14} →\n  {S : Type u_15} →\n    [inst : Semiring R] →\n      [inst_1 : Semiring S] →\n        {σ : R →+* S} →\n          {M : Type u_16} →\n            {M₂ : Type u_17} →\n              [inst_2 : AddCommMonoid M] →\n                [inst_3 : AddCommMonoid M₂] →\n                  [inst_4 : Module R M] →\n                    [inst_5 : Module S M₂] →\n                      (toAddHom : M →ₙ+ M₂) →\n                        (∀ (m : R) (x : M), toAddHom.toFun (m • x) = σ m • toAddHom.toFun x) → M →ₛₗ[σ] M₂","typeReadable":"{R : Type u_14} →\n  {S : Type u_15} →\n    [inst : Semiring R] →\n      [inst_1 : Semiring S] →\n        {σ : R →+* S} →\n          {M : Type u_16} →\n            {M₂ : Type u_17} →\n              [inst_2 : AddCommMonoid M] →\n                [inst_3 : AddCommMonoid M₂] →\n                  [inst_4 : Module R M] →\n                    [inst_5 : Module S M₂] →\n                      (toAddHom : M →ₙ+ M₂) →\n                        (∀ (m : R) (x : M), toAddHom.toFun (m • x) = σ m • toAddHom.toFun x) → M →ₛₗ[σ] M₂","typeReferences":[["RingHom"],["Module"],["RingHom","instFunLike"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddHom","toFun"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Eq"],["DistribSMul","toSMulZeroClass"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["LinearMap"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["HSMul","hSMul"],["AddHom"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["LinearMap","restrictScalars_add"],"typeFallback":"forall {R : Type.{u_14}} {S : Type.{u_15}} {M : Type.{u_16}} {N : Type.{u_17}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.20 : Semiring.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.23 : Semiring.{u_15} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.26 : AddCommMonoid.{u_16} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.29 : AddCommMonoid.{u_17} N] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.32 : Module.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.26] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.36 : Module.{u_14, u_17} R N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.40 : Module.{u_15, u_16} S M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.26] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.44 : Module.{u_15, u_17} S N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.29] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.48 : LinearMap.CompatibleSMul.{u_16, u_17, u_14, u_15} M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.29 R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.23 (SMulZeroClass.toSMul.{u_14, u_16} R M (AddZero.toZero.{u_16} M (AddZeroClass.toAddZero.{u_16} M (AddMonoid.toAddZeroClass.{u_16} M (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.26)))) (DistribSMul.toSMulZeroClass.{u_14, u_16} R M (AddMonoid.toAddZeroClass.{u_16} M (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.26)) (DistribMulAction.toDistribSMul.{u_14, u_16} R M (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.20)) (AddCommMonoid.toAddMonoid.{u_16} M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.26) (Module.toDistribMulAction.{u_14, u_16} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.32)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.40 (SMulZeroClass.toSMul.{u_14, u_17} R N (AddZero.toZero.{u_17} N (AddZeroClass.toAddZero.{u_17} N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.29)))) (DistribSMul.toSMulZeroClass.{u_14, u_17} R N (AddMonoid.toAddZeroClass.{u_17} N (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.29)) (DistribMulAction.toDistribSMul.{u_14, u_17} R N (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.20)) (AddCommMonoid.toAddMonoid.{u_17} N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.29) (Module.toDistribMulAction.{u_14, u_17} R N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.36)))) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.44] (f : LinearMap.{u_15, u_15, u_16, u_17} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.23 (RingHom.id.{u_15} S (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.23)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.44) (g : LinearMap.{u_15, u_15, u_16, u_17} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.23 (RingHom.id.{u_15} S (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.23)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.44), Eq.{max (succ u_16) (succ u_17)} (LinearMap.{u_14, u_14, u_16, u_17} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.20 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.20)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.36) (LinearMap.restrictScalars.{u_14, u_15, u_16, u_17} R S M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.20 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.44) (LinearMap.{u_15, u_15, u_16, u_17} S S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.23 (RingHom.id.{u_15} S (Semiring.toNonAssocSemiring.{u_15} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.23)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3600791839._hygCtx._hyg.44) (LinearMap.{u_15, u_15, u_16, u_17} S S 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3008994475._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3008994475._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3008994475._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3008994475._hygCtx._hyg.41) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₃) (LinearMap.instFunLike.{u_1, u_5, u_8, u_11} R S M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3008994475._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3008994475._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3008994475._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3008994475._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3008994475._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3008994475._hygCtx._hyg.41 σ) (SemilinearMapClass.semilinearMap.{u_1, u_5, u_8, u_11, u_14} R S M M₃ F inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3008994475._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3008994475._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3008994475._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3008994475._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3008994475._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3008994475._hygCtx._hyg.41 σ f inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3008994475._hygCtx._hyg.51 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3008994475._hygCtx._hyg.56)) (DFunLike.coe.{succ u_14, succ u_8, succ u_11} F M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₃) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3008994475._hygCtx._hyg.51 f)","typeFull":"∀ {R : Type u_1} {S : Type u_5} {M : Type u_8} {M₃ : Type u_11} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₃] [inst_4 : Module R M] [inst_5 : Module S M₃] {σ : R →+* S}\n  {F : Type u_14} [inst_6 : FunLike F M M₃] [inst_7 : SemilinearMapClass F σ M M₃] {f : F}, ⇑↑f = ⇑f","typeReadable":"∀ {R : Type u_1} {S : Type u_5} {M : Type u_8} {M₃ : Type u_11} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₃] [inst_4 : Module R M] [inst_5 : Module S M₃] {σ : R →+* S}\n  {F : Type u_14} [inst_6 : FunLike F M M₃] [inst_7 : SemilinearMapClass F σ M M₃] {f : F}, ⇑↑f = ⇑f","typeReferences":[["FunLike"],["RingHom"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["LinearMap","instFunLike"],["SemilinearMapClass"],["Module"],["SemilinearMapClass","semilinearMap"],["LinearMap"],["Eq"],["DFunLike","coe"],["Semiring"]],"valueReferences":[["rfl"],["LinearMap","instFunLike"],["SemilinearMapClass","semilinearMap"],["LinearMap"],["DFunLike","coe"]]},{"isProp":false,"kind":"definition","name":["LinearMap","mulRight"],"typeFallback":"forall (R : Type.{u_14}) {A : Type.{u_15}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.17 : Semiring.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20 : NonUnitalNonAssocSemiring.{u_15} A] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.23 : Module.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20)] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.27 : IsScalarTower.{u_14, u_15, u_15} R A A (SMulZeroClass.toSMul.{u_14, u_15} R A (AddZero.toZero.{u_15} A (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_14, u_15} R A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20))) (DistribMulAction.toDistribSMul.{u_14, u_15} R A (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.17)) (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20)) (Module.toDistribMulAction.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.23)))) (SMulZeroClass.toSMul.{u_15, u_15} A A (AddZero.toZero.{u_15} A (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_15, u_15} A A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20))) (instDistribSMul.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20))) (SMulZeroClass.toSMul.{u_14, u_15} R A (AddZero.toZero.{u_15} A (AddZeroClass.toAddZero.{u_15} A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20))))) (DistribSMul.toSMulZeroClass.{u_14, u_15} R A (AddMonoid.toAddZeroClass.{u_15} A (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20))) (DistribMulAction.toDistribSMul.{u_14, u_15} R A (MonoidWithZero.toMonoid.{u_14} R (Semiring.toMonoidWithZero.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.17)) (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20)) (Module.toDistribMulAction.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.23))))], A -> (LinearMap.{u_14, u_14, u_15, u_15} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.17 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.17)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.23)","typeFull":"(R : Type u_14) →\n  {A : Type u_15} →\n    [inst : Semiring R] →\n      [inst_1 : NonUnitalNonAssocSemiring A] → [inst_2 : Module R A] → [IsScalarTower R A A] → A → A →ₗ[R] A","typeReadable":"(R : Type u_14) →\n  {A : Type u_15} →\n    [inst : Semiring R] →\n      [inst_1 : NonUnitalNonAssocSemiring A] → [inst_2 : Module R A] → [IsScalarTower R A A] → A → A →ₗ[R] A","typeReferences":[["Module"],["IsScalarTower"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["instDistribSMul"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["AddZero","toZero"],["NonUnitalNonAssocSemiring"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["LinearMap","mk"],["AddMonoidHom","toZeroHom"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["AddHom","mk"],["LinearMap","mulRight","_proof_2"],["ZeroHom","toFun"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["LinearMap","mulRight","_proof_1"],["Semiring","toNonAssocSemiring"],["AddCommMonoid","toAddCommSemigroup"],["AddMonoidHom","mulRight"],["RingHom","id"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","instDistribMulAction","_proof_3"],"typeFallback":"forall {R : Type.{u_3}} {R₂ : Type.{u_4}} {S : Type.{u_5}} {M : Type.{u_2}} {M₂ : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 : Semiring.{u_3} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 : Semiring.{u_4} R₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27 : AddCommMonoid.{u_1} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.33 : Module.{u_3, u_2} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.37 : Module.{u_4, u_1} R₂ M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27] {σ₁₂ : RingHom.{u_3, u_4} R R₂ (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_4} R₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.18)} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.65 : Monoid.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.68 : DistribMulAction.{u_5, u_1} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.65 (AddCommMonoid.toAddMonoid.{u_1} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27)] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.72 : SMulCommClass.{u_4, u_5, u_1} R₂ S M₂ (SMulZeroClass.toSMul.{u_4, u_1} R₂ M₂ (AddZero.toZero.{u_1} M₂ (AddZeroClass.toAddZero.{u_1} M₂ (AddMonoid.toAddZeroClass.{u_1} M₂ (AddCommMonoid.toAddMonoid.{u_1} M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.91091314._hygCtx._hyg.27)))) (DistribSMul.toSMulZeroClass.{u_4, u_1} R₂ M₂ 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{R : Type.{u_1}} {S : Type.{u_5}} {M : Type.{u_8}} {M₂ : Type.{u_10}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.18 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.24 : AddCommMonoid.{u_10} M₂] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.30 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.34 : Module.{u_5, u_10} S M₂ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.24] {σ : RingHom.{u_1, u_5} R S 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inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.15)) M M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.30) (LinearMap.comp.{u_1, u_5, u_1, u_8, u_10, u_8} R S R M M₂ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.30 σ σ' (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.15)) (RingHomInvPair.triples.{u_1, u_5} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.18 σ σ' inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.48) g f) (LinearMap.id.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.30)) -> (Function.Surjective.{succ u_10, succ u_8} M₂ M (DFunLike.coe.{max (succ u_10) (succ u_8), succ u_10, succ u_8} (LinearMap.{u_5, u_1, u_10, u_8} S R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.15 σ' M₂ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.30) M₂ (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M₂) => M) (LinearMap.instFunLike.{u_5, u_1, u_10, u_8} S R M₂ M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.332067831._hygCtx._hyg.30 σ') g))","typeFull":"∀ {R : Type u_1} {S : Type u_5} {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M] [inst_5 : Module S M₂] {σ : R →+* S}\n  {σ' : S →+* R} [inst_6 : RingHomInvPair σ σ'] (f : M →ₛₗ[σ] M₂) (g : M₂ →ₛₗ[σ'] M),\n  g ∘ₛₗ f = LinearMap.id → Function.Surjective ⇑g","typeReadable":"∀ {R : Type u_1} {S : Type u_5} {M : Type u_8} {M₂ : Type u_10} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M] [inst_5 : Module S M₂] {σ : R →+* S}\n  {σ' : S →+* R} [inst_6 : RingHomInvPair σ σ'] (f : M →ₛₗ[σ] M₂) (g : M₂ →ₛₗ[σ'] M),\n  g ∘ₛₗ f = LinearMap.id → Function.Surjective ⇑g","typeReferences":[["RingHom"],["LinearMap","instFunLike"],["Module"],["RingHomInvPair","triples"],["LinearMap","comp"],["LinearMap"],["DFunLike","coe"],["LinearMap","id"],["Function","Surjective"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["RingHomInvPair"],["Eq"],["Semiring"]],"valueReferences":[["Function","bijective_id"],["LinearMap","instFunLike"],["And","right"],["RingHomInvPair","triples"],["eq_true"],["LinearMap","comp"],["Function","comp"],["LinearMap"],["DFunLike","coe"],["Function","Surjective"],["LinearMap","id"],["congrArg"],["Function","Surjective","of_comp"],["Semiring","toNonAssocSemiring"],["of_eq_true"],["RingHom","id"],["id"],["Eq","mpr"],["Eq"],["Function","Injective"]]}]